Rolling, sliding, holonomic constraints, and non-holonomic constraints. Landau
confines
the terms rolling, sliding, holonomic constraints and non-holonomic constraints to the
topic of rigid bodies in contact [Lan1, pp.122-124]. Thus whenever we see
these terminologies, we understand what context we are talking about.
The main result of [Go2, pp.1-64, chap. 1 & chap. 2] is the derivation of
Lagrange's equations. The emphasis should have been the perturbation theory of
differential equations. However, in [Go2, chap.1, p.12, l.18] Goldstein
introduces
the concept of non-holonomic constraints which should have been systematically
discussed under the topic of rigid bodies in contact. To improperly throw in a
side topic is a diversion rather than a precaution. To let a concept
belonging to a specific topic appear everywhere is tamtamount to opening Pandora's box.
The result is overwhelming and confusing for readers.
Complex amplitude [Lan1, p.59, l.-7].
Azimuthal and magnet quantum numbers [Eis, p.240, l.-15-l.-11]. The azimuthal quantum number l
and magnet quantum numbers m_{l} result from applying
separation of variables to [Eis, p.237, (7-15), (7-16) & (7-17)]. Then we
relate them to angular momentum and its z-component [Eis, p.253, l.-15-l.-14]. In contrast, Cohen defines l(l+1)ħ
as the eigenvalue of the abstract operator J^{2} [Coh, p.648,
l.14-l.18] and m_{l}ħ_{ }as the
eigenvalue of J_{z} [Coh, p.648, l.-9].
The J^{2} and J_{z} in [Coh, p.648] are similar to [Eis, p.
237, (7-16) & (7-15)] which are the partial results of separation of
variables. Thus Cohen's definition of azimuthal and magnetic quantum numbers
makes it difficult to see the roles that l and m_{l}
play in the Schrödinger equation [Eis, p.236, (7-12) & (7-13)].
Quantization rules. For the theoretic origin of a phenomenon, we
must trace back to its most fundamental cause. Example. For the origin of quantization rules, Planck traces back to a
reason that can explain the discreteness of energy of any simple harmonic
oscillator [Eis, p.20, l.17]. Sommerfeld traces back to a reason that can
explain the
discreteness of angular momentum of any periodic system [Eis, p.165, l.5].
Schrödinger traces further back to the most fundamental reason that can
explain the mysterious requirement [Eis, p.163, l.-6] of integralness for the general case.
(Speed of light, time-space and relativity) We try hard to detach
ourselves from
our senses to observe the universe objectively, but eventually we find
ourselves studying nothing but the restrictions and effects of
human senses. [Rob, p.11, l.20-l.22] says that if the speed of light were infinite,
then time and space would be independent. That is, if human beings had
the sixth sense that is able to detect a signal instantly, then the Galilean
relativity would be the correct type of relativity for human beings.
General axes for the Lorentz transformation [Jack, p.525, (11.19) &
l.−3-l.−1; Rob, p.14, (2.17c) & (2.17d)]. [Rob, p.13, l.11-p.14, l.12] gives a
good reason why r and r' should be separated into their components parallel
and perpendicular to the relative velocity between the two frames.
The original concepts in quantum mechanics.
States:
Classical: A point in [Rei, p.50, Fig. 2.1.1]; Eulerian method [Kara,
p.141, l.-6].
Quantum mechanical: A cell in [Rei, p.50, Fig. 2.1.2] whose volume is
equal to Planck's constant [Rei, p.51, l.14; p.119, l.-16; p.358, l.14].
Remark. The microscopic state of a particle refers to the particle's alterable properties [Hec, p.53,
l.c., l.19-l.27].
The uncertainty principle: The act of observing an electron
disturbs the microscopic system [Eis, p.66, l.−9].
Wave function (Einstein's interpretation of the radiation intensity as
the probability measure of photon density [Eis, p.63, l.−4]; Bohr's analogy
between the wave function and the electric field [Eis, p.64, l.3-l.25]).
Its
probabilistic nature [Eis, p.139, l.-8-l.-3]: The propagation of a wave function is deterministic if the
initial value of the wave function is given. However, we can only
determine the initial probability density rather than initial value
of wave function.
