Lagrange multipliers. Although the geometric proof of the theorem
in [Kap, p.160, l.11-l.17] is elegant and picturesque, it fails to point out its
main usage. In contrast, the algebraic proof in [Rei, pp.620-622,
§A.10] enables us to treat [Rei, p.621, (a.10.5)] as if all differentials dx_{k} were mutually independent.
Thus it shows that the method of Lagrange multipliers makes the original problem
easy to handle. [1]
From the viewpoint of wavefront and phase velocity [Hec, p.101, (4.4)].
Form the viewpoint of the boundary condition [Sad, p.452, (10.98)].
Remark. Jackson's argument to prove [Jack, p.304, (7.34)] is clear [Jack,p.304, l.7-l.9], while Born's argument to prove [Born, p.37, (1)] is not.
Motion in a centric field.
Explaining physics should not be like playing charades. The emphasis must be clear and the explanation must be right to the point.
Good illustration: [Lan1, p.32,l.−20-p.33,l.5 & p.33, Fig.9].
Poor illustrations: [Sym, p.127, Fig. 3.34] & [Go2, pp.90-94, §3-6].
Reading a bad science book makes one feel as though one is falling into a bottomless swamp,
becoming trapped deeper and deeper and eventually getting buried in mud. Reading a bad science book is
similar to building a house on a shaky foundation (Wrong definition: [Sym, p.493, l.−5]; correct definition: [Haw,
§6-2 & §6-11]). No matter how far you may proceed, you must start all over. In
contrast, a good book provides easy access to subtlety, insight and depth.
Besides the definition of tensor, we need to provide efficient tests [Haw, p.99, §6-11] and counterexamples [Haw,
p.196, l.−2] for absolute tensors. Abstraction or generalization can easily make
these subtleties hard to recognize or obscure their motives. [Lan2, p.17, l.16-l.20] gives a partial reason why e ^{ilkm} is not a tensor, while [Haw, p.196, l.−2] tells the whole story.
The insight (The origin of [Haw, p.16, (1-21) &(1-22)]: [Haw, p.15, Fig. 1−12 & p.88, Fig. 5-10]) or key point usually comes from the author's experiences in application. Without understanding the insights, an elegant theory can be reduced to
meaningless manipulating of definitions.
Depth: [Haw, pp.201-204, §13-7] provides proofs for the equations in [Lan2, p.18, l.−6-l.−1].
Example. Expressing the Maxwell equations [Wangs, p.354, (21-30)-(21-33)] in
the
tensor forms so that the equations are covariant with the Lorentz
transformations: [Wangs, p.520, (29-134)-(29-136)].
Remark 1. [Lud, p.54, l.15; l.-9] justify the
rules of lowering and raising indices, while [Lan2, l.16, l.5-l.6] and [Haw,
p.125, 8-3] fail to provide such justifications.
Remark 2. [Lud, p.55, (6.8)] motivates us to define the dual of an antisymmetric tensor (see [Lud, p.55, (6.9)]), while
Landau fails to provide a motive before he defines the dual [Lan2, p.17, l.30].
Landau's definition of dual emphasizes the concept "mutual", so the dual
of a dual is the original. In contrast, Ludvinsen's definition emphasizes the
repeat use of the same rule. Therefore, his dual of a dual is the negative
of the original [Lud, p.55, (6.11)]. The above two seemingly contradictory results do not conflict.
Remark 3. [Lan2, p.17, (6.8)] defines e^{0123}=
+1, while [Lud, p.55, l.-7] defines
e^{0123}=
-1. Landau's definition is just a stipulation, while
Ludvigsen's definition is based on [Lud, p.36, (4.25) & (4.28)]. I prefer
Ludvigsen's definition because it is consistent with the use of the Levi-Cevita
tensor.
Remark 4. The reason why a parameter function l in [Lud,
p.63, l.9] is characterized by the condition v^{a}Ñ_{a}l=1
can be found in [O'N, p.19, Lemma 4.6].
Remark 5. [Ken, chap. 5] emphasizes the origin of tensors. Tensors are developed
as a way to allow physical laws to retain
their form under general coordinate transformation [Ken, p.46, l.7-l.9].
Incidentally, [Ken, p.51, l.-5] explains why the vector and covector
components are the same in rectangular Cartesian coordinates.
Remark 6. Both [Pee, p.237, (8.47)] and [Haw, p.162, (11-3)] fail to provide the physical meaning of
the covariant derivative, while [Ken, p.58, l.16 & p.59, Fig. 6.2] show that the
key to understand the physical meaning of [Ken, p.60, (6.2)] is to use the
concept of parallel transport [Ken, p.59, l.9].
Remark 7. Preserving the tensor notation [Ashc, p.445, l.-8;
p.446, (22.83)] helps us trace the source of symmetry more easily (Compare [Ashc, p.445, l.8] with [Kit2, p.85, l.4]).
Remark 8. The group properties of the tensor component transformations make it possible to formulate a
precise and formal definition of a tensor [Rin, pp.155-156,
§A7].
Remark 9. [Jack, p.270, l.9-l.15] gives a clearer definition about polar or axial vectors than [Lan2,
p.18, l.9-l.12]. [Lan2, p.18, (6.10)] shows how a cross product can be expressed
as a traceless antisymmetric second-rank tensor [Jack, p.269, l.-13-l.-12].
Remark 10. [Reic, p.542] decomposes a tensor of rank 2 into three orthogonal components
using dyads [Sym, p.43].
Remark 11. (Covariant differentiation of tensors of any order [Lau, pp.100-102,
§9.5]) A theory
requires a general solution to the problem rather than just a list of a few examples. [Kre, p.221,
(74.5)] provides a general solution to the problem that [Kre, pp.220-221,
§74] discusses. All the problems of the same type can
be solved using this general solution. Furthermore, the method used in the
general case is consistent with that used in any special case. Thus, the
unification of the methods for special cases is realized. In contrast, [Pee, p.235, l.-9-p.237.
l.15] fails to provide such a conclusive general solution. Although the approach
in [Lau, pp.100-102, 9.5] is more methodical than that in [Pee, p.235, l.-9-p.237.
l.15], it is not as simple and organized as the approach in [Kre, pp.220-221,
§74].
Remark 12. (The geometric meaning of the Christoffel symbols)
[Lau, p.118, Theorem 11.2.1] provides the geometric meaning of the Christoffel
symbols. In contrast, the use of [Pee, p.237, (8.53) & (8.54)] to characterize
the Christoffel symbols [Pee. p.237, l.-14-p.238,
l.13] fails to provide the symbol's geometric meaning.
Remark 13. Spivak's approach: a C^{¥} manifold M [Spi, vol. 1, p.38, l.11]
® the tangent bundle TM [Spi, vol. 1, p.91, l.4-p.93,
l.6; p.101, Theorem 1] ® a section of T*M is called a
covariant vector field [Spi, vol. 1, p.156, l.10] ® a
covariant tensor field A of order k [Spi, vol. 1, p.160, l.-3].
Spivak's approach is cumbersome, but it has advantages. First, it specifies the
base space, a manifold, and thereby justifies the nomenclature "tensor fields". In
contrast, in [Kre, p.102, Definition 30.1], it is unclear whether P refers to
a point in
a surface or a point in the entire space. One is not sure what manifold P
belongs. Second, in differential geometry the
stipulation
given in [Kre, p.102, (30.1)] can be proved as a theorem [Spi, vol. 1, p.158,
l.4]. Third, it is nice to point out that the tensor
field of the type (_{1}^{1})
defined by d_{i}^{j}
is the evaluation map [Spi, vol. 1, p.171, l.1]. Even though Spivak tries to catch
everything by building huge
machinery, he still leaves out several important
perspectives. There is nothing large enough to include everything. We learn a
similar lesson from set theory: there exists no set of which every object is an
element [Bou, p.72, l.15-l.16]. Now we discuss the disadvantages of Spivak's
approach. First, his allowable transformation group is not as flexible as
that in [Kre, p.102, Definition 30.1]. For example, in special relativity we
must specify the allowable transformation group as the group of general Lorentz
transformations [Rin, p.15, l.-7]. Second, his tensor fields are generated by dx^{i}
and ¶/¶x^{j}
[Spi, vol. 1, p.169, l.1]. Conceptually, this definition of tensor fields is too restrictive because the
concept of tensors can apply to not only geometrical quantities but also
physical quantities such as electric fields. In this sense, the rules given in [Kre,
p.102, (30.1a) & (30.1b)] are more flexible. We must realize that the theory
of tensors strides across fields of study and is applicable everywhere.
Any attempt to contain it in a single field of study is simply impossible.
Remark 14. The inner product on a vector space V can be expressed in the tensor
form g_{ij} [Spi, vol. 1, p.409, l.-7],
while the inner product on V* can be expressed in the tensor form g^{ij}
[Spi, vol. 1, p.416, l.1-l.2]. {1} The scalar product of 4-vectors is invariant
under Lorentz transformations [Lan2, p.14, l.16-l.17]. The 4-vectors can be
expressed in contravariant or covariant coordinates [Lan2, p.14, l.-3].
{2}
Covariant coordinates are associated with reciprocal base vectors [Haw,
§1-6; p.14, (1-18)]. Reciprocal base vectors
form the
dual basis [Haw, p.11, l.-5-l.-3].
When we talk about contravariant tensors, our setting refers to V. When we
talk about covariant tensors, our setting refers to V*. Thus,
the dual basis plays an important role in the theory of tensors. I wonder why most textbooks in linear algebra fail to provide the
geometric meaning of the dual basis [Haw, p.15, Fig. 1-12].
Constructing the Lorentz transformation: [Rob, pp.6-9, #2.2 & pp.163-164, #A8].
For classical mechanics, all you need is two textbooks: [Lan1] and [Fomi].
If you throw all of the other textbooks into the trash can, you will not lose much. In
terms of structure, both [Lan1] and [Formi] are well organized. [Lan1] emphasizes
the physical meaning of mechanics, while [Fomi] emphasizes the mathematical
formalism behind mechanics.
Each section of [Fomi] is summarized into theorem form.
The hypothesis and conclusion of each theorem have been clearly specified so that
one can easily apply
the theorem to other readings.
In [Lan1], the material has been condensed and the main results of various topics have been systematically organized so that
we may easily see the big picture and
we may easily move to the front line of research.
In contrast, [Go2] is more like a CliffsNotes
version of [Lan1] except that it is not documented. The regurgitation in the notes style makes it difficult for the reader to understand what is going on if he is not guided by [Lan1]. Although each topic in [Go2] is discussed in great detail, the links among topics are weak. Furthermore, both [Go2]'s structure and reasoning are loose. For example, the meaning of a definition is not always precise and the assumptions of a theorem are sometimes difficult to trace. Thus extracting desired information from [Go2] is often like searching for an auto part in a junkyard.
Let me give some examples to illustrate my points.
[Fomi, p.58, l.-15] defines the canonical
variables in the general sense, while [Go2, p.340, l.17] defines the
canonical variables in a very narrow sense. The origin of the quantization
of energy can be traced to the boundary conditions of the solution of the
Schrödinger equation. Thus, the
canonical variables in the general sense play an important role in
quantization.
[Fomi, p.72] gives a geometric meaning of the Legendre transformation
and shows that the transformation is involutionary, while [Go2] does not.
[Fomi, p.71, l.5] indicates that ¶Φ/¶x
=[Φ,H] is valid on the integral curve
of the system [Fomi, p.70, (11)]. In contrast, the failure to indicate where
[Go2, p.405, (9-94)] is valid makes the calculations in [Go2, p.405, l.-10]
meaningless.
[Fomi, p.76, l.16] relates the generating function to variational
problems, while [Go2, p.382, l.11] does not.
The uncertainty principle [Eis, pp.65-77, §3-3
& §3-4]. [1].
The stability of the ground state of the hydrogen atom: [Eis, p.167,
l.14-l.21 & p.247, l.-6-l.-5].
Symmetric top.
Precession [Lan1, p.106, l.-8-p.107, l.-8; p.116, l.1-l.15]. Remark. In [Lan1, p.112, Fig. 48], let Z-axis rotate
about x_{3}-axis with angular velocity w . If
we are fixed in the rotating (X,Y,Z)-frame, we will see that x_{3}-axis
rotates about Z-axis with angular velocity -
w.
Nutation: [Sym, pp.454-460,
§11.5].
In [Sym, p.457, l.-15], Symon should have pointed
out that we can let 1-axis coincide with Ox in [Sym,
p.455, Fig.11.5] (see [Lan1, p.111, l.19-l.21]).
Let us see how Landau and Eisberg introduce momentum and energy into the
theory of special relativity.
Landau introduces the action integral first. After he finds the
Langrangian, he turns the crank of formalism to obtain momentum [Lan2, p.25, l.-13] and energy [Lan2, p.26, l.3].
Remark 1. [Lan2, p.24, l.-22] says that the
action is invariant under Lorentz's transformations. Landau should have said
that he considers time-space as a 4-vector and the Lorentz transformation as a
metric tensor [Go2, p.288, (7-40) &
Lan2, p.16, l.-3]. The Lagrangian is the unique entity that characterizes the equation of
motion. Although its expression changes with the coordinates we choose, its
value is fixed at the specific location and time.
Remark 2. Certainly, formalism lacks motivation. Even Landau's introduction
of Lagrangian lacks concrete motivation because the Lagrangian itself is abstract
and the concept of tensor is very complicated. Most likely, one derives
[Lan2, p.25, (8.2)] only after special relativity is established.
In contrast, [Eis, pp.A-13-A-17] pays attention to the impact of special
relativity to each individual concept in classical mechanics. Thus Eisberg
generates more interfaces of special relativity with classical mechanics [1] and
creates the motivation behind new concepts. For example, to preserve the
conservation law for momentum, we must allow the mass of a particle to change
with its speed [Eis, p.A-14, l.- 8]. Furthermore, the
direct interface of special relativity with a concept in classical mechanics
rather than the indirect interface with formalism greatly helps us visualize the
new concept in special relativity.
Momentum conservation in special relativity.
Both the example in
[Rob, p.55, l.-16-l.-14]
and the example in [Eis, p.A-13, Fig. A-7] lead to the same result [Rob, p.56,
(5.7) & Eis, p.A-14, (A-18)]. We prefer to use the former example because it is
simpler than the latter example.
