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Physical interpretations in Mechanics

1. Object under study vs. typical phenomenon

   For a system of masses attached by springs, we use [Kap, p.675, (10.61)] as our model. However, when the friction increases, and total mass approaches 0, [Kap, p.675, (10.61] becomes the heat equation [Kap, p.685, (10.87)]. In this limiting case, we should use the heat equation as our model to explain that disturbance is propagated instantaneously [Kap, p.685, l.−2-p.686, l.3; p.688, l.16-l.28], although the object under consideration is still a system of masses attached by springs rather than heat. Thus the property of displaying instantaneous disturbance is not inherent in the object itself. In fact, it is the differential equation [Kap, p.685, (10.87)] alone that characterizes the heat phenomenon described above.
   
   

2. A problem in physics usually has an analytic meaning, a geometric meaning, and a physical meaning. The analytic meaning refers to the problem's mathematical formulation and the subsequent analysis. The geometric meaning shows the picture. The physical meaning explains the origin of the problem. These three meanings should be treated like one entity. Although each of these three aspects is indispensable for a thorough understanding, the physical meaning is most thought-provoking.
   Examples.
   1. A Neumann problem [Jack, p.37, l.−18]. Analytic: Specifying the normal derivative on the surface. Physical: Specifying the electric field on the surface.
   2. Collision.
      1. Analytic meaning of equations of motion.
         Before the collision: The Schrödinger equation has boundary conditions.
         After the collision: The Schrödinger equation has no boundary conditions.
      2. Physical meaning of the solutions.
         Before the collision: [Coh, p.1374, Fig. 2a].
         After the collision: [Coh, p.1374, Fig, 2c].
   3. Fermat's principle.
      Analytic meaning: [Hec, p.109, l.c., l.20-l.23].
      Physical meaning depends on the shape of the mirrored surface [Hec, p.111, Fig. 4.37(c)]: Relative minimum [Hec, p.110, r.c., l.21]; relative maximum [Hec, p.110, r.c., l.24].
   4. The divergence theorem.
      

      		Physical		Analytic
      Electrostatics		Gauss' law		The divergence theorem
      ∥parallel		∥[Jack, p.179, l.−6-l.−5]
      Magnetostatics		Ampère's law		Stokes' theorem

      By the parallelism between electrostatics and magnetostatics, Stokes' theorem may be considered the magnetostatic version of the divergence theorem.

   
   
3. If there are several sets of coordinates available, we only select the one that has significant physical interpretation [Go2, p.461, l.1-l.4].
   
   
4. The wave function for a free particle.
       There are two methods of finding the wave function:
   Method 1: [Rei, p.354, l.- 2-p.358, l.7];
   Method 2. [Rei, p.358, l.- 3-p.360, l.- 9].
   The advantage of method 1: n_x has a physical meaning: the number of nodes of a standing wave.
   The advantage of method 2: [Rei, p.360, l.- 14-l.- 9].
   
   
5. The meaning of electric polarization in [Cor, pp.173-174, ' 9.2] is incomplete because it lacks mathematical justification (see [Born, p.78, l.9-p.79, l.19]).
   
   
6. (Reversible process [Example. Carnot cycle])
       To interpret the physical meaning of a concept properly, we must
   1. understand which viewpoint we adopt,
   2. understand the purpose of the definition,
   3. find a way to make physical quantities well-defined,
   4. specify the system if we say that a system is in equilibrium, and
   5. distinguish an important factor from a trivial one.
   
       Only after solving many physical problems concerning Maxwell's equations, may we understand the physical meaning of vector analysis. Similarly, only after studying the example of Carnot cycle, may we understand the physical meaning of a reversible process.
   