Hyperbolic sine and cosine functions (Compare [Lan1, p.38, (15.12)] with
[Lan1, p.38, (15.10) & (15.11)]).
Angle variable [Go2, p.461, l.−9-l.−7].
The physical origin of [Ru2, p.96, Theorem 4.25] is completely lost. We
must retrieve it from [Hall, p.302, l.10-l.11 & Fig. 17-9].
Origins are rich in connections.
[Go2, p.156, (4-73) & (4-74)] provides the origin of the matrix S_{z} ,
while [Coh, p.393, (4-15)] does not. Only after position vectors are embedded
[Go2, p.150, (4-59)] in the group of the unitary matrices, may we define
observables and wave functions, and understand the wave nature of the motion of a
particle.
Although the sections in [Ru2, pp.195-200, the Inversion Theorem]
constitute a self-contained logical system, an important message is missing:
How do we select auxiliary functions? [Ru2, pp.197-198,
§9.7]. In view of the variational method [Coh, p.1148, E_{XI}], if we
fail to mention Hamiltonian and eigenstates, it would be very difficult to
understand how we select trial functions [Coh, pp.1151-1153,
§2a & §2b].
A wave function is a solution of the Schr`dinger equation. Since the Schr`dinger equation characterizes the equation of motion, it follows that
the wave function contains all the information about the state of the system.
[Mer2, p.9, l.- 14] only lists some physical
consequences of a wave function, but fails to emphasize its mathematical
origin. Therefore, Merzbacher’s argument is not strong enough to convince us
that the wave function contains the complete information of the system
[Mer2, p.28, l.-16].
An observable must be Hermitian [Coh, p.137, l.-4]:
[Mer2, p.193, l.9-l.11] or [Lev2, p.164, l.7-l.13].
The periodic boundary condition [Ashc, p.33, l.1-l.21].
Only by facing the difficulty of solving the problem of
contradiction between experiments and the old theory may we recognize the value of a new principle.
Unless we were to be without water or electricity for a week, we would never understand their importance.
Similarly, unless we trace back to the time when physicists faced the
difficulty of solving the problem of the contradiction with all the tools
available then, we will never appreciate the value of Pauli's exclusion principle. [Iba,
p.115, l.13-l.25] provides such a historical story, while [Hoo, pp.81-84,
§3.2.3] does not.
It would be difficult to understand the true meanings of essential and
accidental degeneracy [Coh, p.784, l.4 & l.9] unless one understands that they
originate from group theory [Tin, p.34, l.22 & l.28].
In [Pee, p.229, l.11], Peebles claims that the origin of the terms "covariant" and
"contravariant" is lost in the mists of time. However,
in [Lan2, p.230, l.9], Landau gives some explanation about their origin. Thus,
origin is something we have to make an effort to search for with imagination
rather than irresponsibly neglect.
The mathematical definition of the pitch of the winding in [Wangs, p.231, l.2]
can be found in [Lau, p.2,l.-10-l.-9].
Inertial frames [Ber, p.31, l.-17-l.-16].
Origins vs. Viewpoints. If our basis is physical, then the origin of [Hoo, p.131, (5.1) & (5.2)] is E
= p^{2}/2m. In contrast, if our basis is
mathematical, then the origin of [Ashc, p.568, (28.2)] is the Taylor series
expansion at a minimum.
The origin provides a study guide because it contains rich and concrete resources.
When we study potential barriers [Coh, p.72, (28-a); p.73, (28-b)], we use the
theory of reflection and refraction of plane waves [Wangs, p.420, (25-67); p.421, (25-71)]] as a guide.
When we study radiation pressure, we use the theory of pressure of an ideal gas as the guide
(compare [Wangs, p.426, Fig. 25-19] with [Rei, p.280, Fig. 7.13.2]).
The dual field-strength tensor [Jack, p.556, l.-4-l.-2].
One may wonder why the relation between frequency and wavenumber is called
the dispersion relation [Hoo, p.38, l.6-l.7]. The following might give a clue
to the nomenclature. The word
"dispersion" is used to describe the fact that the absolute index of refraction is frequency-dependent [Hec,
p.66, r.c., l.15-l.16].