[Rob, p.57, l.-2-p.58, l.2] indicates that
if the total momentum and mass of a system of particles were conserved in one
inertial frame, then they would also be conserved in another initial frame,
while [Eis, pp.A-13-A-15] does not.
Liouville's theorem. [Sym, p.395, l.7-p.396, l.8]
details important similarities between the movement of the phase "particles" and
that of particles in an incompressible
fluid. [Rei, pp.626-628] gives an excellent analytic
proof of Liouville’s theorem. [Go2, p.427, Fig. 9-3] provides a vivid geometric
interpretation; [Go2, p.428, l.1-l.5] gives the intuitive physical meaning of Liouville’s
theorem [1]. It is easier to explain [Rei,
p.54, l.18-l.22] by using [Go2, p.428, l.1-l.5]
than using [Rei, Appendix A.13]. The way that Landau defines a statistical ensemble in
[Lan5, p.9, l.!2-l.!1]
immediately shows that [time average] = [ensemble average].
Remark 1. [Ashc, p.771, Appendix H] extends Liouville's theorem to semiclassical
motion.
Remark 2. [Pat, p.35, (5)] can be explained more intuitively and rigorously
using [Kara, p.152]. Pathria's 3-dim proof [Pat, §2.2] of Liouville's theorem
is a natural approach to the problem, while Reif's 1-dim proof [Rei,
Appendix A.13] is more fundamental in logic.
(Phonons)
noninteracting = there are no cross-terms in the
expression of Hamiltonian. The normal mode is the device to reduce the
complicated problem of N interacting atoms to the equivalent problem of 3N noninteracting harmonic oscillators [Rei, p.408, l.22-l.24].
Phonons are indistinguishable [Rei, p.409, l.-
11].
Phonons obey Bose-Einstein statistics [Rei, p.409, l.-16] not because there is infinite number of them (The reason that Reif gives in
[Rei, p.338, l.3-l.4] is incorrect) but because a quantum number n_{r}
is a state index which is allowed to range from 1 to +¥
without any restrictions [Eis,
p.401, l.1-l.2].
The quantum state of the whole system is specified by the set of 3N
quantum numbers {n_{1}, n_{2}, …, n_{3N}}[Rei, p.408, l.-6]. Each phonon can be in any one of the 3N states with energies
ħw_{r} (r
=1,…, 3N) [Rei, p.409, l.15]. Therefore, from the
viewpoint of quasi-particles, the state of the system = (n_{1} phonons
in state 1, …, n_{3N} phonons in state 3N). From the viewpoint of
standing wave, see [Eis, p.389, l.14].
Remark. According to [Coh, p.602, l.7-l.10], a phonon is actually characterized by a wave vector and an angular frequency
W(k). The discussion about phonons in [Rei,
pp.407-411, §10.1] is restricted to the
condition [Coh, p.603, (82)].
The speed of sound (acoustical waves): [Hoo, p.35,
(2.3)](when atomic spacing << atom displacement << wavelength [Hoo, p.35, l.-12-l.-11]);
[Hoo, p.40, (2.13); Coh, p.603, (84)](chain of identical atoms); [Hoo, p.44, l.-11](chain
of two types of atoms).
Lattice vibrations of 3-dimensional crystals: [Hoo, p.46, l.-12-l.-8].
Aperture and field stops, entrance and exit pupils, and vignetting [Hec,
pp.171-173].
Phase velocity = the speed of a co-phasal surface [Born,
p.18, l.7].
Le Chatelier's
principle.
Good illustration: [Lan5, pp.65-68,
§22].
Poor illustration: [Rei, p.298, l.- 8-p.300,
l.10].
[Associated] Legendre polynomials
[Boh, pp.321-326,
§14.14 & §14.15].
The quality of a textbook
on quantum mechanics can be determined by noting whether or not it includes a
complete analysis of the Legendre polynomials.
Spherical harmonics Y_{n} come from a
solution of Laplace's equation that is a homogeneous polynomial of degree n [Col, chap.IV,
'1.1].
Surface harmonics in Cartesian coordinates and spherical coordinates. Their sign regions. [Col, p.232].
Notice the 1-1 correspondence between [Col, p.232, Fig.IV.2, n=2] and [Coh, p.682, (33)].
For the geometric origin of the generating function, the explanation in [Col,
p.233, l.!4-p.234, l.!7]
is much better than that in [Cou, p.85, l.11-l.18].
The Euler angles [Edm, pp.6-8,
§1.3; Tin, pp.101-103,
§5-3].
Responding to the above experimental evidence, we study the corresponding eigenvalues of operators.
Quantization stems from the formalism of classical mechanics.
[Schi, p.132, (23.2)] and [Schi, p.134, (23.8)] allow us to establish the correspondence [Schi, p.134, (23.9)].
A Poisson bracket does not depend on the canonical variables we choose [Lan1, p.145, (45.9)]. A commutator
bracket does not depend on the basis we choose.
Quantization rules [Schi, p.135, l.15-l.29]. Example: second quantization [Schi, p.342, l.12; p.349, (46.3); p.350, l.1 & (46.6)].
Remark 1. Summerfeld's quantization of action implies both Planck's quantization of energy and Bohr's quantization of angular momentum
[Eis, p.110, l.-5-p.112, l.-7].
The quantization of action can be interpreted in terms of standing waves [Eis,
p.112, l.-6-p.114, l.19]. Schrödinger derives energy quantization
based on the fact that accepted solutions of the time-independent Schrödinger
equation exist only for certain value of the total energy ([Coh, pp.351-358,
Complement M_{III}]Ú[Eis, p.160, l.-15-p.163,
l.-15]Ú[Lan3, p.61, l.1-p.62, l.8]Ú[Mer2,
p.45, l.8-l.20]). Thus, Schrödinger
eliminates the axiomatic requirement of integralness, and traces its
origin directly to the boundary conditions of an eigenfunction ([Eis, p.163, l.-8-l.-5]
& [Lev2, p.69, (4.47); p.70, l.9; p.70, Fig. 4.2]).
Remark 2. In [Eis, p.163, l.-6], Schrödinger
attributes integralness to the finite and single-valued nature [Lev2, p.109,
l.1] of an eigenfunction. According to the theory of differential equations,
it is more appropriate to attribute the discreteness of eigenvalues to boundary
conditions [Chou, p.136, l.-18-p.137, l.4; Bir, pp.288-292].
However, it is important to point out that the discrete spectrum of the
regular S-L system [Bir, p.273, Theorem 5] inspires Schrödinger to
interpret the discreteness in the microscopic world using Schrödinger
equation. At the point when readers encounter [Eis, p.163, l.-6],
Eisberg has not yet provided enough mathematical background for them to appreciate Schrödinger's
statement. However, most textbooks that do build sufficient background forget to mention
that this
clue led to the important discovery. Thus, many textbooks often fail to
accurately reflect history when they explain the theory (see [Bir, chap.10,
§16; Jack, p.77, l.7; Coh, p.663, l.-14]).
The Poisson algebra and the commutator algebra are isomorphic Lie algebras. Proof. The generators of the two algebras produce the same results (Compare [Schi, p.134, (23.10)] with [Coh, p.222, (B-33)].
The rules of Poisson brackets and those of commutator brackets are the same [Schi, p.135, (23.12); Coh, p.168, (10)-(14)].
Through the canonical transformation
in [Fomi, p.93, l.14], we see that the canonical equations of motion and the
Hamilton-Jacobi equation are equivalent [Fomi, pp.88-93,
§23]. Therefore, we may express the general equation of motion in the form of the
Hamilton-Jacobi equation. Finally, we use quantization rules to establish the time-dependent Schrödinger
equation from the Hamilton-Jacobi equation.
Indeed, in the quasi-classical case [Lan3, p.20, (6.1)], the time-dependent Schrödinger
equation reduces to the Hamilton-Jacobi equation of classical mechanics [Mer2,
p.23, (2.39)].
Remark. The ultimate goal of quantization is to derive the Schrödinger
equation. Consequently, the correspondence principle should be formulated in its
strongest form. The correspondence between classical mechanics and quantum mechanics established by [Mer2, p.326, (14.61) & (14.62)] is a
weak form of the correspondence principle because Merzbacher's
formulation cannot
lead to the derivation of the Schrödinger equation. Indeed,
Schrödinger's equation implies [Mer2, p.326,
(14.62)], see [Coh, p.241, l.1-l.9]; Hamilton's equations implies [Mer2, p.326,
(14.61)], see [Lan1, p.135, l.6-l.11].
The beauty of commutator algebra is that we jettison the complicated
calculation required with Poisson brackets and preserve its algebraic essence. The
reduction greatly simplifies qualitative discussion of physical phenomena.
Furthermore, we may apply the commutator algebra and the Schrödinger
equation to a microscopic system.
How to use the concept of wave packets to connect classical and quantum mechanics.
A wave packet is the wave function of a localized particle [Coh, p.26, l.-11-l.-9].
The group velocity of the wave packet = the velocity of the (free) particle
[Coh, p.29, (C-28)].
The correspondence principle: the classical value of a physical quantity =
the expectation value of the corresponding operator [Schi, p.26, (7.9);
p.27, (7.10)].
The uncertainty principle.
Theoretical proof from the wave point of view
(See [Coh, pp.286-289, Complement C_{III}], where
DQ is the variance of a distribution function).
Remark. Bohr's complementarity principle [Schi, p.8, l.4] can be considered as the physical idea behind the proof.
A precise and simultaneous measurement is physically
impossible because of the interaction between the apparatus and the
measured particle. In this case, Dx
represents inaccuracy.
Examples.
Localization experiment.
[Schi, p.9, l.8-p.10, l.11] or [Eis, p.67, l.-16-p.68, l.-11].
When the momentum of the electron is known, the measurement of its position
involves inaccuracy [Schi, p.9, (4.1)] and introduces an uncertainty
into the momentum [the Compton effect: Schi, p.9, l.-10].
Remark. In order to separate the scattered photons from the incident beam, the direction of the incident beam should be
oriented as in
[Schi, p.9, Fig. 2], not [Eis, p.67, Fig. 3-6].
Diffraction experiment [Hec2, p.5, Fig. 1-1]. When the momentum of the photon is known,
the measurement of its position
involves inaccuracy [Hec2, p.5, l.6] and introduces an uncertainty
into the momentum [Hec2, p.5, (1) & (4)].
Momentum determination experiment [Schi, p.10, l.16-p.11, l.18].
When the position
of the particle is known, the measurement of its momentum involves inaccuracy [the
Doppler effect: Schi, p.11, (4.8)] and introduces an uncertainty
into the position [Schi, p.11, (4.7)].
Diffraction experiment with photon indicators [Schi, p.12, Fig. 3].
If the interaction between a photon and an indicator were so weak that would not
destroy the original diffraction pattern, the uncertainty in p_{y}
for a particular photon produced by its encounter with an indicator would have
to be small, as stipulated in [Schi, p.12, (4.9)]. Because [Schi, p.12, (4.12)] contradicts
[Hec2, p.6, (6)], it is impossible to determine through which slit the photons
pass without destroying the diffraction pattern [Schi, p.12, l.-12-l.-10].
Applications.
The limits of geometric optics [Lan2, p.144,
l.-11-l.-1].
The Zeeman effect [Lev2, pp.154-156, §6.8].
The collective states formed by independent, identical fermions using Pauli's exclusion principle.
The shell model of many-electron atoms.
[Lev2, p.338, Fig. 11.6] summarizes [Coh, complements A_{XIV}
and B_{XIV}]. The
central-field approximation [Coh, p.1411, l.-4]
explains why we start with electron configurations [Coh, p.1413, (10);
p.1414, (11)] and why the interelectronic repulsion can be treated as a
perturbation [Coh, p.1412, l.11].
From configuration to terms [Lev2, p.327, Table 11.2]: (a). Equivalent electrons
[Lan3, p.254, l.6-l.28; Coh, p.1423, (23)];
(b). Nonequivalent electrons [Lan3, p.254,l.1-l.5].
The electron gas [ neglect interactions between electrons].
Free electrons enclosed in a box.
There is a one-to-one correspondence between the lattice points in the
k-space and the wave functions of an electron [Coh, p.1434, l.5-l.7].
The ground state of the electron system with the Fermi energy [Kit, p.183,
l.13]: [Coh, p.1392, l.-5-l.-3].
The definition of the Fermi energy in [Coh, 1435, (6)] is more precise than that of
[Kit, p.183, l.13]. Note that Pauli's exclusion principle applies not only to
the electron gas [Coh, p.1434, l.-6-l.-1] but also to the electron system of a solid [Coh,
p.1443, l.-23; p.1161, Fig.4].
Periodic boundary conditions.
The motive of periodic boundary conditions is to simplify calculations [Coh, p.1440, l.-12-p.1441, l.8].
When the interatomic spacing decreases, the splitting increases because the coupling increases
[Coh, p.1159, Fig. 2].
The stationary states of an individual electron are all delocalized [Coh, p.1159, l.2].
The deeper the band's
location, the more narrow it is
[by the tunnel effect; Coh, p.1161, Fig. 4].
Due to Pauli's exclusion principle, only the electrons with energies close to the Fermi energy are
important for the following applications:
deriving the correct formula of specific heat for the electron gas.
deriving the correct formula of magnetic susceptibility for the
electron gas.
explaining why some solids are good electrical conductors while others are
insulators.
Remark. This feature of the restricted number of electrons
allows us to use much simpler concepts (such as the density of states [Coh,
p.1435, l.11] or the location of the Fermi energy [Coh, p.1443, l.-13])
to replace the complicated definition of the ground state of
the electron system [Coh, p.1433, l.-12-l.-5] when we
engage in practical study of the physical quantities associated with the ground state of the
system.
The Michelson-Morley experiment [Rob, pp.28-29,
#3.5].
Electric fields.
Coulomb's law Û Gauss' law [Cor, pp.50-51,
(3.19)-(3.22); Sad, pp.126-127, §4.6.A].
Electric multipoles [Wangs, chap. 8].
A dipole
Its potential: [Wangs, p.114, (8-21)].
Its field: [Wangs, p.120, (8-50)].