   
   1. Points of view: For the discussion of Carnot cycle, [Lan5, pp.57-59, ' 19] considers the system to be the combination of working substance, low-temperature reservoir and high-temperature reservoir (ignore the piston in [Rei, p.188, Fig. 5.11.5]) [Lan5, p.59, l.10], while [Kit, pp.236-240] limits its consideration to the working substance [Kit, p.236, l.- 7].
   2. Purpose: As [Lan5, p.33, l.14-l.15] points out, the purpose of a reversible process is to keep the entropy constant from the beginning of the process to the end.
   3. Making physical quantities well-defined: Entropy is a statistical quantity. It needs to be well-defined. However, only when the system is in equilibrium, may the physical quantities be sharply defined. This is why [Kit, p. 64, l.1-l.2] requires that the system always be infinitesimally close to the equilibrium condition. That is, the external condition must be changed slowly enough to allow the equilibrium to catch up. In other words, the time period that the system remains roughly in the same macrostate must be long enough compared to the relaxation time. If we pull the piston in [Rei, p.188, Fig. 5.11.5, c® d] too fast, gas molecules may not have time to reach the piston. Then the volume and pressure of the gas is not well-defined [Kit, p.64, l.8-l.11]. Thus the equation dU=t ds - pdV used in [Kit, p.236, l.- 7] is valid only when the gas is in equilibrium with itself if it is thermally isolated, and only when the gas is in equilibrium with the reservoir if the gas is in contact with a reservoir. [Lan5, p.34, l.10] claims that, for an adiabatic process, the entropy of the system remains constant. {1}.
   4. Specifying the system that is in equilibrium: [Rei, p.189, l.1] claims that a Carnot cycle is quasi-static. In stage a® b in [Rei, p.188, Fig. 5.11.5], the quasi-static system refers to the working substance alone. In stage b® c in [Rei, p.188, Fig. 5.11.5], the quasi-static system refers to the working substance plus the high-temperature reservoir. Suppose body 1 and body 2 are originally in contact and in equilibrium. If we separate body 1 from body 2, the total entropy will not change because the entropy is additive.
   5. Distinguishing an important factor from a trivial one: At point 1 in [Kit, p.239, Fig. 8.6] the working substance and the high-temperature reservoir have the same temperature. How can heat flow from the reservoir into the working substance? When the piston moves up, the volume of gas increases and the temperature of gas should decrease. To maintain the same temperature, the gas needs to absorb heat. In the two stages of [Kit, p.239, Fig. 8.6.(b)], whether you press the piston or let the piston go down by itself does not matter, as long as there is a required amount of work done to the gas.
   
   
7. A derivation should keep the physical meaning intact.
       R+D=1 [Lan3, p.77, l.13] follows from "incident flux = reflected flux + transmitted flux". Thus Landau's derivation keeps the physical the physical meaning intact. In contrast, Cohen-Tannoudji's computational derivational derivation fragments the physical meaning [Coh, p.70, l.- 5-l.- 4].
   
   
8. When introducing a definition, we should give an appropriate physical interpretation at the earliest stage possible. Otherwise, the definition will lack a motive, and we will be left with no understanding of what is going on.
       In quantum mechanics, to give the concept of angular momentum a complete physical meaning [Lan3, p.82, l.- 5], we must interpret angular momentum as - iS (r´ L ) [Lan3, p.83, (26.2)] instead of [Coh, p.645, (B-6]. It is natural and meaningful to interpret angular momentum as an an operator in this way. Indeed, interpreting angular momentum as - iS (r´ L ) simplifies the discussion of the conservation laws [Lan3, p.83, l.11-l.18] and directly associates a concrete physical meaning with the complicated mathematical expressions in [Coh, p.62, (D-5)] [1]. The only advantage of writing angular momentum as [Coh, p.646, (B-9)] is that it facilitates the discussion of spin [Coh, p.970, (A-4)] and addition of angular momenta [Coh, p.1003, (B-4)].
   
   
9. The physical interpretation helps us understand the inner structure and inspires us to utilize the main feature to allow further simplification. Our final goal is to endow every step of our calculation with a simple and meaningful physical interpretation.
       [Lan3, p.125, l.5-l.15; l.- 8-l.- 6] show that whenever the degree of symmetry increases, the degree of degeneracy also increases. We may use this feature to simplify the calculation of an energy level's degree of degeneracy. (Compare [Lan3, p.125, l.17-p.126, l.7] with [Coh, p.1017, Comment]).
   
   
10. There are several ways to look at orthogonal coordinates:
    1. Energy [Sym, p.357, l.- 13].
    2. Arc length [Lan3, p.129, l.-9; (37.3)].
    3. Vector [Cou, vol.1, p.3, l.- 9].
    4. Quadratic forms [Cou, vol.1, pp.23-26, Chap. I, ' 3.1].
    
