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Consistency in Mechanics

  1. The function of terminology must be consistent in similar models (e.g. waves).
        Suppose nI < nT and E is parallel to the plane of incidence. The reflected wave in [Cor, p.564, Fig. 30-6] and that in [Hec, p.116, Fig. 4.41] have different phases. This is because Corson thinks that ER and EI have the same phase if EIx ERx > 0 [Cor, p.560, Fig. 30-3], while Hecht thinks that the incident light is like a walking person whose left hand represents the direction of E and that ER and EI will have the same phase if its left hand still represents the direction of E after it bounces back from the interface [Hec, p.114, Fig, 4.40] . If we consider the system alone, it will be difficult to determine which of these two opposite views on phase shift is more appropriate. Now let the incident angle be 0o. If a wave goes from a light string to a heavy one, the reflected wave will change phase [Hall, p.309, Fig. 17-15]. Thus Hecht's opinion on phase shift [Hec, p.118, Fig. 4.40(b)] is not consistent with this simple model. Therefore, I agree with Corson's opinion on phase shift.

  2. Given a theory. When we make an assumption for definiteness, we must justify that the assumption is consistent with the theory. Different viewpoints may result in different appearances of a formula.
        In [Wangs, p.412, Fig. 25-9], we assume Ei, Er, Et are all directed out of the page. Will this assumption cause inconsistency? It will not. This is because the title of [Wangs, 23-2] reduces the various electrical fields to two directions and we are allowed to use a minus sign to represent the direction opposite to that we assumed. For example, the minus sign right next to the first equality sign in [Wangs, p.413, (25-39)] reveals the fact that Er is actually directed into the page [Wangs, p.414, l.10].
        Both [Wangs, 23-2] and [Wangs, 23-3] should apply to the case of normal incidence [Wangs, p.416, l.-21-l.-17]. However, [Wangs, p.416, the first equality of (25-50)] differs from [Wangs, p.413, the first equality of (25-33)] by a minus sign. This is because [Wangs, p.412, Fig. 25-9] and [Wangs, p.415, Fig.25-12] have different starting points. The latter takes the opposite directions of Er and Ei into account, while the former does not.

  3. Consistency requires that one make the difference between two statements compatible by establishing a relationship between the two [Eis, p.99, l.25-l.34; p.101, l.13-l.14].

  4. Resolution of the discrepancy between theory and experiment makes the turning point in the development of a new theory [Eis, p.13, l.-24-l.-22].
        [Eis, p.13, Fig. 1-8] prompts Planck to propose his conjecture [Eis, p.13, (1-19)]. He then tries to derive [Eis, p.13, (1-19)] by using [Eis, p.14, (1-21)], a procedure that establishes the equipartition law. Planck finds that Ē kT when DE is small and Ē 0 when DE is large if energy E is treated as a discrete quantity instead of a continuous one [Eis, p.14, l.-10].

  5. The two determinations of Planck's constant h, using completely different phenomena and theories, are in good agreement [Eis, p.31, l.-19].

  6. In [Coh, p.653, (C-40)], Cohen-Tannoudji defines the phase of k,j,m>. In [Coh, p.653, (C-41)], he redefines the phase of k,j,m>. However, Cohen-Tannoudji fails to prove the consistency of the two definitions. [Mer2, p.240, l.22] shows that the two definitions are consistent.

  7. A random sign assignment is unacceptable even if it may lead to the correct final answer. A sign assignment should be consistent throughout the book. If Hr0/Hi0= -Γ [Sad, p.447, l.12], then the sign assignment will not be consistent with the usages in [Sad, (10.31), (10.72) and (10.74)]. Therefore, Hr0/Hi0 should be Γ.

  8. [Dit, pp.95-96, 4.A.9, & p.96, l.-6-l.-4] essentially say that the following two operators do not commute:
    Operator A1: A1(f) = the real part of f.
    Operator A2: A2(f) = the Fourier transform of f.

  9.  
    1. Why does the semiclassical analysis of [Ashc, chap.12] reduce to the Sommerfeld free electron theory of [Ashc, chap.2] when applied to the transport properties of alkalis [Ashc, p.287, l.2-l.4]?
      Answer. alkalis [Ashc, p,287, Fig. 15.3]; noble metals .
    2. Compare the results of the semiclassical model with the results of the Sommerfeld free electron model when we apply them to the transport properties of noble metals.
      1. Sameness [Ashc, p.289, l.1-l.9; Fig. 15.4].
      2. Differences [Ashc, p.290, Fig. 15.5, necks exist; p.292, Fig. 15.7, there are many types of orbits].

  10. (Maxwell generalizes Ampre's law by breaking through its inconsistency in the special case [of steady state]) Ampre's law [Jack, (6.1)(ii)] is not consistent with the continuity equation [Jack, (6.3)] for time dependent fields. Maxwell breaks through the inconsistency by using Coulomb's law to replace J with J + D/t. This generalized Ampre's law [Jack, (6.5)] becomes consistent with the continuity equation [Jack, (6.3)] for time dependent fields.

  11. From a formal derivation to its internal consistency
    1. Polarization charge densities
      1. Polarization charge densities are derived from comparison of formal patterns [Wangs, p.143, (10-6), (10-7) & (10.8)].
      2. The physical derivation of polarization charge densities ([Fur, p.62, Fig. 2.2]; imitate the proof given in [Fur, p.64, l.1-l.7]).
      3. Internal consistency:
        1. Polarization charge densities satisfy the boundary conditions [Wangs, p.145, l.3-l.5].
        2. The total polarization charge is zero [Wangs, p.145, (10-13)].
    2. Magnetization current densities
      1. Magnetization current densities are derived from comparison of formal patterns [Wangs, p.214, (20--9); p.315, (20-10)].
      2. The physical derivation of magnetization current densities [Fur, p.64, l.1-l.7].
      3. Internal consistency:
        1. Magnetization current densities satisfy the boundary conditions [Wangs, p.316, (20-12)].
        2. The total magnetization current is zero [Wangs, p.317, l.5].