The interaction energy of a dipole in an external electric field: [Wangs, p.127, (8-73)]. Remark. [Wangs, p.124, l.-13-l.-6]
gives a physical reason why we are not interested in studying the energy changes
of the external charges.
The torque on a dipole in an external electric field: [Wangs, p.127, (8-75)].
A quadrupole
Its potential: [Wangs, p.115, (8-30)].
Its field: [Wangs, p.123, (8-55)].
The interaction energy of a quadrupole (with an axis of rotational symmetry) in an external electric field: [Wangs, p.130,
(8-81)].
Remark 1. The lines of force are perpendicular to the equipotential surfaces [Sad, p.144, l.-4-l.-3].
Remark 2. If both the monopole moment and the dipole moment are zero, then the quadrupole
moment becomes the dominant feature of a charge distribution [Wangs,
p.115, l.-17-l.-15].
Remark 3. [Wangs, p.80, (5-48)] is the common background used to build [Wangs, p.99, (7.6)] and [Wangs, p.125, (8-62)].
Conductors [Wangs,
pp.83-95, chap. 6]
Remark. [Wangs, p.54, (3-13)] can be easily derived using Gauss' law [Jack, p.28, (1.11)].
The outward force p per unit area at the surface of the conductor is the
product of the surface charge density
and the external electric field [Jack, p.43, l.2-l.4].
Proof. The electric field E_{sheet} generated by a sheet charge distribution is s/2e_{0} above the sheet and is - (s/2e_{0}) below the sheet
[Wangs, p.54, (3-13)]. In order to let E_{total} satisfy [Wangs,
p.83, (6-1); p.85, (6-4)], we must add E_{external} = s/2e_{0}
to E_{sheet}. The field E_{sheet} locally generally by
the sheet cannot exert a force on itself, therefore p = s E_{external }= s^{2}/2e_{0}.
Electrostatic screening [Wangs, p.87, l.-3].
Systems of conductors [Wangs, §6-2,
§6-3; Chou, p.90, l.9-p.94, l.2].
Remark. The definition of [Wangs, p.89, (8-12)] depends on the particular point
chosen on the ith conductor [Wangs, p.88, l.19-l.20].
In order to prove the uniqueness of p_{ij}'s
and the existence of c_{ij}'s, we must use the uniqueness theorem
[Chou, p.91, l.-2-l.-1] to
establish the one-to-one correspondence between F and
Q [Chou, p.91, (2.138)]. Note that [Wangs,
(6-12)] has the advantage that it can be easily translated to a computer program
if the surface charge density distribution is known. Actually, [Chou, p.92, l.-10-l.-5]
shows not only the non-singularity but also the positive-definiteness of the (p_{ij}).
As for the proof of the fact that (p_{ij}) and (C_{ij})
are symmetric, I prefer [Wangs, p.89, l.-6-p.90,
l.13] to [Chou, p.93, l.5-p.94. l.2]. The former proof is constructive and
insightful, while
the latter proof uses formalism.
Electrostatic energy [Wangs, §7-3,
§7-4 §10-8,
§10-9].
A system of charges [Wangs, p.99, (7-10)].
A system of conductors [Wangs, pp.100-101, §7-2].
An electric field [Wangs, p.102, (7-28)].
Electrostatic forces on conductors [Wangs, p.107, l.15-p.108, l.7].
The discussion of electrostatic energy is divided into two classes: constant free charge (if the
system is isolated) and constant potential difference (if the system is
connected to an external energy source) [Wangs, p.105, l.8-p.106, l.-10; p.165,
l.11-l.22].
Remark. [Wangs, p.101, l.-7-p.102,
l.-19] shows that a and c above are consistent.
Boundary conditions.
Static
electric fields
dielectric-dielectric [Sad, p.183, Fig. 5.10)].
conductor-dielectric [Sad, p.185, Fig. 5.12].
Potential continuity between two media [Wangs, p.139, (9-20)].
(Equilibrium) A perfect conductor cannot contain an electric field below
its surface [Sad, 165, Fig. 5.2].
(From disturbance to equilibrium) The relaxation time of a conductor (dielectric) after introducing charge at some interior point [Sad, p.181, (5.49)].
Magnetic fields.
Biot-Savart's law [Sad, pp.263-266, §7.2]
Û Ampère's
circuital law (Proof. Þ:
[Wangs, pp.237-241] or [Jack, pp.178-179, §5.3]. Ü:
[Sad, pp.274-275, §7.4.A].
Remark 1. In proving Ampère's circuital law, [Wangs,
pp.237-241] uses the line integral [Wangs, p.225, (14-2)] for B, while [Jack,
pp.178-179, §5.3] uses the volume integral [Jack, p.178, (5.14)] for B. The
steps of reasoning in Wangsness' proof can easily conjure accompanied physical images, while
those of
Jackson's cannot.
Remark 2. In proving Ampère's circuital law
Þ Biot-Savart's law, we must assume knowledge of the
direction of B due to a current [Wangs, p.242, l.-7]
in order to find the magnitude of B. Thus, we have used a bit of information of Biot-Savart's
law to derive the entirety of Biot-Savart's law. Therefore, strictly speaking, Biot-Savart's
law and Ampère's circuital law are not exactly
equivalent.
Magnetic multipoles [Wangs, chap. 19].
A
magnetic dipole [Cor, pp.337-340].
Its potential: [Wangs, p.302, (19-22)].
Its induction: [Wangs, p.302, (19-24)].
The interaction energy of a magnetic dipole in an external magnetic induction: [Wangs, p.306, (19-36);
p.307, (19-40)].
The torque on a magnetic dipole in an external magnetic induction: [Hall,
pp.541-543, §30-4; Wangs, p.308, (19-42)].
Magnetic energy
[Wangs, chap. 18; §20-6; Jack,
§5.16].
Magnetic forces on circuits: [Wangs, pp.290-295,
§18-3]. Remark 1. It is better to use [Cor,
p.480, l.5] to explain the second of the three equalities in [Wangs, p.291,
(18-38)]. Remark 2. Wangsness uses the sign of a magnetic force alone to determine whether the force is attractive or repulsive [Wangs, p.293, l.22; p.295, l.13]. His strategy is very confusing. However, if we follow Corson's
method by considering the dot product of the magnetic force and the
infinitesimal displacement [Cor, p.480, (26-37)], then it will become much easier to determine whether the force is attractive or repulsive.
Remark 3. The discussion of electrostatic forces on conductors [Wangs, p.104,
Fig. 7.1] is divided into two cases: constant charge and constant potential
difference. The discussion of magnetic forces on circuits [Wangs,
§18-3] is divided into two cases: constant
currents and constant flux [Wangs, p.218, Fig. 13-1]. At first glance, the
divisions seem to depend on the devices we choose. In fact, if we look deeply
into the matter, the choices of divisions are fundamentally determined by the
characteristics of fields [Fan, p.75, l.23-l.25]. The divisions cannot be
made different regardless of device.
[Chou, p.286, l.6-l.10] explains why the second integral of [Jack, p.213,
(5.146)] vanishes without any extra assumption [Jack, p.213, l.-11].
In order to fully understand the concept of magnetic energy, we must perform a series of
clarifications and comparisons.
(Total magnetic energy vs. interaction energy) [Wangs, pp.285-286, Example].
(System: a single circuit with constant current I) If the flux change
through the circuit is dF, then the work done by the
sources (of current) is dW=I dF [Jack,
p.212, l.-11].
(System: a steady-state current distribution [Jack, Fig. (5.20)]) The total increment work done against the induced
emf
[Wangs, p.284, l.-9] by external sources due to a change
dA or dB is [Jack, (5.144)
or (5.147)]. The total work [magnetic energy] to bring the fields (of the
system) up from zero to their final values is [Jack, (5.148)].
Remark. dA or dB refers to the change of the system made by external sources.
(System: a permanent magnetic moment) Compare [Jack, (5.150)] with [Jack,
(5.72)] [Jack, p.214, l.-4-p.215, l.3].
Remark. Note that before placing [Jack, p.214, l.7] the magnetic moment in the external field,
the magnetization M in [Jack, (5.150)] does not exist [Jack, p.215, l.2].
The discussion of magnetic energy is divided into two classes: constant currents (if the system
is connected to an external energy source) and constant flux (if the system is
isolated) [Wangs, p.290, l.-22-p.292, l.5; Jack,
p.214, l.-16-l.-5].
Remark. Whenever we speak of a flux, we must specify the current source that produces the flux. Without such
a specification, people may wonder whether DF in [Chou, p.282, l.-6]
includes the flux from self-inductance.
(General case) In lossy dielectrics [Sad, pp.417-422,
§10.3].
(Special cases)
Plane waves in lossless dielectrics [Sad, p.423,
§10.4].
Plane waves in good conductors [Sad, pp.425-428;
§10.6].
Determination of crystal structures by X-ray diffraction
The Laue condition: Constructive interference occurs if and only if
Dk is a reciprocal lattice vector.
Proof. Þ: [Kit2, p.35, l.-6].
Ü: [Ashc, p.98, (6.4)-(6.7)].
The Ewald construction of diffraction peaks [Ashc, pp.101-104].
The Laue method using a range of wave lengths.
The rotating-crystal method.
The powder (randomly oriented grains) method.
Derive Lagrange's equations using calculus of variations
From the viewpoint of equilibrium [Go2, pp.16-21,
§1-4]
The key idea: (D'Alembert's principle) The infinitesimal work [F-(dP/dt)]·dr
is zero [Go2, p.17, (1-44)] when the system is nearly in equilibrium.
The procedure:
Eliminate the appearance of the forces of constraints [Go2, p.18, l.8].
Transform the constraint coordinates to the generalized coordinates [Go2,
p.18, l.10].
From the viewpoint of action [Fom, §9
& §21].
The key idea: (Hamilton's principle) The infinitesimal action change is zero
near the actual path.
In other words, the action integral along the actual path [Go2, p.36, (2-1)]
is stationary.
Remark. Method A is more difficult than Method B (compare
[Fom, p.46, Theorem 2] with [Fom, p.35, Theorem]) because the former involves
constraints [Fom, p.48, footnote 9]. The procedure A(b) complicates the
problem even more because it fails to use
Lagrange multipliers to exploit the symmetry [Rei, p.621, l.16-l.17].
Normal coordinates in a lattice.
The construction of normal coordinates in [Kit2, p.639, (5)] is much simpler
than that in [Sym, pp.469-471, §12.3] or
that in [Lan1, p.68]. I would like to try to explain why Kittel's
construction is a natural way to decouple the equation of motion
[Kit2, p.641, (19)] and the total energy [Kit2, p.641, (21)] even though my
explanation is not precise and complete.
The reciprocal lattice corresponds to the basis of the Fourier expansion of a periodic function [Kit2, p.32, (5)].
There is a one-to-one correspondence between the lattice points and the reciprocal lattice points
[Ashc, p.87, l.-18].
The energy of a harmonic oscillator is quantized by its frequency [Coh,
p.494, (B-34)].
The direct enumeration of all the wavelike solutions [Hoo, p.37, (2.8)] can be viewed as a method of decoupling [Hoo, p.38, l.17;
p.40, l.11-l.16; Kit2, p.101, Fig. 5] even though decoupling the
equation of motion is traditionally considered the first step toward finding the
solutions.
The dispersion relation [Hoo, p.38, (2.9); Kit2, p.640, (15)] is the major
link between [Hoo, p.37, (2.8)] and [Kit2, p.639, (5)].
The uncertainty principle implies that the Fourier transforms of two
strongly-coupled, broad wave packets in position space are two distantly-separated,
narrow wave packets in momentum space. Thus, the use of Fourier transforms facilitates the
decoupling process of equations of motion in a lattice.
Remark 1. [Coh, p.591, (19) & (21)] motivate us to define the phonon
coordinates as [Kit2, p.639, (5)].
Remark 2. [Mari, p.501, (12.142)] is an extremely powerful device for decoupling
(see [Mari, p.501, l.-1]).
Remark 3. In [Rei, p.408, l.22], Reif says that the concept of normal variables reduces the complicated problem of
interacting atoms to the equivalent problem of noninteracting harmonic oscillators. However, he only justifies his statement from the viewpoint of
energy [Rei, p.408, (10.1.8)]. In fact, whenever we call certain
variables normal variables we must routinely test whether they satisfy the
following requirements in classical and quantum mechanics.
The requirements in classical mechanics.
Decouple the total energy [Coh, p.580, (20)]: Express the total energy in
terms of the energies which can be associated with each of the modes.
Examples. [Coh, p.596, (47) & (50)] (when x and
p are considered as normal variables); [Coh, p.597,
(52) & (53)] (when a is considered as a normal
variable).
Decouple the
equations of motion [Coh, p.577, (11)].
The requirements in quantum mechanics.
Decouple the total energy [Coh, p.600, (76) & (77)].
Decouple the equations of motion
[Kit2, p.641, (19)].
Redecompose
the state space as the tensor product of eigenspaces [Coh, pp.583-584, §c;
especially, p.583, l.2-l.3]: The old
component eigenspace is not invariant under the coupling operator [Coh,
p.598, l.-6-l.-5]. We
must redecompose the state space as a tensor product of new eigenspaces ([Coh,
p.600, (79)] define the new ground state. The new tensor product can be
generated by creation operators [Coh, p.600, l.-11].)
The uncertainty principle [Coh, p.597, (54)].
Any pair of component operators corresponding to different modes commute: Position and momentum operators [Coh,
pp.581-582, (26)-(31); p.597, (54)]; annihilation and creation operators [Coh, p.600, (72-a)]; the
total Hamiltonian [Coh, p.600, (79)].
Liénard-Wiechert potentials and fields for a moving charge [Jack, pp.661-665,
§14.1].
A potential well of arbitrary shape.
bound states: the energies are bounded [Coh, p.357, l.3] and discrete [Coh, p.354, l.-10].
Energy bands for a periodic potential [Coh, p.372, Fig. 2 & p.379, Fig. 4].
Remark 1.
Remark 2. At this general stage, we can only have a qualitative (i. e. geometric) analysis
for the big picture. For
example, we can discuss the structure of its reciprocal lattice. As the case
becomes more specific, more physical meanings can be precisely
associated with [Coh, p.379, Fig. 4].
