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Definitions in Mechanics

  1. The partition function [Kit, p.61, (10)] is the normalizing factor that converts relative probabilities to absolute probabilities [Kit, p.138, l.6-l.8], so Kittel defines the grand partition function accordingly [Kit, p.138, (53)]. Indeed, the formalism helps us guess the definitions concerning the grand canonical ensemble from the corresponding definitions concerning the canonical ensemble. However, the benefit of doing so can only be seen from hindsight. Thus a guess based on formalism actually eliminates the motive that prompts us to look for the grand partition function [Rei, p.346, l.-11-l.-1].

  2. The a defined in [Rei, p.347, l.12] and the a defined in [Pat, p.99, (6)] turn out to be the same [Rei, p.49, (9.6.15) & Pat, p.102, (5)]. These two approaches provide different perspectives of a.

  3. Sometimes the purpose for introducing a definition is far more important than the definition itself.
        [Pat] introduces the concept of canonical ensemble in [Pat, pp.53-61, 3.2] without discussing the important role that the concept plays in application [Rei, pp.219-221, 6.7].

  4. In practice, we would like to express a physical quantity (e.g. entropy) in terms of parameters that are readily measured by experiment [Rei, p.124, l.-9].

  5. (Temperature)
    1. Analytic definition [Rei, p.99, (3.3.10)].
    2. Theoretical measurement with respect to a thermometric parameter of a thermometer [Rei, p.104, l.8-l.12].
    3. Absolute temperature [Rei, p.105, l.8] is independent of the nature of the particular thermometer used and directly connected with theoretical equations [Rei, p.133, l.10-15].
    4. The practical measurement given by a gas thermometer is based on the equation of state for an ideal gas [Rei, p.133, l.-10-l.-9].
    5. Absolute temperature interpreted by
      1. the mean energy per degree of freedom [Rei, p.106, l.6-l.11].

      2. heat [Rei, p.106, l.-13-l.-9].

  6. The advantages of various versions of a definition.
        The definition of entropy in [Rei, p.215, (6.6.5)] has three mathematical advantages [Rei, p.218, l.16-l.24], while the definition in [Rei, p.216, (6.6.8)] has an intuitive physical meaning.

  7. The source of a definition.
    1. Inductive definition.

      An inductive definition is defined by its properties. The definition itself may not tell us how to construct any examples. However, it will naturally lead to the general construction once we endow the definition with a physical meaning. A physical meaning of scalar product: The kinetic energy in terms of generalized coordinates [Gol, p.326, l.2]. General method of reproduction:

      1. Constructing a new scalar product from the fundamental one [Jaco, vol.2, p.176, l.-19].
      2. Any scalar product can be obtained from the above method [Jaco, vol.2, p.176, l.-13-l.-11].


  8. The purpose of a definition is to describe the big picture.

    Differentiation is the inverse of integration. More precisely, its key idea is to take the infinitesimal average [Kap, p.240, (4.49) & p.240, l.-6]. We should not be distracted from this main concern [Kap, p.240, l.-1] by any special type of approach, say, using an abstract measure [Ru2, p.164, Definition 8.3] and a sequence of open balls [Ru2, p.163, l.-2].

  9. If a theorem can be derived from a definition, we should prove the theorem. We should not consider the theorem as another definition because we may easily leave out the important procedure of checking its consistency [Sad, p.13, (1.21) & p.15, (1.30)].

  10. As long as we do not sacrifice the effectiveness of the procedure to check the definition, we should define a concept [Say, curl A] as generally as possible because
    1. we may easily recognize the common pattern and
    2. it then becomes possible to obtain various cases by simply substituting the appropriate data [Kara, p.50, (2.50)]. In contrast, the complication caused by coordinate transformation [Sad, p.77, l.-3-l.-1] is meaningless.

  11. We choose the definition of a concept to best meet the occasion.
    Example. Definition of pressure.
    1. For theoretical purpose: -p = (∂E/∂V)S [Rei, p.161, (5.5.4)].
    2. For the practical measurements made with instruments that are available:
      -p = (∂F/∂V)T [Rei, p.163, (5.5.14)].