    
11. The physical meaning of a Young diagram.
        The Young diagram for a given coordinate wave function represents a simultaneous eigenstate of the following set of operators: the Hamiltonian, the orbital angular momentum, the parity, any permutation within a given row, and any permutation within a given column. The eigenvalue for a permutation within a row is 1; the eigenvalue for a permutation within a column is d_P [Lan3, p.235, l.8-l.9].
    Application.
        Given an electron configuration of an atom with many electrons [Lan3, p.253, l.18], the energy levels perturbed by the interelectronic repulsion can thus [Lan3, p.251, l.4-l.17] be represented by the (spectral) terms [Lev2, p.338, Fig. 11.6].
    
    
12. The interpretation of the same formula [Sad, p.228, (6.32); p.226, (6.22)] can travel different paths if the assumptions are different [Sad, p.236, l.3-l.5; p.237, Example 6.11].
    
    
13. Given an equality, we should be able to use physical considerations to tell which quantities are constants, which quantities are independent variables, and which quantities are dependent variables.
    Example. |k|=w/v [Wangs, p.380, l.-20-l.-16].
    
    
14. [Hoo, p.11, l.1; p.17, l.5 & p.29, l.-13] provide the physical reasons why the crystal structure of solids tends to be close-packed.
    
    
15. Undergraduate textbooks vs. graduate textbooks.
        There are only two kinds of books: good books and bad books. If a learned scholar studies a book designated for undergraduates, he should not feel insulted. Don't you see one who publishes books for idiots or dummies  makes more money than those who publish books for geniuses? In this business world, consumers simply do not have much self-respect.
    1. A good book should be clear and subtle, and should have depth. The following strategy does not automatically elevate a textbook from the undergraduate level to the graduate level: When three sentences are required to clarify a concept, the author only uses one or two.
       Example 1: [Kit2, p.101, Fig.5] should have included detailed interpretations like [Hoo, p.40, l.11-l.16].
       Example 2: Compare [Kit2, p.129, l.-4] with [Hoo, p.63, l.-8-p.64, l.21].
       Example 3. Compare [Ashc, p.217, l.-15: The externally applied fields are treated classically] with [Hoo, p.86, l.-5-p.87, l.14].
    2. Even an undergraduate textbook should be free of vague explanations.
       Example. Compare [Hoo, p.87, l.-18-l.-6] with [Ashc, p.10, l.-6-p.11, l.-13].
    3. A good book [Hoo, p.82, l.7-l.15] should relate the key features of a complicated proof [Ashc, p.42, l.-18-p.47, l.10] to a simple model.
    
    
16. RC charging and discharging
        [Hall, p.531, Table] gives the formulas involved. [Hall, p.529, l.-12-p.530, l.-13] provides the physical meanings for the processes .
    
    
17. It is important to interpret a rule (Hund's rule) in the correct context (Solid state physics; paramagnetism [Hoo, pp.200-211, §7.2]. The origin and applications of the rule should be integrated in one place so that the reader can see its functions. If we isolated the rule from its original context and put in a more abstract setting for the completion of a general theory (For example, in quantum chemistry: many-electron atoms) [Lev2, pp.328-329] or in quantum mechanics: The atom [Lan3, pp.253-254]), then the arrangement will make the rule appear as though it came from nowhere.
        The correct context can provide effective methods to check whether the consequences of the rule will agree with the experiments. For Hund's rule, we can use [Hoo, p.205, (7.15)] or [Hoo, p.205, (7.18)] to test whether the use of Hund's rule will lead to a match between the theoretical values and the values measured from experiment. See [Hoo, p.206, Fig. 7.4; p.208, Table 7.2].
    
    
18. For a charged particle placed in an electric field, we may define momentum using Hamiltonian formalism [Coh, p.1493, (63)]. However, it is difficult to understand the physical meaning of momentum from this choice of definition. Only after we adopt the viewpoint of the conservation of momentum can the physical meaning of momentum become clear [Hoo, pp.211-213, §7.3.1].
    
    
19. [Coh, p.589, l.-13] shows that if k and k' differ by an integral number of 2p/l, their corresponding equations of motion [Coh, p.588, (8)] are the same. [Hoo. p.40, l.12] illustrates the same fact using the wave picture in [Hoo, p.39, Fig. 2,5(b)].
    