Remark 3. How an energy gap arises (the mathematical (quantum) explanation
(Energy gaps must exist somewhere, but we cannot pinpoint their locations.): matching conditions [Coh,
p.369. l.-8; Eis, p.p.458, l.-15];
the physical explanations (At a zone boundary, the symmetric wavefuncton and
the antisymmetric have different energies.): qualitative [Eis, p.459, l.-6-p.460,
l.18]; quantitative [Kit2, p.179, (6)]).
The general theory of an electron in a solid: The main feature of this
approach is that the Hamiltonian is not specified [Coh, pp.1161-1168,
Complement F_{XI},
§2]. The Hamiltonian can refer to a free electron or an electron bound
to an atom.
The allowed energy band: [Coh, p.1163, (9) & Fig. 5].
Stationary states: Bloch functions [Coh, p.1164, (14), (15), and (16)].
Remark. The delocalization of the electron: [Coh, p.410, (C-20)] ®
[Coh, p.1159, l.2] ® [Coh, p.1164, (17)].
Nearly free electron theory [Hoo, pp.100-104,
§4.1]. Remarks.
We can only focus on one theory at a time. Aiming at too many goals will
lead nowhere. [Ashc, chap. 9] and [Kit2, chap. 7] assume that their readers do
not have a background in perturbation
theory, so they try to develop both perturbation theory and nearly free
electron theory at the same time. It turns out that both approaches fail to
provide a clear picture of energy bands. For example, n in [Hoo, p.102, (4.4)]
is associated with the standing wavefunction sin(npx/a)
[Hoo, p.102, l.14] and the n-th term of the Fourier expansion of the
potential. In contrast, the meaning of n in [Kit2, p.187, l.-18]
and [Ashc, p.162, l.17] is not as specific enough as it could be. I like Hook's approach because it
focuses on nearly free electron theory and assumes that his readers have a
background in perturbation theory. Furthermore, I prefer having a complete understanding of
a 1-dim lattice to
having a vague picture of a 3-dim lattice [Ashc, pp.152-166].
Hook's approach
shows insight. Although his approach is not perfect, it is amenable to
improvements.
We may use [Kit2, p.183, l.13] to prove that the only important term in
the lattice potential of [Hoo, p.101, (4.2)] is V_{1}cos(2px/a)
[Hoo, p.129, l.11].
We may use [Ashc, pp.155-156, Case 2] to prove that y has the form
ae^{ikx}+be^{i(k-2p/a)x} [Hoo, p.129, l.13;
Ashc, p.156, (9.22)].
[Kit2, p.179, (6)] is not as good as [Hoo, p.102, (4.4) & (4.5)] because
the former only calculates the energy gap of the first band, while the latter
calculates the energy gap of the n-th band for every n.
The tight binding approximation [Iba, pp.137-142,
§7.3; Ashc, pp.176-184, §General formulation;
§Application to an s-band from a single
atomic s-level].
Remark 1. The caption of [Coh, p.1161, Fig.4] gives a more fundamental
reason than that given in [Iba, p.141, ii)] to explain why a deep lying band
is narrower than the shadow lying band.
Remark 2. Both [Abr, p.10, (1.21)] and [Iba, 139, (7.31)] are based on the Ritz method [Iba, p.139, l.13]. Although the latter formula is more
intuitive, the former formula has the following advantages: (1). The inversion
theorem [Ru2, p.199, Theorem 9.11]; (2). The complicated computation given in [Abr, p.10, (1.23)] is
actually a simple consequence of
[Ru2, p.202, (13)], a fact that Abrikosov probably did not recognize. [Kit2, pp.245-248, §Tight
binding method for energy bands] fails to fully use these advantages of
Fourier analysis. Note that [Abr, p.10, (1.21)] is based on [Ru2, p.192,
(4)], but the physical interpretations of the two formulas can be different:
The domain of w_{n} is cleverly preserved as the
position space, while the domain of f^ is often interpreted as the momentum space (The
position variable disappears because it becomes the dummy variable of
integration).
Remark 3. A scholar should not just discuss trivialities and avoid discussing
difficult issues by pretending not to see them. Most textbooks in solid state
physics fail to
explain why g(R) in [Ashc, p.182, l.-9]
is the same constant for each of the atom's 12 nearest neighbors. Some of the
above books still lack
any improvement on this point even after many editions. [Ashc] is one
exception. However, Ashcroft gives only a vague hint
[Ashc, p.182, l.-12-l.-9].
Ashcroft's argument would be clarified if he were to add that he uses the formula [Ru2, p.186,
§8.27, (1)].
The H_{2}^{+} ion-covalent bonding [Hoo, pp.111-115,
§4.3.2].
The physical meaning of the limits of the first allowed band is given by [Hoo, p.114, Fig.
4.7(a); Coh, p.1159, Fig.2 & p.1179, (48)].
Quantum resonance: [Coh, p.1177, l.-15-l.-9].
The origin and stability of the chemical bond [Coh, p.1179, l.-15-l.-6].
The way to improve the result of the variational method is to enlarge the
family of trial kets [Coh, p.1182, l.-9; p.1183,
Table I; p.1173, Fig. 2].
A 1-dim chain.
The physical meaning of the n-th band refers to the n-th principal quantum
number for a single atom [Ashc, p.183, Fig. 10.4].
Remark 4. The physical meaning of [Ashc, p.141, (8.50)] is given by [Hoo, p.116, l.-6-p.117, l.6].
Remark 5. Both nearly free electron theory and the tight binding approach have similar dispersion
relations [Hoo, p.119, l.-17-p.120, l.-8].
[Iba, p.106, Fig. 6.1] explains why the results derived from the two theories
are consistent.
Remark 6. A scholar should be brave enough to face a challenge and should
not sweep what he does not understand under the rug. [Cra, p.8, l.9-l.17;
Ashc, p.140, footnote 17] explain why E(k) is a continuous function of k. In
contrast, [Kit2, chap.7 & chap.9] and [Abr, chap. 1] do not even mention such a
problem. Unless he or she is extremely careful, an average reader will not be able to
detect these authors have left out something important.
Faraday's law of induction [Wangs, chap. 17].
Faraday's observations [Jack, p.208, l.-16-l.-6].
For a static situation, there is no connection between the electric field and the magnetic field. Faraday's law of induction establishes their connection only for a
nonstatic situation [Wangs, p.263, l.-27-l.-23].
Sometimes we define flux as the product of density and velocity [Wangs,
p.393, l.3-l.6]; sometimes we define flux as the dot product of a vector field and an area
(e.g, the magnetic flux [Wangs, p.251, (16.6)]). What is the relationship
between these two ideas? Answer: [Coh, p.238, l.20-21]. Thus, a generalized
concept keeps only a small number of the
properties of its original concept.
Electromotive force [Hall, pp.518-519, §29-1]; Lenz's law [Hall. pp.577-579, §32-3;
p.580, l.25-l.30; Wangs, p.264, l.-17-p.265, l.-10].
Faraday's law written in the form of [Wangs, p.272, (17-30)] is independent of the motion of the medium [Wangs, p.272, l.9].
[Jack, p.210, (5.137)] is [Chou, p.251, (6.18)]. Its proof is given in [Chou,
§6.3]. d/dt in [Chou, p.251, l.5] refers to
a fixed charge (particle) in the moving circuit (fluid). For its physical
meaning, see [Lan6, p.3, l.1-l.3]. ¶/¶t
refers to a fixed point in space. For its physical meaning, see [Sym, p.313,
l.25-l.26; l.-3-l.-1].
The argument given in [Chou, §6.3] follows
closely the formalism given in [Chou, Appendix I] which has well established physical
interpretations. The proof given in [Wangs, §17-3]
is well tailored to this particular problem, and is simple, direct, and clear in
the mathematical sense. However, there is a gap in the deduction from
[Wangs, p.271, (17-25)] to [Wangs, (17-26)]. The gap can be filled using [Jack, p.209, l.-15-l.-5] ([Wangs,
p.266, (17-8)] is invariant under the Galilean transformation when
v <<c) or using the argument in [Cor,
§23.2 & §23.7] ([Cor,
(23-28)] is invariant under the Lorentz transformation [Cor, (23-61)] when v's magnitude is comparable to that of c).
Infinitely long ideal solenoid.
A has only a jˆ component
[Wangs, p.260, Fig. 16-6]. Inside the solenoid, A is given by [Wangs, p.259,
(16-49)]. Outside the solenoid, A is given by [Wangs, p.259, (16-50)].
Remark. [Cor, p.350, l.5] gives a direct physical reason why A¹0
outside the solenoid.
By symmetry, B is independent of z and of j [Cor, p.355, l.9].
B_{r}=
0 [Cor, p.355, l.14]. Inside the
solenoid, B_{j}
is given by [Cor, p.356, l.15] and B_{z}
is given by [Wangs, p.260, l.-5]. Outside the
solenoid, B_{j}
is given by [Cor, p.356, l.11] and B_{z}
is given by [Wangs, p.260, l.-4].
Remark. The direct
physical reason why B_{z}=0
outside the solenoid can be attributed to the following two facts: 1. [Wangs,
p.226, l.23]. 2. The denominator of the integrand
in [Wang, p.227, (14-11)] is large.
The Hamiltonian is periodic [Ashc, p.134, l.16]. (Proof. Let y=x+R. Then
¶/¶y=¶/¶x.)
Considering the lattice symmetry [Hoo, p.328, l.-9-l.-4], we must require that the wave function satisfy [Kit2,
p.160, (8)]. [Ashc, p.134, l.26-l.30] shows that this physical requirement is
theoretically feasible. If the wave function is degenerate, there will be some difficulty
in proving [Ashc, p.134, (8.12)], but this difficulty can be overcome by the
method indicated in [Coh, pp.141-142, (ii)]. Thus, the first proof is still
valid even without Kittel's extra assumption that y_{k}
is nondegenerate [Kit2, p.179, l.-3-l.-2].
The proof given in [Tin, p.38, (3-26)] is also based on the idea of
diagonalization, but its
discussion is limited to a special case [Tin, p.38, l.11-l.14. Here, the group of the
Schrödinger equation [Tin, p.33, l.-11]
must be cyclic] and uses the language of the group representation theory.
Second proof [Ashc, pp.137-139; Kit2, pp.183-185].
The Fourier series expansion of the wave function: [Ashc, p.137, (8.30)], where {q}=all the values of the wavevector permitted
by the Born-Von Karman boundary conditions (see [Ashc, p.136, (8.27)]; [Kit2,
p.183, (25)]).
Remark. For the physical origin of the Born-Von Karman boundary
conditions, see [Iba, p.83, l.-8-p.84, l.9]. The
advantage of the Born-Von Karman boundary conditions is that we can base
our discussion of a finite crystal on the model of an infinite crystal [Ashc,
p.136, (8.27)] rather than restart the discussion from scratch ([Coh,
Complement O_{III} partially repeats the
discussion of [Coh, Complement F_{XI}])
Decouple the family of {c_{q}} in [Ashc, p.137, (8.30) into subfamilies
c_{k+K} [Ashc, p.138, l.-11], where k's
are defined by [Ashc, p.136, (8.27)]. We label the decoupled wave function
y as y_{k}
[Kit2, p.184, l.7-l.10]. [Kit2, p.235, l.-9-l.-3]
and [Hoo, p.330, l.-13-l.-7;
p.331, l.-7-p.332, l.-1]-332,
§11.4.1] say the same thing, but Kittel's
formulation is more concise and precise.
Third proof [Coh, p.1162,l.1-p.1164, l.-4].
The dashed curves in [Ashc, p.133, Fig. 8.1] are derived from [Coh, p.790,
(C-4)].
The crystal momentum is not the electronic momentum [Ashc, p.139, (8.45);
Kit2, p.205, (11)].
[Ashc, p.141, (8.50)] is clearly explained by the labeling scheme in A.b.ii.
The steps in A.b.ii pinpoint the reason why the second proof
allows the y_{k }to be
degenerate [Kit2, p.185, l.-1].
The origin of the set {e_{nk}
| where k is fixed and n is any integer} in
[Ashc, p.141, (8.50)].
From the viewpoint of eigenvalues: the roots of the determinant of [Kit2,
p.187, (32)].
From the viewpoint of eigenfunctions: [Cra, p.13, l.5-l.7].
The essential ideas of the above three proofs are the same
(decoupling). The second proof uses
Fourier analysis to convert the equation of motion into decoupled linear systems of algebraic equations.
The first proof uses linear algebra to find a basis to simultaneously
diagonalize the Hamiltonian and translation operators. The third proof
specifies the wavefunction [Coh, p.1164, (13)] and is a special case of the first proof
[Coh, p.1164, l.-2-p.1165, l.19].
Empty lattice approximation [Kit2, p.188, l.-4-p.189,
l.-7]: let the potential functions U_{n}(x)
® 0 uniformly in x as n®+¥;
the displacements indicated in the caption of [Kit2, p.236, Fig. 3] are
justified by [Kit2, p.237, (2)].
Energy levels near a single Bragg plane [Ashc, pp.152-159].
In the case of no near degeneracy, by [Ashc, p.155, (9.13)], the shift
in energy from the free electron value is second order in U [Ashc, p.155, l.9].
In the nearly degenerate case, by [Ashc, p.156, (9.19)], the shift in
energy from the free electron value is linear in U [Ashc, p.155, l.10].
Through the careful estimation from [Ashc, p.155, (9.16)], we shift our attention
from [Ashc, p.152, (9.2)] to [Ashc, p.156, (9.19)]. Kittel jumps from [Kit2,
p.186, (31)] to [Kit2, p.191, l.20-l.21] by observing the superficial
similarity between [Kit2, p.177, (5)] and [Kit2, p.191, (49)]. Thus, Kittel's
argument is not as rigorous and careful as Ashcroft's. The argument in [Iba,
p.135, l.-2-p.136, l.19] is also better than
Kittel's.
The reason given in [Hoo, p.101, l.-4] why we should give up the method used for the nondegenerate
case is inadequate because the fact that the first-order energy correction=0
should not stop us from pursuing the second-order energy correction. In
contrast, [Abr, p.14, l.1-l.8] gives a good reason why we should switch to
nondegenerate case. Furthermore, [Abr, p.14, l.9] gives more choices than those
given in [Hoo, p.101, l.-2-p.102, l.5]
The caption of [Coh, p.409, Fig. 11] says that the two perturbed levels "repel each other". The meaning of this statement is not clear.
