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Theory and Its Application in Mechanics

  1. Classical mechanics is a special case of quantum mechanics. Their common features [Hall, p.798, (43-1), (43-2)] make the two theories compatible.
    1. We introduce new axioms when classical mechanics fails.
      I. Adding new blood.
      (i). Failures at a basic level because angular momentum needs to be quantized [Hall, p.790, l.8-l.11].
      (ii). Failures at a more refined level before we bring in Schrödinger equation (See quantized energy [Hal, p.804, Example 2] & probability distribution [Hall, p.805, Fig. 43-8]).
      II. Switching between dual viewpoints.
      (i). Wave property leads to the uncertainty principle [Hall, p.809, (43-18)].
    2. The scope [Coh, p.9, l.-11] for an effective application.
          For large-scale oscillators [Hall, p.778, Example 1], the quantum numbers are enormous and the quantized nature of energy will not be apparent. For effective application, the quantum effect is negligible.
    3. Correspondence principle.
          The development of a new theory is guided by the old one.

  2. Theory versus reality.
    (1). We ignore the difference between theory and reality.
    Example. Ideal gas.
    (2). We isolate the exceptions and make necessary repairs.
    Example. Orthonormal bases versus cylindrical (or spherical) coordinates [Kap, p.200, l.11-p.201, l.-7].
    (3). The theorem concerning change of variables has various forms. The standard form [Kap, p.350, l.4] is [Kap, p.349, Theorem III] or [Ru1, p.206, Theorem 9.32]. However, this special form is not good enough to deal with various situations.
    (A). Compare the assumption of [Kap, p.346, Theorem I] with that of [Kap, p.349, Theorem III]: See [Kap, p.346, l.-3-p.347,l.2].
    (B). Compare the assumption of [Kap, p.348, Theorem II] with that of [Kap, p.346, Theorem I]: The degree of the mapping of Cu v into Cx y is not necessarily equal to ±1[Kap, p.348, l.15].
    (C). Possible generalizations for [Kap, p.348, Theorem II]:
    (i). From a simple closed curve to a set of curves [Kap, p.349, l.2].
    (ii). From [Kap, p.346, b), c)] to [Kap, p.349, b'), c')].
    (D). The generalizations in (B) & (C) may involve special topics like algebraic topology or measure theory.
    (4). To choose a starting point for the axiomatic approach, the practical side of a definition is not necessarily as good as its theoretical side. Example: Tensor (Compare [Kara, p.186, l.-11] with [Coh, p.154, l.17]).

  3. The development of a complicated theory is often guided by a simple idea.
        To concretely represent the restricted Lorentz transformations [Go2, p.296, (7-65)], we use two-way approach. That is, we approach our central goal from both directions: In alternating steps, we gradually make a rotation more like a pure Lorentz transformation, and make a pure Lorentz transformation more like a rotation. Although the entire process is complicated, the guiding principle is quite simple: A 4-dimensional pure Lorentz transformation is like a 3-dimensional space rotation.

  4. Ideal vs. reality
    Example 1. Theory vs. application.
    Example 2. Model vs. Experiment.
    Example 1 vs. Example 2.
    1. Modeling vs. (Going from practical construction to hypothetical theory) [Wan3, p.114].
    2. Realization [Win, p.52, Fig.2.15®Fig.2.16] vs. (Embodiment of a general theorem) [Wan3, p.115].

  5. How a theory and its application are separated from each other
        [Pat, p.119, the second equality of (16)] should have pointed out which theorem it is based upon; see [Coh, p.158, (F-24]. [Coh, pp.157-158, II.F.§a.α] should have given practical examples. For example, the energy and particle number of a system can be simultaneously measured, so the Hamiltonian operator and the number operator [Pat,p.119, l.13] commute.

  6. The legitimacy of assumptions.
        The advantage of Jackson's division of E into two cases [Jack, p.305, l.7 & p.306, l.12] can only be seen from hindsight. To prove the legitimacy of this formal classification, we must resort to the argument in [Born, p.30, l.!2].

