Organizing and Fine Tuning in Mechanics

  1. To show that System A reduces to System B under a certain condition, all we need to do is to point out that the key component in System A reduces to its counterpart in System B under the above condition.
        When kT>>ħω, the partition function of the quantum-mechanical system [Pat, p.76, (15)] reduces to its classical counterpart [Pat, p.75, (3)]. It is unnecessary to show that μ, P, S, U, CP, and CV in a quantum-mechanical system reduce to their classical counterparts by computing their limits when kT>>ħω. This is because the limiting process and differentiation can be performed in either order. It is wrong to mislead the readers with the impression that the cumbersome calculations in [Pat, p.77, l.−13] have any significance.

  2. Let f (E)=e−βEg (E) [Pat, p.71, l.4] and B={E** F (E) is stationary at E=E*}. Pathria claims that the set B has a unique element [Pat, p.71, (6)] before he actually describe the shape of the function f [Pat, p.71, (8)]. He should prove that f has the bell shape before he jumps to the conclusion.

  3. Logical rigor
        It is appropriate for a squid to release black dye when it is in danger. However, when a physicist constructs a proof and is not absolutely sure about its correctness, he should not publish it. This is because a seemingly plausible argument may lead his readers into darkness. In other words, he should check his strategy and reasoning carefully.
        The argument in [Pat, p.131, l.4-l.13] is incorrect because Pathria mistakenly manages to manipulate energy [Pat, p.130, l.2], coordinates [Pat, p.130, l.−1], and momenta [Pat, p.131, l.7]. He should have manipulated the physical kets [Coh, p.1386, the symmetrization postulate] of a system of identical particles instead. Using the closure relation [Coh, p.137, (D-35)] and the definitions in [Coh, p.1397, (D-3); p.1398, (D-9)], we can derive the right-hand side of [Pat, p.131, (12)] directly from the left-hand side of [Pat, p.131, (3)] without going through [Pat, p.131, l.2-p.132, l.15].

  4. (General theory vs. special cases)
        If we can figure out a formula for the general case, it is unnecessary to derive the same formula for the special case. All we need to do is substitute the data into the formula for the general case.
        Based on the definitions in [Pat, §6.1 & §6.2], a canonical ensemble can be generalized to a microcanonical ensemble by stretching each energy level to a band [Pat, p.137, Fig. 6.1] and can also be generalized to a grand canonical ensemble by allowing particle exchange (compare [Pat, p.53, (1)] with [Pat, p.100, (1)]). [Pat, p.143, (22)] is derived from the properties of a grand canonical ensemble while [Pat, p.139, (18a)] is derived from the properties of a microcanonical ensemble. These two formulas derived from different ensembles produce the same result for a canonical ensemble because an ensemble which is simultaneously a grand canonical ensemble and a microcanonical ensemble is a canonical ensemble. By the same token, [Pat, p.145, (5)] surely agrees with [Pat, (5.5.20)] because a canonical ensemble is a special case of a grand canonical ensemble.

  5. (Timeliness)
        The physical meaning of α and β in [Pat, p.140, (21)] should have been introduced immediately after [Pat, p.138, (14)] so that the reader would have a clue to recognize their identities using [Kit, p.41, (26) & p.132, (35)]. Otherwise, we will lose the important timely clue for their identification. By the way, it is a good idea to preserve the subscripts of a partial derivative (see [Kit, p.41, (26) & p.132, (35)]) because they help track which variables are fixed when there are many variables involved.

  6. Considering the simplicity of method and clarity of formulation, [Bow, p.24, l.−2-p.25, l.4] is better than [Jack, p.529, l.−8-p.530, l.11].

  7. Although [Eis, chap.7 & chap. 8] contains a good theoretic development of one-electron atoms, Eisberg still makes several conceptual mistakes.

    1. In [Eis, p.281, l.- 10], Eisberg mistakes the constant of motion [Coh, p.247, (D-58)] for an eigenvalue of L.
      Remark 1. In terms of quantum mechanics, a Hermitian operator has a definite eigenvalue in an eigenstate. In contrast, in terms of classical mechanics, a constant of motion does not change in time. Although the constant of motion sometimes may help predict whether an operation has an eigenvalue, the prediction is not as reliable as one may think [Eis, p.258, l.23]. A good introduction to the eigenvalues and the eigenfunctions of L2 can be found in [Coh, p.662, (D-8-a)].
      Remark 2. Cohen-Tannoudji should have related the 2nd equality of [Coh, p.247, (D-58)] to the simultaneous measurement of A and H.

    2. In [Eis, p.281, l.- 7], Eisberg mistakes the expectation value for an eigenvalue of Lz.
      Remark. The measurement of a dynamic quantity in an eigenstate should have a definite value. Furthermore, unless we define eigenfunctions first, the probability has no meaning. Although the probability concept plays an important role in quantum mechanics, the concept should be applied only when necessary. A good introduction to the eigenvalues and the eigenfunctions of Lz can be found in [Coh, p.662, (D-8-b)].

    3. [Eis, p.258, l.25] says that , due to the uncertainty principle, no two components of angular momentum can be simultaneously measured with complete precision. The mathematical meaning of a simultaneous measurement is equivalent to diagonalizing the two matrices simultaneously. Therefore, the above fact is due to [Coh, p.140, Theorem III & p.142, Comment (ii)] rather than the uncertainty principle.