In contrast, [Ashc, p.155, l.4-l.7] defines the phrase "two energy levels repel
each other" clearly and mathematically.
Semiconductor crystals.
The equation of motion in k space of an electron in an energy band.
in a uniform electronic field: [Kit2, p.204, (4)].
in a uniform magnetic field: [Kit2, p.204, (7)]. Remark 1. [Kit2, p.204, l.-3-p.205. l.3] illustrates [Ashc, p.229, Fig. 12.6].
Remark 2. The projection of a real space orbit in a plane perpendicular to the field is
an orbit of the same shape and rotation direction as the k-space orbit, but
rotated 90° around the field direction [Ashc,
p.230, l.10-l.13; Hoo, p.375, Fig. 13.7].
A hole.
wavevector: [Kit2, p.206, (17)].
energy: [Kit2, p.207, (18)].
velocity: [Kit2, p. 208, (19)].
effective mass: [Kit2, p.208, (20)].
equation of motion: [[Kit2, p.208, (21)].
Remark. [Kit2, p.209, Fig. 9] is derived from [Kit2, p.204, (4)].
Heat conduction by phonons [Iba, pp.94-99, 5.6; Hoo, pp.67-74].
Remark 1. [Hoo, p.69, (2.73)] is the mathematical proof of the physical formulas [Hoo, p.67, (2.68) & (2.69].
Remark 2. [Hoo, p.70, (2.75)] and [Iba, p.96, (5.43)] are the same. The former formula is derived from elementary kinetic theory, while the latter formula is derived from the consideration of
the canonical distribution [Rei, p.205, l.10] [1].
Remark 3. Normal processes versus umklapp processes [Hoo, p.67, Fig. 2.17(b) versus Fig. 2,17(c); Iba, p.97, Fig.5.6(a) versus Fig. 5.6(b); Kit2, pp.134-135, Fig. 16a,c versus Fig. 16b,d].
The heat capacity of electrons in metal.
The tangents in [Iba, p.114, Fig. 6.6 & p.115, Fig. 6.7] help clarify
the procedure for estimating the small fraction of the free elections that can absorb thermal energy.
The estimate in [Hoo, p.82, (3.16)] is better than that of [Iba, p.115, (6.36)].
See [Iba, p.117, (6.46)].
General relativity.
Remark. The tensor design serves to keep the measurement of physical quantities covariant with coordinate transformations so that
physical laws will retain the same form.
The strong equivalence principle [Ken, p.11, l.-12-l.-10;
p.12, l.7]. Remark 1. The
strong equivalence principle based on the weak equivalence principle [Ken, p.10,
l.-15] is an extension of the first postulate of special
relativity [Ken, p.11, l.-6-l.-3].
Remark 2. A frame in free fall can cover the space-time manifold locally but not globally [Ken, p.12, l.3;
p.40, l.7-l.-9; Pee, pp.231-233,
§ The Metric Tensor].
Remark 3. The principle of generalized covariance [Ken, p.63,
§6.4] can be considered the tensor version of
the strong equivalence principle.
If we express the physical laws in special relativity in terms of tensors,
they will retain the same form in any other accelerated frame. In
particular, if a formula involves derivatives, the derivatives in the
corresponding formula under a coordinate transformation should be replaced by
covariant derivatives [Ken, p.81, l.-15-l.-10].
Mass affects the metric of the space-time manifold: The Schwarzschild metric
equation [Ken, p.44, (4.10)] reduces to the Minkowski metric equation [Ken,
p.44, l.8] in the limit of zero mass.
Remark 1. Geodesics in (space ® Minkowski space
® curved space-time) [Ken, p.41, l.10-l.18]. The
length of a geodesic in space is a minimum, while the length of a geodesic in
the space-time of special relativity is a
maximum [Ber,
p.56, l.4-p.57, l.-5]. This is because the metric
tensor in space is positive definite, while the metric tensor in space-time is
indefinite. Remark 2. The Schwarzschild metric tensor can be derived from [Pee, p.271,
(10.84); p.273, (10.92)]. Another proof can be found in [Ken, p.15, l.-3-p.17,
l.15; Ber, p.75, l.1-l.8]. The first equality in [Ken, p.16, l.12] can be
derived from [Rin, p.40, (17.1)]. The second derivation of the Schwarzschild
metric tensor can be made rigorous by using Einstein's field equation [Ken,
Appendix D, pp.195-196].
Newton's second law [Ken, p.63, (6.11)].
The conservation laws of the four vector momentum [Ken, p.81, (7.13)].
In the Newtonian limit, Einstein's equation [Ken, p.83, (7.19)] will reduce to Newton's law of
gravitation [Ken, p.85, l.7].
The tangential acceleration vs. the normal acceleration [Cou2, vol. 1,
p.396, (41) & (42)].
Electromagnetic properties of matter [Fur, §2.4; Wangs,
pp. 546-568]
It is easier to recognize the outline of electromagnetic properties of
matter in [Fur, §2.4] than in [Wangs, pp.
546-568]. Furthermore, few books explain [Fur, p.104, Fig. 2.18] as clearly as [Fur]. However, it
is better to study [Kit2, pp.380-392] before one reads [Fur, §2.4].
This is because [Kit2, pp.380-392] provides rigorous definitions of
applied electric field, the macroscopic electric field, and the local electric
field. The prerequisite to understanding [Kit2, pp.380-392] is [Wangs, chap. 10 and chap.
23].
For clarity, [Wangs,
p.548, Fig. B-1] should be supplemented with [Hall, p.472, Fig. 26-12].
Hysteresis [Cor, pp.375-377, §21.2 & p.422, Example].
Remark. [Wangs, p.338, l.22-l.32] illustrates the theoretical advantage of using a Rowland ring, while [Cor,
p.375, footnote] explains why Rowland rings are no longer used in practice.
The Lorentz condition [Wangs, p.365, l.12-l.16].
The basis of the Debye interpolation scheme.
[Ashc, p.466, l.3] To be consistent with the Dulong and Petit law at high temperatures, the area under the theoretical curve g_{D}(w)
[Ashc, p.466, l.1] must be the same as that under the experimental curve [Rei, p.410,
Fig. 10.1.1].
[Ashc, p.466, l.4] To obtain the correct specific heat law at low
temperatures, the theoretical curve must agree with the experimental curve in
the neighborhood of
w = 0. Thus, the Debye scheme should adopt the simplification
given in [Ashc, p.456, l.1-l.10; Fig. 23.1].
The energy [Lan1, p.14, l.10], momentum [Lan1, p.17, l.-14-l.-12], and angular momentum
[Lan1, p.20, l.-8-p.21, l.9] of a closed system.
Conservation laws for the energy-momentum tensor of the electromagnetic field:
Special relativity: [Lan2, p.82, (33.6)].
General relativity: [Ken, p.81, l.16].
Conservation of crystal momentum [Ashc, p.786, (M.7)]:
Isolated insulator [Ashc, p.787, (M.18)].
Scattering of a neutron by an insulator [Ashc, p.788, (M.22)].
Isolated metal [Ashc, p.788, l.-7-l.-5].
Scattering of a neutron by metal [Ashc, p.788, l.-4-p.789, l.2].
Remark 1. In terms of reduced symmetry, the conservation of crystal momentum in Case C.a
is similar to the the conservation of
angular momentum in the examples given in [Lan1, p.21, l.1-l.9].
Remark 2. In order to understand the precise meaning of Noether's theorem,
one needs an elaborate
analysis like Sagan's. Compare [Sag, p.120, Definition A2.16] with [Fomi, p.80,
Definition].
Thermal conductivity.
The formula for thermal conductivity [Ashc, p.500, (25.30) & (25.31)].
The derivation of the formula can be found in [Rei, p.479, Fig. 12.4.2 (1-dim)
® Ashc, p.500, Fig. 25.3 (3-dim)]. The arrow
implies that the basic idea in the two cases is the same.
The reasons why a perfectly harmonic crystal would have an infinite thermal conductivity.
The phonon states are stationary states [Ashc, p.496, l.-11]
Û There are no collisions between different phonons
[Kit2, p.133, l.23] (i.e. there is no thermal resistivity).
Remark. The scattering of phonons means that the wave functions of phonons evolve with time.
A nonvanishing mean velocity is given by [Ashc, p.141, (8.51)] (see [Ashc, p.497, footnote 15]). This mean velocity is
not driven by a temperature gradient [Kit2, p.134, Fig. 16a, l.5].
[Pei, p.40, (2.56); l.10-l.11].
At high temperatures (T >> Q_{D}):
[Ashc, p.501, (25.32) & (25.33)].
At low temperatures (T << Q_{D}):
As the temperature decreases, the conductivity will increase [Ashc, p.504,
(25.40) & p.500, (25.31]. The phonon mean free path will increase up to the
length limit imposed by lattice imperfections, impurities, or size. Hence, the phonon
mean free path will become independent of temperature. Thus, the temperature
dependence of the conductivity is determined by the specific heat. Specifically, the
conductivity will rise as T^{3}. The rise will
reach a maximum at a temperature where umclapp processes [Ashc, p.502,
l.1] become frequent [Ashc, p.504, (25.39)] enough to yield a mean free
path shorter [Ashc, p.504, (25.40)] than the temperature-independent one.
Beyond this temperature, the conductivity continues to decline as exp(T_{0}/T)
[Ashc, p.504, (25.40)] up to temperatures well above Q_{D}.
After this the exponential decline is quickly replaced by a slow power law [Ashc,
p.501, (25.33)].
Remark. When studying thermal conductivity, I had a hard time understanding
both [Ashc] and [Kit2]. Ashcroft repeats the same word "the mean free path"
five times in a single paragraph (see [Ashc, p.504, l.-19-l.-10]).
His
act of repeating the same words as though they were an incantation and his
consideration on the impacts on
the mean free path due to an overwhelming number of factors only obscure,
rather than clarify, the
key point. [Kit2, pp134-135, Fig. 16] and its illustrations occupies almost
half of the space of the entire section [Kit2, pp133-135,
§ Thermal resistivity of phonon gas].
However, this figure is only a minor point in understanding thermal
conductivity. Thus, Kittel's emphasis is misplaced.
The Hartree-Fock approximation.
The expectation value of the Hamiltonian: [Ashc, p.333, (17.14) or Ost,
p.111, (3.2)].
The Hartree-Fock equations: [Ashc, p.333, (17.15) or Ost, p.114, (3.14)].
Remark. For the purpose of the derivation of the above equalities, Ostlund's simplified notions are more appropriate.
Rayleigh scattering [Jack, p.466, (10.35)] explains why the sky is blue, why the sunset is red,
why it is easy to get a sunburn at midday, and why
infrared is good for seeing distant stars through the dust in the Milky Way.
The dispersion relation in a plasma [Kit2, p.274, (15)] explains the transparency of alkali metals in the ultraviolet
and the reflection of radio waves from the ionosphere [Kit2, p.274, l.-3-l.-2].
The Robertson-Walker metric [Ber, p.105, (6.1.3)]. Remark. Peebles'
oversimplified introduction to this metric [Pee, p.54, (5.9)] fails to stress
its insight: the Gaussian curvature is invariant. The proof of Theorema
egregium in [Ber, Appendix B, pp.160-163] is awkward. A better proof can be
found in [Lau, p.65, Theorem 5.5.1].
Special relativity.
How we synchronize clocks [Rin, p.9, l.-13-l.-9].
Why the transformation from one inertial frame to another is linear
[Rin, p.11, l.17-l.27].
The analogy between a rotation
and a Lorentz transformation [Cou2, vol. I,
§4.1.j].
Transport theory.
Reif's approach goes from the easy to the complicated: using average velocity v
to express the collision frequency [Reif, p.470,
(12.2.7)] ® using
the distribution function f(r,v,t) to formulate the Boltzmann equation
[Reif, p.525, (14.3.8)]. In contrast, Huang's approach jumps to the
complicated directly [Hua, chap.3]. Thus, Huang leaves out the following two important
turning points of the theory's development:
Flux: [Reif, p.470, (12.2.6)]
® [Reif, p.497,
(13.1.3)].
The Boltzmann equation: [Reif, p.509, (13.6.2)] ® [Reif,
p.525, (14.3.7)]. The equivalence [Reif, p.510, l.1] of the two formulations
enables us to jump from a crude approach [Reif, p.504, l.3-l.6] to an more
accurate approach.
Huang fails to prove du=du' [Hua, p.60, l.-13], while Reif gives a rigorous proof
[Reif, p.521, l.15].
Reif establishes a relationship between s(v_{1},v_{2}ïv_{1}',v_{2}') and
s(W) [Reif, p.520,
(14.2.4)], while Huang does not. Therefore, the statement in [Hua, p.69, l.-8]
is not clear. Similarly, [Reif, p.524, (14.3.3) is clear, while [Hua, p.66,
(3.29)] is not.
The generalization from an inversion [Reif, p.522, l.1-l.2]
to a rotation or reflection [Hua, p.63, l.9-l.10] is unnecessary because it does not have any
other useful application than the inverse collision.
[Reif, p.523, l.-18-l.-15] imposes an essential assumption on f(r,v,t)
to justify the format of the mathematical expression in [Hua, p.56,
(3.2)]. The reasons given in Huang's justification [Hua, p.56, l.8-l.14] are
related, but are not essential.
The equality in [Hua, p.96, l.8] is derived from [Reif, p.529, (14.4.20)]
and its corresponding formula for the inverse collision.
The Collision frequency: [Reif, p.470, (12.2.7)] is too crude and [Reif,
p.470, (12.2.8)] is too sophisticated. [Reic, p.660, (11.14)] gives an
appropriate interpretation of the Collision frequency.
[Hua, p.106, (5.72)] is correct, but Huang's argument for its derivation
is incorrect. It would be better to use brute force to calculate each coefficient of
L_{kl} in the
summation on the right-hand side of [Hua, p.106, (5.71)].
(Unifications) The conservation theorem [Hua, p.96, (5.14)] unifies the conservation laws of mass, momentum, and energy [Hua, p.98, (5.21)-(5.23)].