    
20. We cannot give a new definition without analyzing its inner physical structure.
    Example: the crystal momentum of an electron that interacts with the periodic potential of the crystal lattice [Hoo, p.126, l.12]. It is not enough to formally define the crystal momentum based on [Hoo, p.126, (4.24)] alone. We must divide the total momentum into the momentum of the electron and the lattice momentum as in [Kit2, p.205, (11)] and then use physics to show that the total momentum change is indeed ħDk [Kit2, p.206, (15)].
    
    
21. The same physical relation [Wangs, p.551, (B-21)] can be interpreted in quite different ways according to its applications. For example, see [ Wangs, p.551, l.5-l.10].
    
    
22. Interpretation should focus on what the concept is, rather than what it is not.
        In [Par, p.315, l.10], Park says that ds/dW does not denote "differentiation with respect to W". However, in [Par, p.316, l.-11-l.-6], Park obviously treats ds/dW as an inverse operation of integration. If ds/dW is not some kind of differentiation, then Park has contradicted himself. Therefore, Park should have specified the category of differentiation to which ds/dW belongs. When a physicist raises a mathematical question, he should pursue the answer until it is complete. Otherwise, he only exposes his ignorance. In this case, ds/dW is the differentiation with respect to the Lebesgue measure on a unit sphere [Ru2, p.164, Definition 8.3].
    
    
23. Through Reif's elementary approach, the physical meaning of the pressure tensor is clear [Reif, pp.473-474, (12.3.2)-(12.3.5)]. In contrast, Huang's more general approach using the Boltzmann transport equation [Hua, p.97, l.-1] obscures the physical meaning of the pressure tensor.
    
    
24. Summation and integration are equivalent concepts in physics. Using the limiting process, one can prove that summation implies integration. Using the properties of the delta function, one can prove that integration implies summation [Jack, p.27, l.5-l.10].
    
    
25. The physical meaning of differential changes of state [Zem, pp.34-36, §2-6].
    
    
26. There is no shortcut for gaining deep understanding. Haste only makes waste. Corson uses [Cor, p.682, Fig. 37-1] to explain [Cor, p.681. (37-39)(iii)]. His explanation is confusing because readers do not know the specifics of the situation he is describing and what motivates him to prove the formula. In contrast, the explanation in [Wangs, p.479, l.3-l.16] is convincing because it considers consistency from multiple viewpoints and also because it presents the assumptions clearly and carefully [Wangs, p.474, l.12; p.479, l.18-l.19]. Thus, every little bit of information adds to the formula's rigor and significance.
    
    
27. In [Reif, §12.1], Reif fails to give clear definition of the notation t. He should have said that the time t is measured from the instant of the last collision [Chou, p.179, l.12-l.13]. Once the meaning of t is clarified, the meanings of [Reif, p.465, l.4] and [Chou, p.179, (4.13)] become clear.
    
    
28. [Chou p.250, l.8-p.251, l.-3] gives a rigorous mathematical proof of [Chou, p.251, (6.18)], but fails to provide a physical interpretation. Jackson's approach is much better, because [Jack, p.210, l.10-l.13] provides the key physical idea behind the mathematical proof before delving into details.
    
    
29. Our interest in the concepts involved with a current is based on the significance of their relationship to the type of the current [Fan, §1.2].
    1. Convection-current density: J=rv.
    2. Conduction-current density: J=sE.
    
    
30. [Wangs, chap. 8] gives a mathematical construct of a multipole term [Wangs, p.112, (8.13)], but fails to explain its physical meaning [Fan, p.115, l.17-p.116, l.1].
    
    
31. [Jack, §1.13] discusses the relaxation method for two-dimensional electronic problems, but fails to explain its physical basis [Fan, p.130, Fig. 4.8].
    
    
32. One should provide a direct and insightful physic interpretation of the phase velocity [Matv, p.25, Fig. 2] rather than present a roundabout interpretation that requires formal calculations [Hec, p.19, (2.32)]. Hecht's strenuous effort is much ado for nothing.
    
    
33. The operational meaning of linear superposition: reducing a complicated problem to two simple problems [Jack, p.61, l.3-l.13].