Comparisons in Mechanics

  1. The purpose of comparison is to reveal the critical point where classical and quantum mechanics deviate from each other. It is unnecessary to memorize every detail of a new theory. All we need to know is the major corrections that the new theory makes to the old one.
    1. [Rei, p.375, (9.13.8)]
      The mean energy density between w and w +dw
      = (the mean number of photons in energy state e =Sck)´ (# of photon states per unit volume between k and k+dk)´ 2´ (the volume of spherical shell of radius k and k+dk).
    2. [Eis, p.12, l.- 11-l.- 8]
      The mean energy density between n and n +dn
      = (# of standing waves between n and n +dn )´ (the average energy per standing wave).

        Reif's factorization can only show why quantum mechanics is correct. In contrast, Eisberg's factorization not only shows why quantum is correct, but also explains why classical mechanics goes wrong [Eis, p.12, (1-17)]. In this sense, Reif fails to recognize the essential point of Planck's blackbody spectrum [Eis, p.17, (1-27)] even though his proof is logically correct. Kittel also fails to locate the critical point where the classical argument the classical argument starts to go wrong, although his factorization allows him to do so.

  2. The purpose of comparison is to identify the two different approaches in view of the big picture.
        [Mer, p.22, l.-6-p.24, l.7] and [Lan3, pp.19-21, §6] give two methods of passing to the limiting case of classical mechanics. By [Cou, vol. 1, chap. 4, §3.1], the problem of the variational calculus is equivalent to Euler's equation. In other words, the solution of the first problem is the solution of the second problem, and vice versa. For example, Hamilton's principle is equivalent to Lagrange's equation [Cou, vol. 1, §10.1] and Fermat’s principle is equivalent to Lagrange's integral invariant [Born, p.127, (1)]. The discussion in [Lan3, pp.19-21, §6] is based on the viewpoint of variational calculus, while the discussion in [Mer, p.22, l.-6-p.24, l.7] is based on Euler's equation (i.e. Schrödinger's equation [Mer p.22, (2.35) & l.-2]) derived from variational calculus. This allows us to identify (Compare the ways that [Mer, p.22, (2.36)] and [Lan3, p.20, (6.1)] are introduced) the approach in [Mer, p.22, l.-6-p.24, l.7] with that in [Lan3, pp.19-21, §6].

  3. Comparing two solutions' power series coefficients is equivalent to comparing the coefficients of the two differential equations [Mer2, p.270, l.17-l.23].

  4. Solution to the eigenvalue problem: The variational method [Mer2, pp.212,-214, §10.3] vs. the method of solving the characteristic equation [Mer2, p.209, (10.13)]
    1. [Coh, pp.132-136, Chap.II, §D.1] proves that a Hermitian operator is diagonalizabe, while [Mer2, p.210, l.1-l.-10] proves that the diagonalizability can be extended to any normal operator. This is not the whole story. For further generalization, see [Jaco, vol.2, p.192, Theorem 13; p.185, Theorem 8; p.193, Theorem 14].
    2. The advantages of the variational method.
      1. When the eigenvalues of the Hermitian operator has a lower bound, we may directly find the minimum eigenvalue.
      2. When the basis of the vector space is unknown, we may use the Rayleigh-Ritz trial functions instead [Mer2, p.139].
      3. When the dimension is large, the variational method is more effective than the method of solving the characteristic equation.
      Remark. The minimax principle in [Halm, p.181, §90] is the variational method. The discussion in [Halm, p.181, §90] fails to help us recognize the above advantages and, thereby, obscures the essential meaning of the minimax principle.
    3. The drawback of the variational method: it only applies to Hermitian operators instead of normal operators.

  5. Only through comparison can a concept's distinguishing feature be revealed.
        Only through comparing [Eis, p.457, (13-11a) & (13-11b)] with [Eis, p.454, (13-8)] can the physical meaning (i.e. the impact of a periodic potential on the wave function of a particle) of [Ashc, pp.133-134, (8.3) & (8.4)] become clear [Eis, p.457, l.16-l.26].

  6. Just as we can compare results by substituting different data into the same formula, once we determine the only differences between two parallel theories, we can design a rule which enables us to modify the formulas in one theory to directly obtain the corresponding formulas in the other theory.
    Example. The only differences: [Ashc, p.467, l.3-l.8].
    Formula 1: [Ashc, p.457, (23.20)] « [Ashc, p.467, (23.38)].
    Formula 2: [Ashc, p.454, (23.11)] « [Ashc, p.467, (23.40)].

  7. After comparing the momentum conservation law for a particle [Lan 1, p.15, l.-4-l.-3] with that for a crystal solid [Ashc, p.472, (24.6)], we would like to proceed further by explaining why they are different [Ashc, p.472, l.-18-l.-13].

  8. Intrinsic properties vs. extrinsic properties.
        In the pure case or at a high temperature the two concepts are the same. Why do we need to distinguish them? How do they become identical for the above two cases? [Ashc, p.565, Fig. 28.2] shows why and how. How does the theory of semiconductors explain their differences? See [Ashc, p.564, l.22-l.26]. The detailed calculations in [Hoo, pp.142-146] blur these insightful key ideas.

  9. It can be said that quantum statistical mechanics is more refined than classical statistical mechanics. This is not only because quantum theory is based on more fundamental principles, but also because it produces more results. In some cases, a quantum result is more refined than its counterpart in classical mechanics. In order to identify the influences of the quantum principles we must point out what results can be derived from both theories [Hua, p.256, l.-8-l.-7] and what results can be derived only through quantum theory [Hua, p.256, l.-9-l.-8; p.257, l.-6].

  10. In [Wangs], the electric field of a uniformly charged sphere and that of an isolated spherical conductor are unrelated. Compare the example in [Wangs, p.64, l.13-p.65, l.3] with the example in [Wangs, p.85, l.-3-p.86, l.3]. In contrast, Choudhury uses the same formula [Chou, p.46, (2.32)] to discuss both cases [Chou, p.44, l.-8-p.48, l.2]. Choudhury's approach saves space and avoids repetition.
    Remark. The concept of an electric field and that of a scalar potential are mathematically equivalent. Wangsness should not divide these two equivalent concepts into separate chapters (see [Wangs, chap.3 & chap.5]). He further confuses his readers by using different examples for each concept.

  11. [Man] uses statistical mechanics as the starting point to define entropy [Man, §2.1-§2.3]. Thus, the contents of [Man] are interwoven by the methods of thermodynamics and the methods of statistical mechanics. In contrast, [Zem] starts with thermodynamics and does not discuss statistical mechanics until Chapter 11. By separating the two theories, we may understand the limitations of thermodynamics and the advantages of statistical mechanics through comparison. For example, using statistical mechanics we gain the microscopic viewpoint of entropy [Zem, (11-15) & (11-27)(i)]and obtain the formula for s0 [Zem, (8-5) & (11-29)].

  12. [Zem, §12-2] discusses Einstein approximation and Debye approximation in the same section and puts illustrations of the approximations side by side [Zem, p.308, Fig. 12-2]. This design makes it easy to compare their key ideas, identify their differences, and see how close they are to reality.

  13. (Photons)
        Most books discuss electrons in one chapter and discuss photons in another chapter, and make only rough comparisons between the two. In contrast, [Zem, §17-11] discusses photons using the electron model because its goal is unification. Zemansky first recognizes the intrinsic differences between photons and electrons [Zem, p.442, l.-7-p.443, l.21], and then makes necessary modifications [Zem, p.444, l.4 & l.10] as he develops his formula for photons.

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