Huang derives the conservation theorem from the Boltzmann transport equation [Hua,
p.67, (3.36)], which involves the concept of differential cross section. In
contrast, [Reic, pp.534-537, §10.B.1]
derives the conservation laws of mass, momentum, and energy without
using the concept of differential cross section. The entropy source [Reic,
p.539, (10.26)] helps define the generalized currents and forces [Reic,
p.539, l.-11-l.-5].
Thus, the discussion in [Reic, pp.537-541, §10.B.2]
is an indispensable step toward recognizing that transport coefficients
are the generalized conductivities of a hydrodynamic system [Reic,
p.539, l.-4-p.540, l.1;p.541, l.5-l.10; p.543,
(10.29)-(10.31)]. Putting transport coefficients and conductivity into the same
category is a kind of unification that stride across different fields.
Classical statistical mechanics
Ensemble [Hua, p.141, l.8].
Remark. If we can prove a statement directly, we should not take a detour. [Hua, p.141,
(7.6)] can be directly derived from the definition of a partial derivative.
Huang's detour approach [Hua, p.141, l.-13-l.-12] indicates that he does not understand the definition of a partial derivative very well.
The density of states [Man, pp.324-335, Appendix B].
Remark. [Man, p.334, (B-38)] can be proved using [Kit2, p.87, Fig. 18].
A system in a heat bath [Man, pp.52-64, §2.5].
Remark 1. The proof of [Man, p.57, (2.29)] can be found in [Reif, p.213, (6.5.8)].
Remark 2. We need not repeat the historical approach. The classical method of
counting states must be fully justified in terms of quantum mechanics.
Compare [Reif, p.51, l.8-l.16] with [Man, p.174, l.-7-l.-1].
Fluctuations
Energy [Reif, p.110, (3.7.14); p.213, (6.5.8) or Man, p.58, (2.31) (the canonical ensemble)].
Occupation numbers [Hua, p.82, (4.54) (the ideal gas)].
The canonical ensemble evolves from the microcanonical ensemble:
The drawback of the microcanonical ensemble with respect to calculations [Hua, p.153, l.7-l.12].
The new constraint imposed on the canonical ensemble [Hua, p.157,
l.2-l.4].
The ideal gas.
Remark 1. When considering ideal gases, the first thing one
has to do is to throw away all the world's documents about ideal gases. One
should
study only [Reif, §9.1,
§9.2, §9.6,
§9.7, §9.8,
§9.10], except for any discussions about
Maxwell-Boltzmann statistics they contain. If one needs the required background on
quantum mechanics, one should read only [Coh, §XIV,C.3.d].
This approach relieves one of the burden of studying a tremendous amount of incorrect physics. To stop the
practice of torturing physics students, the future authors of physics
textbooks should follow my advice.
Remark 2. Mandl
points out that [Hua, p.146, (7.22)] is based on [Hua, p.152, (7.52)].
However, Mandl's strategy to prove [Man, p.188, (7.70)] does not work. It is
better to follow Huang's calculation scheme [Hua, p.152, l.9-l.10].
Remark 3. In classical mechanics, a
rigorous definition of physical states
[Coh, p.1392, (C-9)] for a system of identical particles does not exist.
Therefore, for the partition function of the ideal gas, the classical method
of counting states requires a correction when compared with the correct
quantum result. The only book in classical mechanics that contains a clear definition of a macrostate is [Zem,
p.279, l.11], but the concept is borrowed from quantum mechanics. The goal of counting states in
classical mechanics is to lead to a rigorous
definition of physical states. A good classical method of counting
states should facilitate accomplishment of this goal. For example, the first term of [Man,
p.168, (7.9)] corresponds to [Coh, p.1390, (C-11)] and the second term of
[Man, p.168, (7.8)] corresponds to [Coh, p.1390, (C-10)] (up to the
normalization factor). Even so, the classical derivation of [Man, p.169,
(7.10)] is not as rigorous as the quantum mechanical derivation of [Reif,
p.361, (0.10.3)].
(Paramagnetism) The discrepancy between [Reif, p.208, (6.3.7)] and [Pat,
p.81, (14)] is due to different averaging methods. The former averages the
magnetic susceptibility over
the two spin states [Reif, p.207, (6.3.3)], while the latter averages the
magnetic susceptibility over all
solid angles [Pat, p.80, (7)].
(Microcanonical ensemble) Reichl fails to explain why C_{N}
= N! h^{3N} in [Reic, p.348, (7.16)]. N! can be
explained using the strategy given in [Reic, p.359, l.8]. h^{3N}
can be explained by [Man, p.174, l.-1]. Similarly,
Huang fails to explain the Gibbs correction factor in [Hua, p.195,
(9.42)]. Intrinsically, Microcannonical ensemble is a classical design. Its
shortcomings are discussed in [Man, p.182, l.9-l.18]. The problem with [Hua,
§9.5] is that Huang throws quantum
particles into a classical design without explaining why it is justifiable to
do so. A theory should be built with its essential features. Building a theory should not be like making a
pizza with every topping on it. it should not involve throwing all the knowledge into one pot. A
cumbersome theory that has no
application or that is designed to solve every problem is trash. Even just looking at [Hua, p.194, Fig. 9.1] makes
one dizzy. The discussion that goes with this figure is even more confusing.
One should apply a method in a
flexible manner instead of being entrapped in its mathematical structures.
Furthermore, a model
should be as simple as possible. Consequently, the discussion of an
ideal gas in [Reic, p.348, Exercise 7.2] is better than that in [Hua, p.196,
l.-3-p.197, l.16]. The same remark applies to [Pat,
§6.1], Cliff's notes of [Hua].
Remark. Landau uses the fact that levels broaden into bands [Lan5, p.15, l.12;
p.22, l.-12-l.-9] to explain why the
microcanonical ensemble, a classical design, still applies to quantum
statistics.
The grand partition function
The justification of the definition given in [Hua, p.190, (9.27)]: [Reif,
p.347, l.1-l.18].
Its simplified relationship with the partition function [Reif, p.347,
(9.6.6)].
Equalities [Hua, p.198, (9.61) & l.-3-l.-1].
Remark. Reichl derives [Reic, p.382, (7.121) & (7.123)] using terminology
that is less
intuitive but conveys the same idea (i.e.,
step a and step c). However, the advantage of the Lagrange multipliers, trace,
and the number representation allows Reichl's argument to go directly from
[Reic, p.378, (7.109)] to [Reic, p.382, (7.121) & (7.123)]. It is unnecessary
to pass through the middle stage given in [Hua, p.190, (9.27)], and then change the basis
(see step c) to 1obtain the desired result.
Einstein's gravitational field
equation [Pee, p.268, (10.65)]
® Gravitational field equations for nonrelativistic material [Pee, p.269, (10.69); Rin,
p.103, l.12-l.14].
Electromagnetic field equations [Rin, p.103, (38.3); p.104, (38.5) & (38.7)]
Û the Maxwell equations [Rin, p.107, (38.20), (38.21) & (38.23)].
Polarizability.
The Lorentz relation [Chou, p.76, (2.92)].
The Clausius Mossotti relation [Chou, p.76, (2.95)].
Remark. For details of this topic, consult [Wangs, pp.546-554, Appendix B-1].
Boundary-value problems in electrostatics.
Formal solutions of the Poisson equation [Chou, p.31, l.-2-p.32,
l.9].
The existence of solutions of the Poisson equation with Dirichlet or Neumann boundary conditions [Chou,
§3.2].
The uniqueness of solutions of the Poisson equation with Dirichlet or
Neumann boundary conditions [Chou,
§3.2].
Remark 1. The assumptions of this problem are carefully written in [Chou,
§3.1]. The argument in [Jack, §1.9]
cannot be considered rigorous because Jackson fails to state these assumptions
clearly.
Remark 2. All of the following statements are justified by the uniqueness theorem.
[Chou, p.115, l.-10-l.-8].
[Chou, p.117, l.2-l.5].
[Chou, p.120, l.1-l.2].
The method of images [Chou, §3.4; Sad,
§6.6; Jack, §2.1-§2.5]].
The image charge must be external to the region of interest [Jack, p.57,
l.-9-l.-8].
The solution of the Poisson equation is provided by the sum of the potentials of the charges inside the region
of interest [Jack, p.57, l.-6-l.-5].
Remark 1. [Wangs, p.175, l.18-l.6] gives three methods to find the force on q.
By the uniqueness theorem, all we need is the method given in [Chou, p.117,
l.2-l.5]. Other methods are unnecessarily complicated. Even trying to find a
method other than the method of images is meaningless in the first place
because anyone who understands the uniqueness theorem thoroughly should not
raise such a question. [Jack, p.60, l.-20-l.-13],
produced by a Berkley professor, is also of no value.
Remark 2. [Sad, p.240, l.9-p.241, l.6] provides a piece of general guidance that helps
to solve boundary-value problems using the method of images. This guidance,
based on experiences, can only be described by guidelines rather than specific
details. The success of applying these guidelines relies on one's skill and
experiences.
Although guidelines are valuable advice, they do not guarantee the success of
problem solving. Driving is an example. Driver A with ten year experience is accident free now. Driver B just received his driver
license for the first time. Even if Driver A gives Driver B good guidelines
about safe driving, the latter still has to go through many accidents during
the first year to learn to become a safe driver. Similarly, to become
a skilful problem solver, one has to practice constantly and compile
new guidelines from one's own experiences. [Lau, p.67, l.12-l.16] provides
another interesting example.
Complex-variable methods [Chou, §3.5].
Conformal representation [Chou, §3.6].
Solutions for the spherical boundary conditions: [Jack,
§2.6].
Boundary-value problems with azimuthal symmetry
Dielectric sphere in a previously uniform electric field: [Cor, pp.231-233, Example], [Wangs,
pp.194-197, Example] and [Chou, pp.148-150, (ii)] all give the formula of the
electrostatic potential. Of the three
proofs of this formula, Corson's proof is the best. His concept is clear and his analysis is
rigorous. In [Wangs, p.197, Fig. 11-13], Wangsness says the lines belong to
the E field. In fact, they belong to the D field (see [Cor, p.232, Fig.
12-2]).
A useful device: [Jack, p.101, l.-2-p.104,
l.4].
Remark 1. Although Jackson elegantly uses the uniqueness theorem to prove [Jack,
p.102, (3.38)], the proof of [Wangs, p.112, (8-12)] is a natural
approach which links Legendre polynomials
more closely to their generating function [Guo,
§5.3].
Remark 2. Although Jackson applies the device only to the boundary-value problems, the key idea of this device is actually based on [Ru2, p.226, Corollary].
Mixed boundary conditions (e.g., conducting plane with a circular hole) [Jack,
§3.13}
Remark. The equations of [Jack, p.132, (3.179)] can be solved using [Guo, p.406, (8) & (9)].
Boundary-value problems with dielectrics [Jack,
§4.4].
Remark. For a plane, we use the method of images [Jack, p.154, l.-2-p.157, l.9];
for a sphere or spherical cavity, we use separation of variables in spherical
coordinates and expand the solution in a series using the basis of the Legendre polynomials. These methods are
essentially the same as those of finding the green functions [1].
Electric property of dielectrics
The Ewald-Oseen extinction theorem [Born,
p.101, l.4-l.7] The dipole field is the sum of two terms [Born, p.102,
(21)]: one cancels out the incident wave
[Born, p.102, (23)], whereas the other satisfies the wave equation with velocity c/n
[Born, p.100, (10)].
Remark 1. (Internal references) The validity of the
statement in [Born,
p.101, l.4-l.7] is not well documented, so one may not catch its
meanings immediately until one finishes reading [Born,
§2.4.2]. However, if Born had pointed out
where the readers can find the proof for each phrase of the statement in this
long section [Born, §2.4.2] as I did
above, the readers would catch the meaning at the first reading and would have
a clearer picture for understanding the rest of material in [Born,
§2.4.2]. The mathematics textbooks written
by N. Bourbaki are famous for their internal references: The validity of
almost every statement is well documented whether the proof is given before
the statement or after.
Remark 2.
For various electric fields in dielectrics only [Kit2, pp.380-392] provides
clear definitions. Therefore, it is important to identify
the effective field E' in [Born, p.85, l.23] with the local field E_{local}
in [Kit2, p.386, (14)] or the polarizing field E_{p}
(producing the displacement of charges) in [Wangs, p.547, l.-14]
and identify the mean field E in [Born, p.85, l.24] with the total macroscopic
electric field E in [Kit2, p.384, (7)].
Remark 3. The assumption [Born, p.104, (32)] is not used in [Born, p.104,
l.8-p.107, l.12]. [Born, p.104, (32)] is proved by [Born, p.107, (49); p.105,
(41)] with the assumption [Born, p.104, (33)]. The purpose of presenting
[Born, p.104, (32)] before its proof is to help create [Born, p.104, Fig. 2.4] so
that we know what is going on.
Remark 4. The extinction theorem provides the insightful relationship between the incident field and the dipole field. This relationship
based on the microscopic viewpoint (dipoles) is so powerful that it can be used to derive both the law of refraction [Born,
p.107, (52)] and the Fresnel formulae [Born, p.107, (55a) & (55b)].
Molecular polarizability and electric susceptibility [Jack,
§4.5]
The Clausius-Mossotti equation [Jack, p.162, (4.70)].
The Lorentz-Lorenz equation [Jack, p.162, l.-1].
Remark.
[Born, p.87, (17)] serves to link the microscopic quantity
a [Born, p.92, (30)] to the macroscopic phenomena (e
and n). The Lorentz-Lorenz equation implies that the refractive index depends
on frequency [Born, p.92, (31)]. The proof given in [Born,
§2.3.3] is valid only for the first
approximation [Born, p.85, l.12]. In contrast, the proof given in [Born, §2.4.2]
is rigorously derived from an integral equation concerning polarization
[Born, p.100, (4)].
Electrostatic Energy in dielectric media [Jack,
§4.7].