  7. The Development of the Special Theory of Relativity
    1. Goal: Maxwell's equations should take the same form in all inertial frames [Bow, p.10, l.13].

    2. The insufficiency of the Galilean transformation [Bow, p.4, l.-5].

    3. Main result: The Lorentz transformations are reciprocal and are themselves covariant [Bow, p.15, l.-14].

  8. As technology advances, we must jettison some old concepts along the way
    1. because old models are incorrect [Sym, p.56, l.-4-p.57, l.3] or
    2. because the old tools are unnecessarily inconvenient and complicated [Rob, p.3, l.14-l.22].

  9. If we fail to understand the underlying theory (Tensors: [Haw,p.125, §8-3]), we must memorize some strange rules [Lan2, p.16, l.5-l.6]. However, the rule in [Lan2, p.16, l.5-l.6] does give an effective derivation of [Lan2, p.17, (6.9)] (Compare the derivation with the method of contraction [Lan2, p.18, l.-4]).

  10. The development of quantum mechanics
    1. From quantized energy to the dual nature of radiation

      Classical regime Quantum regime Explanation for quantum effects
      Black-body radiation The Rayleigh-Jeans law [Man, p.253, (10.21)]. However, experimental results contradict the Rayleigh-Jeans prediction [Eis, p.13, Fig. 1-8]. Wien's law [Man, p.254, (10.23)]. Planck's radiation formula [Man, p.250, (10.14)].
      1. Planck's derivation
        1. Using interpolation to directly obtain the form of the law [Man, p.363, l.-6-p.364, l.4].
        2. Using the wave picture [Eis, pp.17-18, Example 1-4; Man, p.251, l.3-l.13]
          New idea: E = nhn [Eis, p.17, l.15]; theoretical basis of the new idea: In [Eis, p.16, Fig. 1-10], discrete energies E= nhn successfully explain the discrepancy between [Eis, p.13, (1-18)] and [Eis, p.13, (1-19)].
      2. Einstein's derivation using the particle picture [Man, p.240, l.-9-p.251, l.1]
        New idea: E=hn [Man, p.249, (10.9); Wu, p.35, l.1]; theoretical basis of the new idea: [Eis, p.29, l.-4-p.31, l.15].
      The wave-particle duality The Rayleigh-Jeans law [Wu, p.41, l.2]. Wien's law [Wu, p.41, l.1]. Considering the energy fluctuations of Planck's law, Einstein derives [Wu, p.40, (1-13)] which is the sum of two mean-squares of fluctuations: one from the wave property and the other from the particle property. Thus, Planck's law reveals the simultaneous presence of both the wave and particle properties of radiation. Based on [Lan2, p.79, l.10; p.80, (32.15)], Planck proposes [Wu, p.41, (1.14)(ii)]. See [Wu, p.41, l.11-l.15]. [Wu, p.41, (1-14)] directly relates a electromagnetic wave to to its particle properties [Wu, p.41, l.17-l.19].
      The photoelectric effect 1. The maximum kinetic energy of a photoelectron is independent of the intensity of illumination.

      2. Existence of a cutoff frequency.

      3. The absence of a time lag.

      		 E=hn [Eis, p.30, l.- 7-p.31, l.15].

      K= hn - w [Eis, p.30, (2-3)].
      The Compton effect l ® ¥ Þ Rayleigh scattering dominates [Eis, p.38, l.-4]. l ® 0Þ Compton scattering dominates [Eis, p.39, l.3]. E=hn [Eis, p.34, l.27-p.35, l.6].
      X-ray production h® 0 Þ l min® 0 [Eis, p.42, l.-15]. Existence of a minimum wavelength [Eis, p.41, Fig. 2-10]. E=hn [Eis, p.41, l.- 3-p.42, l.10].
      Pair production hn ³ 2m0c2 [Eis, p.44, l.8-l.11]. [Eis, p.47, (2-17) & Fig. 2-15].
      Simple harmonic oscillators [Eis, p.138, l.2 & p.165, Fig. 5-19]]. [Eis, p.137, l.-19]. The time-independent Schroedinger equation only allows certain energy values [Eis, p.161, Fig. 5-14].

      Remark.
      1. Mechanics is in the classical regime
        Û h is comparatively small
         Û The quantum numbers specifying the state of the system are large [Eis, p.117, l.21-l.27]
        Û The energy function is continuous [Coh, p.494, (B-34)].
        That is, when h® 0 or n® ¥ , quantum formulas are reduced to their classical limits.
        Note. The quantum regime agrees with its classical limit only in the sense of average behavior because the small-distance fluctuation cannot be detected experimentally.

      2. Mechanics is in the quantum regime Û Mechanics fails in the classical regime.

      3. Why quantum effects cannot be detected in the classical regime:
        1. Quantized energy [Eis, p.21, l.18].
        2. The Compton shift [Eis, p.40,l.8-l.10].
      4. The expressions E=hn and p=h/l are used to explain the Compton effect. These expressions reveal the dual nature of electromagnetic radiation [Eis, p.40, l.26-31].
    2. From the particle-wave duality to probabilistic measurement.
      1. Small scale measurement requires the concept of probability.
            de Broglie's theory of a matter wave [Eis, p.56, (3-1a) &(3-1b)] ® Experimental confirmation [Diffraction: Eis, p.57, l.-9] ® The uncertainty principle [Diffraction: Eis, p.67, Fig. 3-6] ® small-scale measurement [Eis, p.78, l.12-l.8].

      2. The wave function gives the quantitative expression of probability.
            Einstein's interpretation of the intensity of radiation as a probability measure of photon density (Using the duality relation [Eis, p.63, l.-4]) ® Bohr's analogy between the wave function and the electric field [Eis, p.64, l.3-l.25].
    Remark. [Dit, p.15, Fig. 1.6] gives a simple picture of the development of quantum mechanics. Especially useful is the fact that it indicates the factor that triggered each new stage of development. Quantum mechanics serves to make the concept of waves and that of particles compatible. The crucial difference between waves and particles lies in the way the energy is transported [Dit, p.5, §1.10]. Therefore, quantum mechanics uses Schröedinger's equation which features the Hamiltonian to solve the problem of the contradiction between particles and waves.

  11. The convenience of a mathematical device for application is determined by its flexibility [Lan5, p.109, l.10].

  12. Application (Using notation as a device).
        The visual display of a series of matrix multiplications is cumbersome. Dirac's ket and bra operations [Coh, chap. II.B] reduce it to a simple expression. The trick of inserting 3i *ui><ui* is nothing but inserting the identity matrix. Dirac's notation functions amazingly well because of the associtivity of matrix multiplication. For a fixed orthonormal basis *ui>, Dirac's device makes it easier to interpret the identity operator as 3i *ui><ui*, which serves many purposes in application. Notably, the device inspires us to predict some results before we actually prove them [Coh, p.123, (C-11) & p.148, (E-15)].

  13. Experiments motivate us to develop a new theory [Eis, p.103, l.9] and a new theory stimulates new experiments that help show its validity [Eis, p.103, l.-18-l.-15].

  14. The empirical equation for photoelectric effects [Eis, p.30, (2-4)] confirms Planck's conjecture [Eis, p.31, (2-2)].

  15. Bose-Einstein's theory of photon gas puts the cavity radiation formula [Eis, p.17, (1-27)] on a more solid foundation [Eis, p.34, l.1-l.3 & l.7-l.8].

  16. Some physical quantities cannot be directly measured by experiment. We have to determine their values by using formulas and the other physical quantities whose values can be directly measured [Lev2, p.133, l.13-l.15].

  17. A theory's clarification and refinement requires hands-on experiences (e.g. hydrogen-like orbitals [Lev2, §6.8]).
        Physicists treat hydrogen-like atoms as the theoretical expansion of the hydrogen atom. Chemists have to treat them as unique elements. Consequently, they must confirm the theory by experiments.
        Pictures should not be used as a show, an advertisement, or a piece of art. The discussion in [Coh, Complement EVII, §2] resembles superficial travel notes. For example, it fails to describe how to construct the figures [Lev2, p.150, l.-16-p.152, l.-1]. In contrast, the deeper discussion in [Lev2, §6.8] is more like a serious laboratory report.

  18. Perturbation theory does not automatically teach us how it applies to the helium-atom. First, even to figure out the unperturbed Hamiltonian for the helium-atom requires an ingenious device: treat the helium atom as the sum of two hydrogen-like atoms [Lev2, p.252, l.-3-p.253, l.6]. Second, in order to reduce calculations, we must find effective cancellations [Lev2, p.254, l.-7-l.-1] when evaluating the integral.

  19. Merzbacher fails to stress important practical applications of [Mer2, p.175, (8.148)]. See [Ashc, p.141, l.15-l.20].

  20. There is no perfect theory for every problem. When solving a problem, we must continue to search for a new device as we proceed.
        Both the cases in [Cou, vol.1, p.291, l.-6-p.292, l.2] and the cases in [Cou, vol.1, p.295, l.2-p.296, l.4] belong to the category of the boundary conditions [Bir, p.256, (2)] of a sturm-Liouville system. Thus, from the viewpoint of Hilbert space [Bir, p.313, Theorem 11], all the above boundary conditions are unified. However, the matching conditions for a single motif and those for a periodic well must be discussed separately (Compare [Coh, pp.352-355, Complement MIII] with [Coh, pp.369-370, §b]) because the wave function of a periodic potential is not a bound state (see [Coh, Complement NIII]).

  21. (The trend of a theory's development)
        To facilitate computer calculations, graphical methods [Jen, p.131, §8.2] for ray tracing have been replaced by matrix methods [Hec, pp.248-253].
    Input: lens parameters (thickness, index, and radii) [Hec, p.249, r.c., l.8].
    Output: the cardinal points [Hec, p.249, r.c., l.9; p.250, (6.36) & (6.37)].
    Example: [Hec, p.251, Fig. 6.10].

  22. Mathematical arguments pave the way for experimental investigations.
    1. Ampere's circuit law including the term for displacement current [Sad, p.382, l.1-l.8].

  23. [Sad, p.457, (10.123)] is theoretically possible, but it is rarely used in practice.

  24. A theory's development (induction).
    1. Distill important feature from careful observations [Hall, p.575, l.-10-l.-8; p.576, l.6-l.9].
    2. Make brave guesses [Hall, p.576, l.11-l.13].
    3. Preserve the important features in further refined experiment [Hall, p.579, l.20-l.23].

  25. When a theory and experiments disagree, we must refine the approximation in our theoretical model [Rei, p.434, footnote & Fig. 10.7.3].

  26. Technology helps overcome the difficulties encountered on both the theoretical side [Cra, p.24, l.-14-l.-10] and the experimental side [Cra, p.24, l.20-23] in determining the Fermi surface.

  27. Roads to the fundamentals.
    1. From cohesive energy to the band structure: cohesive-energy calculations provide the earliest motivations for accurate band structure calculations [Ashc, p.409, footnote 17]. Later, the fundamental role  played by the band structure in explaining physical properties establishes it as important on its own.
    2. The classification of solids formerly relied on the nature of cohesion. Now we choose to emphasize the spatial electronic arrangement in describing the categories of solids [Ashc, p.374, l.-10; p.396, l.15-l.17] because bonding is only one of many properties strongly affected by this spatial distribution [Ashc, p.379, l.5-l.6].

  28. (Electromagnetic momentum)
        [Wangs, p.361, (21-73)] is equivalent to [Lan2, p.82, (33.6)]. To understand the inner structure of electromagnetic momentum, we have to follow Landau's path: Action [Lan2, (27.4)] ® the equations of motion [Lan2, (32.4)] ® the conservation of the 4-momentum [Lan2, (32.6)] ® the physical interpretation of the energy-momentum tensor (an arbitrary field: [Lan2, (32.15) & p.80, l.19-l.20]; the electromagnetic field: [Lan2, p.81, l.-7-p.82, l.3]).
    Remark. Landau's Deductive approach based on formalism helps us directly (i.e., effectively in the logical sense) toward our goal. However, if we reverse the process and try to synthesize the fundamentals based on intuition, we often have to discuss other unrelated material to show patterns, generalize results, or search for fundamentals, etc.

  29. When we develop the classical theory of the ideal gas, we have to make sure that our argument does not contradict quantum mechanics [Hua, p.56, l.3-l.20] and the assumptions of our classical model should be compatible with the quantum effects [Hua, l.-17-l.-11].

  30. If an author's basic concept is incorrect, he will leave a series of errors in his theory. This is because an incorrect premise will not lead to a correct answer unless he commits another error.
    Example. In [Reic, p.186, (2)], J should be replaced by ïJ ï (see [Jack, p.120, l.-2]). Because of  this error, when Reichl tries to prove dx1…dxn=da1…dan in [Reic, p.187, l.-9], he has to make another incorrect assumption: det(Ō)=1 [Reic, p.187, l.-10]. Actually, det(Ō) can be -1.

  31. How a starting point affects the quality of a theory.
    Example. Using the concept of collisions rather than the relaxation time as the starting point to approximate the transport theory [Reif, p.516, l.-14-l.-9] facilitates estimations of the errors committed.

  32. A theory is not a collection of experimental results [Wangs, p.218, (13-1)]. We have to organize them in a logical manner. For example, [Jack, p,177, (5.8)] can be derived from [Jack, p.175, (5.4)] and [Jack, p.177, (5.7)].

  33. In the preface of [Zem], Zemansky considers thermodynamics the root of thermal physics. His outstanding opinion makes great contrast to those of the contemporary physicists who consider quantum mechanics the root of every branch of physics.

  34. A theory's starting point should be based on a familiar model.
        The theory of magnetostatics in [Chou, chap. 5] starts from Maxwell's equations [Chou, p.191, (1.51a,b)]. Due to the similarity between [Jack, p.25, (1.3)] and [Jack, p.175, (5.4)], the theory of magnetostatics in [Jack, chap. 5] starts from [Jack, p.175, (5.4)] by considering a current element. Like [Jack, chap.5], the theory of magnetic fields in [Cor] starts from [Cor, p.328, (18-5)] by considering an entire circuit.

  35. (Maxwell's equations) A brave guess vs. its theoretical base
        A brave guess is usually inspired by inconsistency of the forms of equations [1]. The procedure of the guess is simple. However, developing a theory to prove the validity of the guess can be very complicated. Maxwell was convinced that the displacement current must exist from the viewpoint of boundary conditions [Wangs, p.352, l.11-l.13]. Otherwise, we will have a case in which the tangential components of H will not be continuous across a boundary, contradictory to boundary conditions. On the other hand, if we assume the existence of the displacement current, the contradiction will not occur [Wangs, p.353, l.5]. In [Chou, p.293, l.-10-p.295, l.-9], Choudhury shows that the effect of the displacement current is the same as that of the conduction current from the viewpoint of magnetic induction. In [Hall, §37-3], Halliday shows that in the case of [Hall, p.652, Fig. 37-1 (b)], the displacement is equal to the conduction current. Planck's radiation law is another example [Man, p.363, Problem 10.3; p.250, (10.14)].

  36. Experimental confirmation
        [Matv, p.48, §Experimental proof of the electromagnetic nature of light] The tone of this section makes it seem that Matveev is implying that without Wiener's experiment there would not be sufficient evidence to establish the electromagnetic theory. In fact, the correct attitude should be that Wiener's experiment confirms the predictions of the electromagnetic theory. In the process of the experimental confirmation, we also gain some new results [Matv, p.48, text, l.-4-l.-2].

  37. (Theory vs. application) A theory [Wangs, Appendix B: Electromagnetic properties of matter] is like a tool room. It is difficult to know a tool's functions until we utilize it to understand its roles in applications [Hec, §3.5 Light in bulk matter]. An application helps identify the tool's location, i.e., which book or page contains the required solution. When an application lacks a clear explanation [Hec, p.71, (3.73)], the theory enters to fill the gap [Wangs, p.565, (B-83)]. The first equality of [Hec, p.50, (3.44)] is a consequence of [Matv, p.61, (7.12)] and [Wangs, p.357, l.-5].

  38. Convention of signs.
    1. The convention of signs must cover all the cases. They must be organized in one place, e.g., at the beginning of the theory [Jen, p.50, l.4-l.14].  To facilitate easy reference, the sign conventions should not be spread throughout the theory.
      Remark. The meaning of each notation in [Hec, p.154, Table 5.1] can be found in [Hec, p.163, Fig. 5.22].
    2. No matter how complicated a definition is [Hec, p.155, l.c., l.-4], we must trace to its root [Hec, p.155, l.c., l.-3], and give a proper sign based on the convention.
      Remark. fo (resp. fi) is a special case of so (resp. si) [Hec, p.155, l.c. l.-3; r.c., l.4]. [Hec, p.163, Table 5.2, row 3] gives a more effective method of determining the sign of a focal length.  This method can be justified by [Hec, p.155, (5.9) & (5.10); p.159, l.c., l.-6-r.c., l.2].
    3. No matter how deep we go into the theory, we have to make sure that the sign of every physical quantity complies with the convention [Morg, p.60, l.-11-l.-5].
      Remark. Why does Furtak use the symbol ¬ to represent the positive distance D, the symbol ® to represent the positive distance D' in [Fur, p.158, Fig. 3.23], and use the symbol « to represent distance Dl in [Fur, p.156, Fig. 3.21]? He fails to give a good explanation in [Fur, pp. 155-160]. In order to give a clear explanation we must use the concept of object space and that of image space given in [Jen, p.70, §4.11]. Symbol « is a combination of  symbol ® and the symbol ¬. Symbol ® representing Dl in [Fur, p.156, Fig. 3.21] refers to the object distance from the second refracting surface, while symbol ¬ representing Dl in [Fur, p.156, Fig. 3.21] refers to the image distance from the first refracting surface. According to the convention of signs given in [Fur, p.146, l.1-l.2], both symbols ® and ¬ for Dl are positive.
    4. The convention is a systematic design which automatically prevents twisting of a symbol's meaning out of context. In [Morg, p.30, (2.11)], n refers to the medium for the incident light and n' refers to the medium for the transmitted light. If one mistakenly interprets n in [Morg, p.30, (2.11)] as the refractive index of the medium where the virtual object is located [Jen, p.80, Fig. 5B], then one may mistakenly put n of [Morg, p.62, l.-3] in the denominator rather than the numerator. If one does so, one will violates the convention that the light is incident from the left.

  39. Due to the similarity between TE waves and TM waves, Born provides an exchange rule [Born, p.52, l.17-l.18] to obtain the equations for TM waves from the corresponding equations for TE waves. In my opinion, one should not apply this rule unless one's concepts are absolutely clear. This rule gives a shortcut  for experts rather than beginners. It is better to derive the equations for TM waves step-by-step by following the arguments that prove the corresponding equations for TE waves. If we must use this exchange rule, we should apply this rule expediently. Suppose we want to obtain the equations for TM waves from the corresponding equations for TE waves. If the equation is located in [Born, p.52, l.22-l.28], we should replace e by -m. If the equation involves U, V, and W as defined in [Born, p.54, (17)-(19)], we should interchange e and m. Then we may ask why not just replace e by m. Finally, if we want to obtain [Born, p.60, (51)]'s counterpart for TM waves, should we replace the factor pl /p1 in [Born, p.60, (51)] by ql /q1? The answer is no.

  40. Links {1}.