  12. When we propose a new definition that is more general than the old one, we must show that there exists an example that satisfies the new definition but not the old one. If the two definitions were equivalent, it would be unnecessary to give another name for the same concept.
    Example.
    Old definition: Pure state [Coh, p.297, l.-14].
    New definition: Mixed state [Coh, p.300, §4].
    [Coh, p.439, comment (ii)] provides an example of mixed state that cannot be represented as a pure state.
    Discussion. The difficulty in finding such an example is due to the fact that the two types of averaging inherent in a mixed state are inseparable [Lan5, p.18, l.-7].

  13. A definition serves to illustrate a point; it does not ramble on about the trivial. [Lan3, p.5, l.−10] shows that a complete set serves to collect enough simultaneous measurements to characterize a physical state. In contrast, Cohen's definition of c.s.c.o. [Coh, p.143, §b] emphasizes mathematical formalism and logical rigor, but somehow misses the original point.

  14. The generalization of a definition should preserve the characteristics of old concept in the best possible manner under the weaker conditions.
        When generalizing a concept [e.g., the differentiation of an operator with respect to time] from classical mechanics to quantum mechanics, we would like to preserve the characteristics of the old concept under a weaker hypothesis. To do so, we may encounter some difficulties [Lan3, p.26, l.16]. A revolutionary idea is often required to conquer the difficulties.
        In practice, it is sufficient to use the derivative of the average value of an operator to solve problems. Thus it seems unnecessary to define the differentiation of an operator with respect to time. Perhaps this is the reason why [Coh, pp.240-241, III.D.1.d.a ] does not discuss the differentiation of an operator. However, even from the theoretical point of view, only through the use of average value may we obtain the best possible result  under the weaker hypothesis [Lan3, pp.26-27, 9].

  15. Eliminating ambiguity improves the quality of a definition [Lan5, p.25, l.1-l.28].

  16. A definition or an axiom should not be used as a means for the author to avoid the responsibility of explaining the geometric origin of the concept (spin). [Coh, chap. 9, A.2] fails to define the spin variable [Lan3, p.200, l.-16] and spin component [Lan3, p.200, l.-11] of a wave function. Cohen-Tannoudji also fails to relate spin to infinitesimal rotation [Lan3, p.201, l.14-l.19; Go2, p.173, (4-105')] which unifies the concept of spin angular momentum and that of orbital angular momentum. Thus, Cohen-Tannoudji should have used [Coh, p.984, (5)] as the definition of spin angular momentum to start his theory on spin. The purpose of mechanics is to emphasize insight along with the big picture, not to descend into formalism.

  17. The formulation of a definition (Angular momentum)
        The final version of a definition that appears in a textbook usually has been revised many times by experienced researchers in the field. Consequently, the formulation of a definition is like a preview of a theory’s development. In light of past experience, the future improvement of formulation should be oriented toward the following goals:
    1. Like a football team, it must build a strong defense. In other words, it must sustain any kind of test or attack: it must be as general and inclusive as possible and be able to provide a satisfactory answer when a question is raised.
          For example, the definition of angular momentum [Lan3, p.82, l.16-l.19] automatically includes the case of interaction. The same is true for and the definition of the sum of angular momenta. Furthermore, the definition of total angular momentum agrees with the concept of operator sum.
      Proof. y (ra,rb)= y (R-1 ra, R-1 rb), where R is a rotation.
         y (xa,ya,za; xb,yb,zb) = y (xa - e ya, ya +e xa, za; xb - e yb, yb +e xb, zb)
      =y (xa,ya,za; xb,yb,zb)+e[xa(y/ya)-ya(y/xa)]+e[xb(y/yb)-yb(y/xb)] =y(xa,ya,za; xb,yb,zb)+e [Lz(a) +Lz(b)]y .
      Remark. The classical definition of angular momentum does not have this advantage of dealing with interaction.
    2. It should have a vision and allow the consistency of different approaches. In other words, it must contain devices delicate enough to detect nuances, clarify the ambiguities, and make seemingly contradictory statements compatible.
      1. The definition of total angular momentum [Coh, p.983, (1)] automatically includes the case of spin-orbit interaction.
            According to [Coh, p.984, l.1; (3),(4),(5)], (r)Ru(a ) acts only in Er and (s)Ru(a ) acts only in Es. Does this fact mean that Ru(a ) cannot apply to the case where the spin-orbit interaction exists? No. The spin-orbit interaction only affects the wave function y (r,s ) (where s is the spin variable) and implies that y (r,s ) cannot be expressed as the product f (r)c (s ). The rotation operator Ru(a ) can apply to the general wave function y (r,s ), so the application is not necessarily limited to the specific form f (r)c (s ).
      2. According to [Go2, p.173, (4-118)] or [Lan3, p.201, l.14-l.19], [exMx, eyMy]=exe yMz. However, according to [Lan3, p.216, (58.3) & (58.4)], Uy(ey)-1Ux(ex)-1Uy(ey)Ux(ex) =Uz(exey). The former statement uses the difference operator, while the latter statement uses the inverse operator. Do these two statements concerning infinitesimal rotations contradict each other? No. The M's in the former statement refer to the infinitesimal rotation generators [see Go2, p.168, (4-102); p.173, l.12 & l.-5], while the U’s in the latter statement refer to the ordinary sense of infinitesimal rotations.
    3. Variety (Addition of angular momenta)
          One version of a definition good for one purpose may not be good enough for another purpose. Thus, in order to fully understand a concept, we must study it from various perspectives.
          For example, using the state space E (j1, j2) to discuss J 2 and Jz [Coh, p.1013, l.1] has the advantage of dealing with the dimension [Coh, p.1017, (C-39)] and irreducible group representation. In contrast, using symmetric tensor to express the wave function of a particle with spin s [Lan3, p.210, l.21-l.34] has the advantage of discussing both the relationship between rotation and symmetry [Lan3, p.210, l.27-l.29], and the subspace generated by an eigenvector using rotations [If y1 ay1+by2 and y2 cy1+dy2, then y1f1 (ay1+by2)(af1+bf2)=a2y1f1+ab(y2f1+y1f2)+b2y2f2].

  18. The Cross Section
    1. Key idea: measuring the impact of the projectiles [Mer2, p.279, l.3-l.12].
    2. Experimental measurement
      1. Method: using probability [Mer2, p.280, (13.1)].
        Example [Mer2, p.279, Fig. 13.1].
      2. Differential cross section [Mer2, p.281, l.16-l.22].
        Example [Mer2, p.281, (13.5); p.283, Fig. 13.3].
      3. The difference between the laboratory frame-of-reference and the center-of-mass frame [Mer2, p.282, (13.6); p.286, l.2].
    3. Theoretical prediction (Scattering cross section)
      1. Classical formula ([Mer2, p.283, (13.7); p.284, (13.9)] or [Lan1, p.53, (19.5)]).
      2. Quantum uncertainty limits the usefulness of the classical formula [Mer2, p.284, l.9-l.11].
      3. Establish the formula of the differential cross section [Mer2, p.290, (13.23)] using stationary scattering states ([Coh, p.909, l.12; p.910, (B-9)] or [Mer2, p.287, (13.13)]).
    4. The experimental measurement must match the theoretical prediction.

  19. The meaning of electric displacement should not be limited to its mathematical expression [Sad, p.122, (4.35)]. We must take into account its purpose [Sad, p.122, l.-5] and its counterpart in the magnet field [Sad, p.123, l.4-l.5].

  20. (Stress tensor)
        [Sym, 10.6] defines stress tensors through three physical models: an ideal fluid a viscous fluid an elastic solid. Each successive model is more complex than its predecessor. The features of each model are well-preserved in its successor. We can clearly see the reason why the concept of tensor is required in defining "stress" for a viscous fluid [Sym, p.436, (10.182)]. In contrast, [Lan7, 1] defines the stress tensor from a theoretical point of view [Lan7, p.5, (2.2)]. The reasoning in [Lan7, p.4, l.-13] is far-fetched and [Lan7, p.6, l.12-l.15] only lightly touches on the physical meaning of the stress tensor. Thus, we cannot see the big picture. In addition, we cannot pinpoint the physical reason why we need the concept of tensor to define "stress".
    Remark. The physical meaning of sik in [Lan7, p.5, l.4] is clear, while the physical meaning of Pyx in [Sym, p.438, l.-9] is not.

  21. The purpose of the definitions pertaining to  a subject serves to
    1. provide the necessary tools to solve the problems that physicists have experienced.
          To discuss the magnetic force on a current, we must distinguish the internal magnetic field from the external magnetic field [Hall, p.559, l.16-l.3].
    2. provide an easy way derived from synthesis to analyze more complicated situations.
          For a simple case, such as [Hall, p.541, Fig. 30-6], we can easily determine the direction of the magnetic dipole moment by [Hall, p.544, (30-11)]. However, for the more complicated case, such as [Hoo, p.216, l.8], we must follow the rule given by [Hall, p.543, l.20] to determine the direction of the magnetic dipole moment.

  22. A useful definition of the energy in a dielectric enables us to have control over the charge distribution in order to store energy. Then we will be able to retrieve the energy from the system as reversible work. Since we have no control over the bound charges, their built-in energy will remain the same. Therefore, when defining the energy of a dielectric, we consider the energy contribution from the controllable  free charges only [Wangs, p.161, l.10-l.20].

  23. The definition of an electric field does not come from direct measurement, but from a limiting process whereby the ratio of the force on the test body to its charge approaches a fixed value as the charge becomes smaller and smaller [Jack, p.24, l.2-l.11].

  24. (Effective mass)
        A new definition must agree with the old one and make the theory consistent. For example, to preserve the form of Newton's second law [Ashc, p.229, (12.31)], we define the effective mass as [Ashc, p.228, (12.29)]. Furthermore, the second-order term in the Taylor expansion of E(k) at a fixed point reduces to the kinetic energy in the free-electron case [Cra, p.22, l.10-l.17]. Thus, the energy for a band agrees with the energy for a free electron. This gives the physical reason why we define the effective mass in this form.
    Remark. [Cra, p.22, (1.3.16)] is derived from [O'N, p.74, l.-7].

  25. (Time)
        Sometimes a concept is an inseparable part of the entire system and the key to defining the concept is to indicate what role the concept plays in the entire system [Ken, p.1.-17-p.2,l.19]. In particular, the definition should specify the concept's relationship to the entire system. If we were to isolate the concept from the system, then its resulting definition would become meaningless [Lud, p.8, l.19-l.-10]. Thus, this broad and objective view of defining a concept only comes from hindsight. Any other old and ingrained view would be biased and incorrect.  Any question inconsistent with the entire system would be wrongly proposed [Lud, p.8, l.16-l.19].

  26. The caption of [Ashc, p.567, Fig. 28.4] gives a clear definition of the threshold for direct or indirect transitions, while the caption of [Kit2, p.202, Fig.4] fails to give a precise definition of the thresholds. The lack of clarity means [Kit2] fails to live up to the standards implied by its book title: Introduction to Solid State Physics.

  27. When an author defines a geometrical figure, he or she should include a picture.
        The Wigner-Seitz cell for the body-centered cubic Bravais lattice is a truncated octahedron [Ashc, p.74, Fig. 4-15]. Webster's New Explorer Dictionary does not even give the definition of an octahedron. The Oxford American Dictionary' s definition is "a solid figure with eight faces". This definition is not exactly what a mathematician has in mind. Only Webster's New World College Dictionary (3rd ed.) provides a figure of an octahedron. Only after comparing it with [Ashc, p.74, Fig. 4-15] do we understand how to cut the corners to obtain the latter figure. Although the Oxford English Dictionary online gives original and historical examples, it fails to provide photos and videos, even  when they are vital for clarity. The name of a city should have a map showing its location. The name of a bird requires a colorful photo and a sound track so that we can identity it. The definition of a scientific experiment requires a video to illustrate its procedure. The priority of a dictionary is clarity rather than saving troubles.

  28. When justifying the definition of the differential scattering cross section, we should make a thorough investigation [Reic, p.678. l.3] instead of being content with limited progress [Go2, p.106, l.15].

  29. (Average occupation numbers) Even though a mathematical expression fills the bill of a term in an application, the evidence thus gathered is not strong enough to fully justify calling the expression by that term. For example, the justification of the definition given in [Reic, p.383, (7.127)] is not as good as that given in [Hua, p.199, (9.65)].

  30. (Self-inductance) We should give the simple definition first, and then the complex one; this approach enables us to avoid roundabout arguments when introducing the simpler definition.
        In [Chou], Choudhury defines the self-inductance for a single circuit [Chou, (6.52)] first, and then defines the coefficients of mutual inductance for two circuits [Chou, (6.55)]. In contrast, Wangsness defines the coefficients of mutual inductance for two circuits [Wangs, p.278, l.4] first, and then the self-inductance for a single circuit [P.280, l.17] by repeating the argument from [Wangs, (17-44)] to [Wangs, (17-46)].

  31. (Flux linkage) A definition is a necessary tool for clarification.
        F12 in [Chou, p.267, l.4] refers to the flux linkage through the first coil due to the current in the second coil. [Chou] defines the concept of flux, but lacks the definition of flux linkage. Without using the definition of flux linkage or Riemann surface, the meaning of F12 in [Chou, p.267, l.4] is very confusing. In contrast, [Cor, p.349, (19-2)] defines the flux linkage and [Cor, p.421, l.-4-l.-2] explains how the concept provides the key idea to eliminate confusions.

  32. (Reflectance and transmittance) A definition should be based on its natural physical origin rather its artificial mathematical content. The definitions in [Hec, p.119, Fig. 4.47; p.120, (4.54) & (4.55)] are insightful, while the definitions in [Wangs, p.420, l.13-l.14; (25-65) & (25-71)] impede our understanding. Furthermore, the latter definitions are only consequences of the former definitions.

  33. For the definitions of the following concepts: aperture and field stops, entrance and exit pupils, the Huygens eyepiece and the Ramsden eyepiece,
    1. read [Hec, 5.3.1, 5.3.2, and 5.7.4] if you want isolated definitions that are abstract and vague;
    2. read http://hep.ph.liv.ac.uk/~rinnert/phys210/phys210_lecture10.pdf if you want to integrate these concepts into a single natural setting so that you can apply them skillfully.

  34. Even though the concept of principal points is motivated by seeking a simple form of the thick-lens equation [Morg, p.58, (5.2)], we should not define principal points by using [Morg, p.59, (5.9) & (5.10)]. A concept should be defined in a way that gives easy access to the main results in a theory. For example, in order to prove [Fre, p.132, (5.39)], all we need to do is identify [Fre, Fig. 5.16] with [Fre, Fig. 4.7]. Obviously, the geometrical properties of the principal points are more important than the algebraic expressions of the coordinates of these principal points. Furthermore, the geometrical definition of principal points given in [Hec, p.243, Fig. 6.1] is much easier to deal with. Therefore, we prefer the geometrical definition to the algebraic definition even though this choice comes from a hindsight. In fact, [Jen, p.83, l.-15-l.-4; p.85, Fig. 5G] show that we can graphically locate the focal points and principal points for a given thick lens.

  35. (Focal points) When its meaning can be clarified in words [Jen, p.61, l.1-l.2], we should not give a definition using a formula [Hec, p.158, l.c., l.-5]. If a formula applies to  two cases [Jen, p.60, l.-4-l.-3], then the definition using a formula can be deceptive and ambitious. If one's consideration is limited to one case, the other will be left out. If one simply visualizes a figure without formulating the concept in a rigorously manner [Hec, p.159, Fig. 5.15], one's understanding is incomplete. By the way, the definition given in [Hec, p.158, l.c., l.-5] has an advantage: we can easily determine the sign of the primary focal length based on the sign convention regarding the object distance.

  36. The definition of an electric field involves a limiting process [Jack, p.25, l.2-l.11] and an idealization in macroscopic electrostatics [Jack, p.25, l.-3-l.-2].