Remark. The material in [Jack, p.166, l.-2-p.167, l.17] is treated by Jackson
as part of the content of his advanced textbook. However, in [Wangs,
p.164, l.1-l.4], Wangsness treats the same material as an exercise of his
elementary textbook. It is too difficult for the reader of an elementary
textbook to encounter an exercise that is accorded extensive coverage in an
advanced textbook. There is difference between writing a paper and doing an
exercise. Similarly, it is not proper to put an exercise from an elementary textbook into
the content of an advanced textbook. Considering the intended reader, it is
clear that one of the above two authors must be seriously wrong.
Remark. The concept of various fields in the first paragraph of [Jack, p.160]
is clear. In comparison with [Jack, (4.71) & (4.95)], the signs of forces are
carefully explained in [Wangs, (7-36), (10-99) and (B-7)]. In contrast with
the abstract theory given in [Jack, §4.5,
§4.7], [Wangs, p.166, Fig. 10-18] gives
a concrete example. To reap benefits from both books, it is important to
the differences in terminology they employ. Let us attach a subscript J to a notation if the
notation is used in [Jack] and a subscript W to a notation if the notation is
used in [Wangs]. By comparing [Jack, p.161, (4.67)] with [Wangs, p.548,
(B-9)], we see E_{p;W}
= E_{;J}+E_{i;J}
and the sum in [Wangs, p.163, (4.73)] = a in [Wangs,
p.548, (B-9)]. The following identities show that there are no inconsistencies
in concepts between [Jack,
§4.5] and [Wangs, Appendix
§B-1] even though the same notation or
terminology in the two books may mean different things: E_{ I;W} = E
_{near;J} and
E_{;J}
= E_{ p;W}
- E_{
i;J}
= (E_{ ;W} + E _{O;W} + E
_{I;W}) - (E _{
near;J}
- E_{ P;J})
= E_{ ;W} + E _{O;W}
+ E_{ P;J }. The above comparison is only a
temporary remedy. In the future, we must unify the terminology in this area so
that physicists can speak the same language.
Force on a localized current distribution in an external magnetic induction
[Jack, p.188, l.-12-p.189, l.-7;
Wangs, pp.531-538, §A-2]
The proof of [Jack, p.189, (5.69)] is incorrect. The notation m´Ñ in [Jack, p.189, (5.67)] is problematic. This notation is
not defined
in any math textbook because
multiplication and differentiation are not commutative: m(¶f/¶x)¹¶(mf)/¶x.
Furthermore, [Jack, p.189, (5.68)] is incorrect. For a correct formula of
Ñ(m×B), see [Wangs,
p.34, (1-112)]. Thus, [Jack, p.189, (5.67)] should have been
F_{i} = S e_{ijk}[m´ÑB_{k}(0)]_{j};
[Jack, p.189, (5.68)] should have been
F = m´(Ñ´B) + (m×Ñ)B
=
Ñ(m×B).
Remark 1. Jackson's serious mistakes reveal the urgent need to
strengthen the teaching staff in today's institution of American higher education.
Remark 2. Using [Chou, p.214, (5.62)], Choudhury gives an elegant proof of [Chou, p.214, (5.63)].
Remark 3. There is no inconsistency between [Jack, p.189, (5.69)] and [Wangs,
p.537, (A-35]. We can use [Wangs, p.533, (A-20)] to explain why the two
formulae look different.
The Magnetic Hyperfine Hamiltonian
A classical treatment [Jack, p.190, (5.73)].
A quantum mechanical treatment [Coh,
pp.1247-1256, Complement A_{ XII}].
Remark 1. [Coh, p.1251, Fig. 2; p.1252, l.1-l.19; p.1253, l.-17-l.-4]
provide a better explanation of the second term of [Jack, p.188, (5.64)]
and of the contact term of [Jack, p.190, (5.73)].
Remark 2. [Jack, p.145, l.-7-p.146, l.-5] gives a rigorous proof of the statement in [Coh, p.1066,
l.6].
Remark 3. The formula given in [Coh, p.1249, l.-2]
and the formula given in [Jack, p.190, l.-7] are
the same. The latter formula is derived from [Jack, p.176, (5.5)] by replacing
the x in [Jack, p.175, Fig. 5.1] with -x (where x
is the position of the electron).
Remark 4. [Jack, p.190, l.-13-l.-11] tells us what the hyperfine
interaction is, while [Coh, p.1248, (5)] traces to the origin of the hyperfine
interaction.
Magnetization
A substance with permanent magnetization: B and H [Jack,
§5.10].
Without an external field [Jack, p.198, (5.105) & (5.106)].
In an external field [Jack, p.200, (5.112)].
Remark. The difficulty of the method given in [Jack, p.199, l.1-l.8] lies in
calculating ò _{[}_{0,
a]}:
ò _{[}_{0,
a]}
= ò _{[0, r]}^{ }
+ ò _{[r, a]}.
The difficulty of the method using the vector potential [Jack, p.199, l.-12-p.200,
l.6] lies in calculating the curl. The method given in [Jack, p.198, l.6-l.-1]
does not have these difficulties, so it is the simplest.
A paramagnetic or diamagnetic substance: the magnetization is the result
of the application of an external field [Jack, p.200, (5.115)].
A ferromagnetic substance: the phenomenon of hysteresis allows the creation of permanent magnets
[Jack, p.201, Fig. 5.12].
Remark. For the basics of hysteresis, see [Sad, p.328, l.5-p.329, l.-1].
Magnetic shielding
Analogies between a conductor and a body with high permeability
The field lines outside and near to the surface [Jack, p.201, l.-17-l.-15].
Cavities [Jack,
p.201, l.-15-l.-13].
The dipole moment and the inner field [Jack, p.202, (5.121); p.203, Fig.5.14].
Thermodynamic equilibrium [Zem, §1-5
& §2-1]; equations of state [Zem, §2-5;
§2-8-§2-12]; macroscopic states and thermodynamic variables [Zem,
p.26].
Quasi-static transformations [Hua1, p.4, (f)]. In order to
warrant the use of an equation of state, we must perform a quasi-static
process. Methods of performing a quasi-static process [Zem, p.51, l.-9-l.-8;
p.56, l.14-l.16; p.57, l.6-l.7; p.85, l.-14-l.-4].
Remark. Slow free expansion is quasi-static [Hua1, p.4, l.11-l.12]; fast free expansion is not quasi-static [Zem, p.113, l.11].
Reversible transformations
Adiabatic reversibility [Zem, §7-1-§7-7].
Reversibility involving heat transfer: reversibility in this case refers to the
universe [Zem, p.85, l.-13-l.-4; §8-5-§8.6].
Remark.
[Zem, §7-7] proves that the solutions of the
differential equation [Zem,
p.165, (7-1)] are reversible adiabatic
hypersurfaces. The illustration builds a solid foundation for the following concepts: Carnot
cycles [Zem, p.173, Fig. 7-8], Kelvin temperature scale [Zem,
§7-10] and entropy (Compare [Zem, p.179, l.-3]
with [Zem, p.174, l.14, the first equality]; [Zem, p.180, (8-3)]). Because they lack this
indispensable proof, the statements about the above concepts given
in both [Hua1] and [Kit] are unclear. For this reason, their foundations of
thermal physics are seriously flawed.
Ideal-gas temperature = Kelvin temperature [Zem,
§7-8-§7-11].
Speed of a longitudinal wave [Zem, §5-7].
The second law of thermodynamics
The following three statements are equivalent:
No process is possible whose sole result is the absorption of heat from a reservoir and the conversion of this heat into work.
No process is possible whose sole result is the transfer of heat from a cooler to hotter body.
Whenever an irreversible process takes place the entropy of the universe increases.
Proof. A Û B [Zem, §6-7].
A Þ C [Zem, §8-5-§8-8].
C Þ B [Man, p.115, l.-9-l.-1].
Remark 1. If you compare [Zem, p.154, 1] with [Hua1, p.10, l.-18-l.-7],
one can easily find that T_{1} in [Hua1, p.10,
l.-8] should have been T_{2}.
This error remained undetected through two editions of [Hua1] (1963 & 1987).
This reveals the fact that in these forty years, either no one reads the
publications of MIT professors or no one cares about the books published by MIT
professors.
Remark 2. [Hua1, p,19, l.13-l.-6] discusses some subtle points that we should pay attention to
when we apply the second law.
The Clausius theorem
The most concise and insightful proof is given by [Reic, p.28, l.5-p.31, l.4]. [Reic,
p.30, Fig. 2.5] clarifies the confusion contained in other proofs. The proof of [Zem, p.180, (8-3)] is based on [Zem, p.173, (7-13)]. The
inexact differential in [Zem, p.173, (7-13)] has a specific form. The inequality in [Hau1, p.14, l.-8]
conveys Clausius' subtle point about the inexact differential
đQ. However, the proof of the Clausius theorem
in [Hua1, pp.14-15], omits too much detail. A detailed proof using the same
argument can be found in http://en.wikipedia.org/wiki/Clausius_theorem.
The Clapeyron equation [Zem, p.31, l.-4-p.35, l.12]
[Zem, p.247, l.-12] mentions the fact that the vapor pressure P(T)
is a function of T only, but does not provide the proof. [Hua1,
p.33, (2.3)] does give the proof.
Chemical equilibrium
Let us compare [Reif, §8.2,
§8.3 and §8.10]
with [Zem, §14-8].
Note that [Zem, §14-7] emphasizes the following subtle points:
Even though the initial states of the phases are not in chemical equilibrium, it is still possible to describe them in terms of thermodynamic coordinates. This is because these phases are in mechanical and thermal equilibrium
[Zem, p.366, l.11-l.17].
The functions that express the properties of a phase when it is not in
chemical equilibrium must reduce to those for thermodynamic equilibrium
when the equilibrium values of the n's are substituted [Zem, p.388, l.-7-l.-4]. In other words, in
thermodynamic equilibrium the n's are fixed values, but when the system is
not in chemical equilibrium, these n's are
variables.
Remark 1. [Reif, §8.2 and
§8.3] are reduced to twelve lines in [Zem,
p.368, l.-6-p.369, l.6].
Remark 2. The proof of [Reif, p.314, (8.7.18)] is excellent, while the proof
given in [Zem, p.372, l.-7-p.373, l.6] is very
confusing.
Degree of reaction [Zem, §14-12].
Equation of reaction equilibrium [Zem, §14-13].
Law of mass action [Zem, §15-1].
Heat of reaction [Zem, §15-3].
Affinity [Zem, §15-5].
The phase rule
Without chemical reaction [Zem, §16-2 &
§16-3].
With chemical reaction [Zem, §16-4 &
§16-5].
Displacement of equilibrium [Zem, §16-6].
Thermocouples [Zem, §17-6-§17-10].
Black-body radiation
Why do we use cavity radiation to represent black-body radiation?
Because
Cavity radiation is in thermal equilibrium so that the thermodynamic coordinates can be defined
[Man, p.246, l.-15].
A small hole in a wall has the same absorbing and emitting power as a black-body. Key idea: [Man, p.246, l.-6-l.-3].
Proof: [Zem, p.91, (4-17)].
[Man, Appendix B] proves the formula for the density of states using both particle [Man,
§B.2] and wave [Man,
§B.3] approaches. [Man,
§10.3] proves
Planck's law using both the particle and wave approaches.
In order to prove Wien's displacement law, [Reif, §9.13] obtains the maximum by drawing the graph of the function
[Reif, p.375, Fig. 9.13.1], while [Zem, §17-14] obtains the maximum
using calculus.
Planck's radiation equation [Zem, p.446, (17-27)] reduces to
The Rayleigh-Jeans law [Man, p.253, (10.21)] or the equipartition theorem
[Man, p.253, l.-6] in the limit of low frequencies.
Wien's law [Man, p.254, (10.23)] in the limit of high frequencies.
Remark. Studying the problem from the viewpoint of entropy [Wu, p.33, (1-7)], Planck originally used the method of interpolation to derive his
radiation equation from
the Rayleigh-Jeans law and Wien's law [Man, p.363, l.-6-p.364,
l.3].
We may prove the formula for radiation pressure [Zem, p.451, (17-32)] using
The kinetic theory [Zem, p.451, l.9-l.-1].
The partition function [Man, p.255, l.5-p.256, l.11].
Remark 1. How is the concept of standing waves related to cavity radiation?
Ans. [Eis, p.7, l.16; p.8, l.20; p.14, l.3]. [Eis,
§1-1-§1-4] can serve as both a good
introduction and a good summary of black-body radiation because its
formulation of the theory is closely and clearly related to basic concepts.
However, there is a mistake in [Eis, p.11, l.31-l.32]. N(n)dn
(the number of allowed frequencies) ¹ N(r)dr (the
number of quantum states [allowed k-vectors]) unless we regard a
frequency as a vector. In contrast, [Man, p.328, l.3] adopts a unified and better
convention.
Remark 2. [Man, Appendix B] gives a comprehensive discussion on the density of states.
There are two points worth noting: First, the density of states is independent
of boundary conditions [Man, p.330, l.1-l.2]. Second, the discussion of density
of states naturally leads us from the wave equation [Man, p.324, (B-1)] to the Schrödinger equation [Man, p.331, l.-6-l.-2;
Reif, p.353, l.-3-p.354, l.9].
The homopolar motor and the homopolar generator [Cor, pp.399-404].
Remark. [Wangs, p.276, l.15-p.277, l.8] can be used as a stepping stone to understand the two examples in [Cor, pp.399-404].
Quasi-static electromagnetic fields and the skin effect [Chou,
§6.4].
Remark. [Wangs, p.450, l.-2-p.451, l.8] discusses the physical significance of the neglect of the displacement current from the viewpoint of energy loss and the viewpoint of the time needed for propagation of signals, while [Chou, p.255, l.15-p.256, l.2] discusses the physical significance from the latter viewpoint alone.
Derivation of the macroscopic Maxwell equations [Chou,
§7.2].
Remark 1. Both Jackson and Choudhury call the details of the proof gory [Jack, p.255, l.-10; Chou, p.304, l.-1].
If this trivial proof is considered gory, I wonder what adjective should be used
to describe Tycho Brahe's or Kepler's work.
Remark 2. A formula should be written in its simplest form. [Chou, p.309, (7.44)] can
still be reduced to [Jack,
p.256, (6.96)]. Both [Chou, p.309, (7.44)] and [Jack, p.256, (6.96)] are
incorrect as they stand. S_{n}
should have been inserted in front of S_{r,s}
in [Chou, p.309, l.2]. The expression inside the [ ] in [Jack, p.256, l.7]
should have been (Q_{n}')_{ag}(v_{n})_{b}-(Q_{n}')_{gb}(v_{n})_{a}.