  8. Eisberg's explanation of p(E1)p(E2)=q(E1+E2) in [Eis, p.C-3, l.-11-p.c-4, l.4] is not clear. A better explanation can be found in [Kit, p.61, (9)].

  9. Eisberg should have used entropy instead of detailed balancing [Eis, p.381, (11-4)] to characterize thermal equilibrium [Eis, p.381, l.14] because transition rate is an outdated and problematic (e.g., what is the initial state or final state during a measurement? See [1]) concept in quantum mechanics. A better way to prove [Eis, p.382, (11-14) & p.383, (11-20)] can be found in [Kit, p.154, (4) & p.158, (10)].

    1. What does “classical limit” mean?
    2. Why is a low concentration or a high temperature in the classical regime?

        For the above questions, neither [Rei, §9.8] nor [Kit, p.160, l.-11-p.161, l.11] provides satisfactory answers. However, the use of both [Rei, 247, l.14-l.17] and [Pat, p.145, l.3-l.8] enables us to answer the questions.

  11. In [Kit, p.155, l.6-l.8], Kittel should have explained the legitimacy of applying the system-reservoir concept to orbitals by using [Rei, p.211, l.-9-p.212, l.2].

  12. In proving that |L1- L2| is the least possible value of L [Lan3, p.100, l.-15], it suffices to find (2L1+1)(2L2+1) independent eigenvectors using [Coh, p.1017, Comment]. The lengthy formal derivation [Coh, p.1016, l.11-p.1017, l.13] of [Coh, p.1017, (C-38)] is unclear and confusing.

  13. [Coh, p.659, (C-60) & (C-61)] and [Bri, p.17, (2.4)] both discuss the matrix elements of angular momentum. In terms of mathematical structure, the exposition in [Coh, Chap.VI, ' C.3.c, pp.657-660] is poorly organized. The discussed material has a point to make, but [Coh, Chap.VI, §C.3.c, pp.657-660] fails to indicate what the main point is. Thus, it lacks the final touch and thereby fails to hit the heart of the matter. In fact, the discussion of matrix elements of angular momentum requires the concept of irreducible group representation [Bri, pp.13-18, §2.1-§2.2] to tie up the loose ends.

  14. The requirement |s| < 1 in [Mer2, p.246, (11.68)] is incorrect (see [Col, p.234, l.7 & (IV.10)]).

  15. Although [Mer2, p.209, l.6-l.11] attempts to characterize a normal operator using eigenvectors, the logical structure of Merzbacher's statement is unclear. For a correct formulation , see [Halm, p.160, Theorem 1].

  16. Introduction and conclusion.
    1. An introduction is a preview and guide, while a conclusion is a summary.
    2. An introduction stresses that which is fundamental.
          Only through discussing a theory's [Bragg reflection] fundamentals [Iba, pp.35-38, §3.1] may we understand both its inner structure (Compare the derivation of [Iba, p.41, (3.27)] with that of [Ashc, p.99, l.3]) and how the fundamentals help link with other concepts ([Iba, p.47, (3.39)] provides a formula for the structure factor, while [Hoo, p.325, (11.21)] is unable to provide such an explicit formula because Hook fails to address the inner structure).
          Suppose we discuss chemical bonding in a solid. When we introduce the concept of metallic bonding, we should give a fundamental reason why it is different than covalent bonding and why metal tends to crystallize in relatively close packed structures [Iba, p.12, l.-1-p.13, l.3]. The description of the facts in [Kit2, p.76, l.-14-l.-12] does not explain their physics.
    3. An introduction emphasizes the big picture: We should treat the field as a whole and determine how each concept affects other concepts and further studies.
          The practice of reading the introduction of each book on a certain topic and then writing a summary does not produce a good introduction because what one write will lack a clear goal. An introduction should not to cram everything vague, shallow or insignificant into one place. Only those who master the field and update their knowledge about recent developments are entitled to write an introduction. An introduction should emphasize the interplay among subtopics. We should not treat subtopics as isolated entireties. Thus, a crystal lattice ie associated with a point group (Compare [Iba, p.29, l.11] with [Ashc, chap.4 &chap.7]). When introducing symmetry, we would like to discuss its physical significance [Iba, pp.25-28, §2.4].
    4. An introduction should emphasize the origin and the background rather than specific details (quasicrystals: compare [Iba, p.23, l.16-l.19] with [Kit2, pp.48-49, §Quasicrystals]).
          Suppose one writes an introduction. If he desires to flash an unforgetful picture [Kit2, p.28, Fig.1] which sticks in his mind, it is best to explain what it means and why it is important. Otherwise, it might be more proper to put it in the main text or conclusion rather than the introduction.  A picture displayed in the introduction cannot be left as a mastery. It must be followed by a more detailed explanation (Compare [Kit2, p.29, l.4-l.10] with [Iba, pp.48-51, §3.7; pp.54-63, Panel I & II].
          When introducing the concept of diffraction, Landau emphasizes that diffraction phenomena are consequences of small deviations from geometric optics [Lan2, p.145, l.-16].
          Ashcroft lists the shortcomings of the static lattice model in [Ashc, chap. 21] before he introduces the concept of lattice vibrations [Ashc, chaps. 22-27]. The list emphasizes that a new theory is needed to explain the inconsistencies between the old theory and the experimental results.
    5. [1].

  17. A refinement serves to elaborate a qualitative, vague idea into a quantitative, precise statement.
    Example. [Cra, p.15, l.1-l.9] ® [Cra, p.18, l.-15-l.-1].