Scalar and vector potentials
[Chou, p.315, l.3-l.7] can be directly derived from [Chou, p.585, l.3-l.4]. The argument in [Chou, p.314, l.6-p.315, l.1] basically repeats the argument in [Chou, p.584, l.7-p.585, l.1].
Debye's theory of solids
In order to decouple the equations of motion [Hoo, p.38, l.17], we transform
from the position space [Hoo, p.37, (2.7)] to the momentum space [Hoo, p.38,
(2.9); Man, p.324, (B.1)] using [Hoo, p.37, (2.8); Man, p.330, (B.19)]. This
method of finding normal coordinates has a physical origin [Hoo, p.39, Fig.
2.5]. [Kit, pp.102-106] fails to point out the main purpose of phonons:
decoupling the equations of motion.
In the one dimensional case,
g(w)
is given by [Hoo, p.53, (2.33)]; the assumption
w =
v_{S
}k is equivalent [Hoo, p.59, l.20-22] to taking
g(w) as given by the broken line on [Hoo,
p.54, Fig. 2.11].
[Wu, p.45, (I-17)] is easier to derive and evaluate than [Man, p.160, (6.27)].
Moment of inertia [Sea1, §9-6,
§9-7].
Remark. Some earlier editions of this book use summation instead of integration.
I never like this practice. After I read [Sea1, §9-6,
§9-7], I realized that the presentation of the
6th edition using integration is much better than [Hall,
§12-5] or anything about moment of inertia
existing on the web.
The energy values of the bound states of the hydrogen atom are discrete.
[Coh, chap.VII, §C.3.c] gives a detailed
explanation. [Mer, p.266, l.-14 & l.-12]
make a few improvements.
The Doppler effect for electromagnetic waves
[Rob, p.21] discusses the Doppler effect from the viewpoint of time dilation. [Matv,
p.33, l.1-l.36] uses the tensor approach. The tensor approach automatically
shows that all the formulas on relativity are covariant with Lorentz transformations, and
effectively leads to a quick answer. It also
condenses three cases (The source is moving toward the observer, away from the observer,
or along a line normal to the line to the observer) into one
formula [Matv, p.33, (2.62)]. However, Matveev's approach is not as insightful
as Bobinson's approach. For example, it is easier to see that [Matv, p.33,
(2.65)] is a purely time dilation effect from the context of [Rob, p.21] than
that of [Matv,
p.33, l.14-l.36].
Optical properties of metals [Hec, §4.8;
Wangs, §24-3 &
§25-6; Matv, p.120, §Color of bodies,
Sec. 19 & Sec. 20] discuss the optical properties of metals. All the above books
facilitate our understanding in some aspects, but none of their discussion are
complete. We must piece together their discussions to see the entire picture.
Color of gold: For a chunk of gold, we can only see the reflected light [Wangs,
p.423, l.-2-l.-1; Hec,
p.131, l.c., l.-14-l.-11]. By [Hec.
p.129, r.c., l.-12-l.-8],
the gold appears reddish yellow. When the light source is on the other side of a
thin foil of gold, we can only see the transmitted light, so the gold appears
greenish [Wangs, p.423, l.14-l.20].
Some alkali metals are transparent to ultraviolet [Hec, p.129, l.-3-p.130,
l.c., l.19].
Remark. For the proof of w_{p}
= (Nq_{e}^{2}/e_{0}m_{e})^{1/2},
see [Wangs, p.401, (24-138)].
A metal (s =
+¥) is an extension of an dielectric (s
= 0).
The dispersion equation (compare [Hec, p.71, (3.72)] with [Hec, p.129, (4.79)]).
For plane waves, insulators and conductors are two extreme limiting cases and have corresponding discussions [Wangs, p.387, l.10-p.388,
l.-12].
Any problem in geometrical optics can be solved either using formulas or
using graphs. The latter method not only has the visual advantage, but also can
be used as a check for calculations from the former method. Example: [Jen,
pp.86-87, Example 2].
Virtual objects
Illustrated in a figure: [Hec, p.155, Fig. 5.11].
Described in words: An
object is virtual when the rays converge toward it [Hec, p.155, r.c., l.-7-l.-6].
Characterized by the object distance:
s_{o} < 0 [Hec, p.163, Table 5.2].
Virtual images
Illustrated in figures: [Hec, p.152, Fig. 5.5(c); p.155, Fig. 5.10].
Described in words:
An image is virtual when the rays diverge from it [Hec, p.155, r.c., l.-8-l.-7].
Characterized by the image distance: s_{i} < 0 [Hec,
p.163, Table 5.2].
Only after understanding the meaning of [Jen, p.55, Fig. 3J] may one understand the construction of [Hec,
p.151, Fig.5.3(a)].
Comparing the proof of [Jen, p.56, (3n)] with the proof of [Hec, p.154,
(5.8)]: Although the former proof is simpler, the latter proof is more methodological.
If m>0, the image will be virtual [Jen, p.54, l.15]. This can be seen by [Morg,
p.30, (2.11)(ii); Jen, p.49, Fig. 3E].
The focal plane of a lens [Hec, p.160, Fig. 5.17 (where the radius
of s is determined
by [p.155, (5.10)]), and Fig. 5.18].
Geometrical optics uses a lot of modeling. By comparing the arrangement of
sections in [Jen, chap. 3] and that in [Jen, chap. 4], we see that the theory of
thin lenses is parallel to the theory of refracting surfaces. The theory of
spherical mirrors is a special case of the theory of lenses [Morg, p.35, l.18].
The following three theories-thin lenses, thin-lens combinations, and thick lenses- are
parallel because they use the same parallel-ray method to form images (compare
[Jen, p.69, Fig. 4H] with [Jen, p.80, Fig. 5B(b)]; compare [Jen, p.75, Fig. 4M]
with [Jen, p.83, Fig. 5E]). Consequently, the corresponding formulas in these
three theories are the same if we properly choose the reference points. For
example, [Jen, p.84, (5k)] and [Morg, p.67, (5.24)] can be considered identical. A systematic approach to
the problems in geometrical optics entails mastering all the above patterns.
A system of lenses can be treated as a thick lens [Morg, p.67, l.-7-l.-1].
Treating mirrors as lenses
Reflection is considered as refraction [Matv, p.163, l.4-l.14].
(Sign conventions) Identify [Hec, p.184, Table 5.4] with the combination
of [Hec, p.154, Table 5.1] and [Hec, p.183, Table 5.2].
(Properties) Apply the same graphical constructions used for lenses to
mirrors (e.g., identify [Jen, p.101, Fig. 6E] with [Jen, p.63, 4D]; ray 8 in
[Jen, p.106, Fig. 6I] is constructed based on [Jen, p.47, Fig. 3C]), apply the
same formulas for lenses to mirrors, and identify [Hec, p.185, Table 5.5] with [Hec,
p.163, Table 5.3].
Thick mirrors [Jen, §6.5] can be considered as thick lenses.
Remark. [Jen, p.107, Fig. 6J] is consistent with the convention given in [Matv,
p.163, l.4-l.8], while [Jen, p.133, Fig. 8C] is consistent with the convention
given in [Hec, p.252, l.c., l.-8]. In my opinion,
Jenkins should have adopted the former convention as a standard and stuck to it.
For a detailed and systematic study of the effects of stops, see [Jen, chap. 7].
In [Hec, §6.2], the method of ray tracing applies only to paraxial rays. That is, it is used
only for the first-order approximation. Actually, in principle, the graphical method of ray
tracing [Jen, chap. 8] and the matrix method [Jen, p.143] can be exact.
[Matv. Sec. 21-Sec. 23] condense geometrical optics into 21 pages and are ready
for practical application using computers. In addition, Matveev proves every
statement that he presents in these sections. His rigorous reasoning and
ability to organize are impressive. In contrast, [Hec, chap.5 & chap. 6] use
132 pages to discuss geometrical optics and leave many statements unproved
(e.g., [Hec,
(6.1)-(6.4); (6.34); (6.36)-(6.37)]). In one place, Hecht claims he has proved
[Hec, (6.34)]. Actually, he uses unproven [Hec, (6.2)] to prove [Hec, (6.34)].
Thus, all he has done is state the formula [Hec, (6.2)] twice. For a detailed proof, see [Matv, p.167, (23.19)]. However, logic is
not the only tool to facilitate our understanding. For example, the definitions of
principle points in [Matv, p.166, l.-1] is not as
good as the definitions given in [Hec, p.243, Fig. 6.1]. The
graphical constructions given [Jen, §3.6
and §3.7] should not be deemphasized for
they have an visual advantage. Matrix methods are a useful tool only for
computer calculations. A tool is used when needed. If we use methods to
discuss topics other than computer calculations, the tool will become a burden
rather than an advantage. In view of [Fur, chap. 3], the theory of
geometrical optics are indeed made more organized and compact by the matrix
method. All the necessary information on rays is essentially contained in a single matrix.
However, the theory's formulation given in [Fur, §3.1-§3.4]
is not as well prepared for application [Fur, §3.5]
as that given in [Hec, §5.1-§5.6]
for application [Hec, §5.7].
[Hec, p.154, (5.8)] is derived from Fermat's principle, while [Fur, p.145, (3.26)] is derived from Snell's law.
Although both approaches consider a bundle of rays, the latter approach is more
natural and straightforward.
The essence of the theory of wave packets can be summarized in three stages:
Superposition of two plane waves [Born, p.19, Fig. 1.5].
Superposition of oscillations with equidistant frequencies [Matv, p.97, Fig. 53].
Group velocity: the velocity of the maximum of the wave packet [Coh, p.30, (C-31); Fig. 6].
Maxwell equations
In vacuum or microscopic fields: [Fur, p.44, l.6-l.9].
In matter (macroscopic fields)
Average over a volume that is macroscopically small but microscopically large: [Fur, p.60, l.1-l.4].
In terms of (controllable) free charges and free current densities: [Fur, p.65, l.14-l.17].
Suppose r_{f} = 0 and J_{f} = 0. In terms of
the material parameters: [Fur, p.68, l.3-l.6].
Foundations of geometrical optics [Born, chap. III]
I = |<S>| [Wangs, p.357, (21-58);
Matv, p.61, (7.12)].
Proofs of the eikonal equation: The proof given in [Born, p.112, l.1-l.9]
uses the first-order approximation, while the proof given in [Born p.112, l.16-p.113, l.2] uses
the second-order approximation. Therefore, the latter proof is more refined.
Proofs of the law of refraction The proof given in [Hec,
§4.4.1] is restricted to plane waves and
planar interfaces. The method lacks potential to be applied for
generalization. The proof given in [Hec, p.107, l.c., l.1-l.14] uses Fermat's
principle which is artificial; The proof given in [Jack,
§7.3] uses the basis of a vector space which
is also artificial. In
addition, the way the boundary conditions are used in [Jack, (7.34)] differs
from the way they are used in [Born, p.5, Fig. 1.2], which
complicates matters. The proof given in [Born, §3.2.2]
is the most natural proof because it is directly derived from the boundary
conditions. The method meets the requirement for axiomatization: any theorem in
electromagnetism should be able to be derived from Maxwell's equations. This enables us to trace the theorem to its source. In addition, the derivation of [Born, p.125, (17)] is the same as
that of [Born, p.6, (23)], which is good for unification. Furthermore, the proof
given in [Born, §3.2.2] applies to the
general case [Born, p.125, l.-11-l.-7].
It is the only proof that establishes a strong link to electromagnetism. I wonder why other textbooks leave out such an insightful proof.
[Born, p.126, (23)] represents a normal congruence; [Born, p.126, (22)] represents a normal rectilinear congruence.
For the proof of the former statement, see [Wea1, vol I,
§105]. [Sne, p.21, Theorem 5] provides a
proof of the identity given in [Wea1, vol. I, p.202, l.-12].
Remark. In the early twentieth century, the textbooks of optics discussed the
topics in differential geometry and the textbooks of differential geometry
discussed the topics in optics. Each subject solidified the other's foundation
and stimulated the other's growth. Now in the twenty-first century, optics and
differential geometry have become mutually exclusive subjects. The textbooks in
optics are devoid of questions about differential geometry and the textbooks in
differential geometry are devoid of questions about optics.
The integral given in [Jack, p.42, (1.58)] is equal to 4p [Jack,
p.42, l.18-l.19].
Proof. Ñf (r,q,f) =
(¶f/¶r)r^+r^{-1}(¶f/¶q)q^+(r sin q)^{-1}(¶f/¶f)f^
[Wangs, p.33, (1-102)].
Ñ(|r+n|^{-1})×(r/r) =
Ñ(|r+n|^{-1})×r^
= ¶(|r+n|^{-1})/¶r.
ò _{[0,+¥]} ¶(|r+n|^{-1})/¶r dr = |r+n|^{-1}
ú_{r = 0}^{¥} = -1.¬
Vibration of membranes
Tension per unit width = Constant T; Vertical deflection = w(x,y,t).
Consider the displacement of an element of area dxdy at time t.
x-direction: width = dx; left slope = tan q»q; right slope = q+dq.
q =
¶w/¶x Þ dq = [¶(¶w/¶x)/¶x]dx = (¶^{2}w/¶^{2}x)dx.
Vertical component of the tension in the x-direction: left end: -T(dy) tan » = -T(dy)q; right end:
T(dy) (q+dq).
Net vertical force from x-direction tension is T(dy)dq =
T(¶^{2}w/¶^{2}x)dxdy.
Similarly, the net vertical force from y-direction tension is T(¶^{2}w/¶^{2}y)dxdy.
Total vertical force on dxdy is T(¶^{2}w/¶^{2}x+¶^{2}w/¶^{2}y)dxdy.
Let r be the mass per unit area. Then (rdxdy)(¶^{2}w/¶^{2}t)
= T(¶^{2}w/¶^{2}x+¶^{2}w/¶^{2}y)dxdy.
Therefore, r(¶^{2}w/¶^{2}t) =
T(¶^{2}w/¶^{2}x+¶^{2}w/¶^{2}y).
The following figure is viewed along the positive y-direction: