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## Axioms in Mechanics

### Logic serves to organize the content of a subject and axiomatization serves to put us in a vantage point to begin our odyssey of logical reasoning.

1. Supporting evidence for the fundamental postulate of equal probabilities [Rei, p.54, l.- 13]:
1. Theoretical basis: a consequence of the H theorem [Rei, p.58, l.20].
2. Theoretical simplicity [Rei, p.54, l.18-l.22].
3. Small interactions [Rei, p.57, l.- 11 & p.58, Example 1].

2. The justification of [Coh, p.222, Sixth postulate] is not that it has been deduced entirely from information already known experimentally, but that it correctly predicts results which can be verified experimentally. For the case of a free particle, we can validate this postulate because Schrödinger's equation is consistent with the de Broglie-Einstein postulate and the corresponding principle [Eis, p.129, l.1-l.6]. However, if the potential energy of the system is V(x,t), we cannot verify the postulate in the general case, but we can postulate it to be true.
Remark. For the case of free particles, [Eis, p.129, l.1-132, l.2] gives a long argument which leads to the Schrödinger equation. In contrast, if we use photons radiated from harmonically oscillating sources [Wangs, p.472, (28-24)], the argument becomes much simpler.

3. [Lan2, p.24, §8] (free particles) starts with action to derive the formulas for momentum [Lan2, p.25, (9.1)] and energy [Lan2, p.26, l.2] using a formal scheme [Lan2, p.25, l.18 & p.26, l.2]. The derivation is not intuitive because {action, Lagrangian} is closely related to the abstract perturbation theory of differential equations.

### In contrast, [Jack, pp. 533-539] uses the conservation of momentum and energy to derive the formulas for momentum [Jack, p.536, (11.46)] and energy [Jack, p.536, (11,51)]. Jackson’s derivation is more intuitive, insightful, and to the point than Landau’s mainly because {energy, momentum} are more concrete than {action, Lagrangian}.

4. Confirming an axiomatic theory by experiments

### Originally, a postulate (e.g. the 2nd postulate [Jack, p.518, l.1] of Special Relativity) may be just a brave guess and insightful starting point of an axiomatic theory. In order to apply the theory to practical problems, confirmation of distant consequences (e.g. time dilation [Bow, p.19, l.15-l.20]) of the postulate is not sufficient. This is because we still cannot determine whether Postulate A is true even if the theory validates Statement "AŽB" and the experiment confirms the truth of Statement B. Certainly, we have made some progress, but the goal of completely validating the theory has not yet been achieved. Thus we must confirm the validity of the postulate itself by experiment. That is, we must establish the following fact [Jack, p.523, l.16]:

The speed of the emitted light is still c even when the speed of the light source is close to c.

5. Postulates are used as the foundation of a mathematical theory, but they have more functions in physics. In physics, we attempt to relate electromagnetic waves to light, so we assume the electric field and magnetic field proceed in a wave pattern [Hall, p.664, (38-2) & (38-3)]. Thus we make a brave guess to fill the reasoning gap and then derive a theoretical value of the speed of light [Hall,p.667, [38-1]]. At last we verify the theoretical result by experiments. Therefore, the postulates have the function of relating and predicting phenomena in physics. Furthermore, the experimental verification enhances our belief in the proposed postulates (See [Rei, p.48, l.11-l.16]).

6. A postulate is given mostly because it is a good starting point for a formal theory in the logical sense. However, this does not mean that a postulate can be given without any physical ground. For example, [Coh, p.222, Sixth Postulate] is supported by
1. the fact that time evolution can be described by a generating function [Go2, p.414, l.−15 & p.416, (9-120)] and
2. formal correspondence [Go2, p.401, l.−2].

7. Axioms can be used to determine if a principle is fundamental.
Suppose someone discovers a principle. We would like to develop an axiomatic theory to explain it. If it only takes a few steps to derive the principle from the axioms, we say that the principle is basic. If many useful results can be derived from the principle, then we say that the principle is useful. If the principle is basic and useful, then we say that the principle is fundamental. For example, Pauli's exclusion principle is fundamental [Coh, p.1389, l.−10-l.−5].

8. When a new axiom is introduced into a theory, we should modify the old axioms to accommodate the new one and verify that no contradictions arise. Furthermore, in the situations in which the new axiom can be ignored, we must show that the modifications to the old axioms are no longer unnecessary.

Example.
(Introduction to the symmetrization postulate)
1. Accommodation [Coh, p.1393, §4]
1. Measurements can be restricted in the physical subspace [Coh, p.1394, l.−7].
2. A physical state will remain in the physical subspace as time evolves [Coh, p.1396, l.14].
2. Undo the accommodation when the new axiom can be ignored [Coh, p.1407, l.8; p.1408, l.8].

9. Sometimes an axiom serves as a steppingstone for applying the old rules to new territories.
The status of an axiom should not prevent us from searching for its supporting evidence. In fact, its supporting evidence often inspires our thoughts.
In [Lan3, p.25, §8], we first show that ψ satisfies [Lan3, p.25, (8.1)], where H is a linear operator. Secondly, we show that H is Hermitian. Thirdly, we demonstrate that H is Hamilton's function in the quasi-classical case. Thus we can prove many essential facts about the sixth postulate [Coh, p.222, l.5] of quantum mechanics [1]. In this case, the axiom is only used as a steppingstone for applying the old rules to new territories. Any further proof to justify such a usage would be pointless.

10. A new axiom should be based on experimental facts even though the old theory fails to explain them [Eis, p.99, l.- 16-l.- 13].

11. In physics, it is difficult to determine whether the concept of eigenvalue or the concept of expectation value is more basic. In [Mer2, p.43, (3.72)], Merzbacher uses eigenvalue to introduce the concept of expectation value. In [Mer2, p.54, l.1-l.19], Merzbacher uses expectation value to introduce the concept of eigenvalue. Thus, the two concepts are intrinsically related and of equal rank in the logical sense. Each concept motivates us to study the other. Which concept should be introduced first will depend on the choice of familiarity as well as convenience.

12. Motivation for the Fourth Postulate of Quantum Mechanics [Coh, pp.216-218].
1. Through analogy [Coh, pp.14-15].
2. Through expanding the measurement from a definite value to the probabilistic case [Coh, p.19; Mer2, p.31, l.-1].
3. Through stipulating a special case [Lan3, p.6, l.15] of the axiom from which we may derive the rest cases of the axiom [Lan3, p.8, l.-1-p.9, l.13].
Remark. A particle's probabilistic behavior:
1. The diffraction experiment in [Schi, p.5, l.-8-l.-1] dictates the way we think about photons [Schi, p.6, l.1-l.2].
2. Due to the uncertainty principle, a particle's initial conditions of motion cannot be exactly specified [Eis, p.66, l.31-l.32].

13. The supporting evidence for the first postulate [Coh, p.215] of quantum mechanics.
1. The linear harmonic oscillator [Mer2, p.88, l.-6-l.-1].

14. Cohen-Tannoudji uses the definition of an observable to avoid presenting [Lev2, p.191, Postulate 4] as an axiom. Consequently, in general, there is no effective method for checking this definition. Since we cannot find a counterexample of [Lev2, p.191, Postulate 4], we had better treat it as an axiom.
Supporting evidence:
1. The eigenvectors of a Hermitian transformation in a n-dim vector space can be chosen to be orthonormal [Lev2, p.232, l.3].
2. [Bir, p.313, Theorem 11].
Remark. The differential operator (d/dx)[p(x)d/dx] [Bir, p.256, (1)] is Hermitian. The proof is similar to [Lev2, p.165, l.-3-p.166, l.10].

15. The symmetrization postulate [Coh, p.1386, l.-6-l.-1].
1.  To remove difficulties of path determination [Coh, p.1374, l.-1] and exchange degeneracy [Coh, p.1376, l.5], Cohen-Tannoudji proposes the symmetrization postulate [Coh, p.1387, chap. XIV, §C.2]. His supporting evidence tends to come from the negative side. Levine offers more constructive information about this postulate. First, the wave function of n identical particles must be either completely symmetric or completely antisymmetric [Lev2, p.287, l.22-23]. Second, the correct zeroth-order wave function for the first excited state of the helium-atom strongly suggests the exact form of the antisymmetric wave function of identical particles [Lev2, p.289, l.12]. Third, she narrows down the postulate to the Pauli principle [Lev2, p.287, l.-11]. Fourth, failure to consider the Pauli principle would yield calculations that contradict experimental evidence [Lev2, p.291, l.-2].
2. The construction rule of physical kets should not be stipulated as in [Coh, p.1388, §3.a, especially, p.1384, formula (B-50)]. It should be established by trial and error using a simple example [Lev2, p.292, l.12-p.293, l.-10].
3. The theoretic development of permutation operators [Coh, chap. XIV, §B] originates from the simple idea in [Lev2, p.287, l.19-l.21].

16. If only a small part of the statement of an axiom needs to be assumed to be true, we should be able to recognize that small part and understand why its truth needs to be assumed without proof.
Example. The second axiom in special relativity is needed merely to separate the two possibilities of relating initial frames and to determine the value of c [Rin, p.19, l.-7-l.-6].

17. The first law of thermodynamics
1. The adiabatic case [Zem, p.73, l.8-l.10]: For a change from a definite state to another state of the system, the same amount of work is required irrespective of the mechanism used to perform the work or the  intermediate states through which the system passes. Experimental evidence: Joule's experiments [Man, l.11, l.1-l.15]; theoretical basis: [Zem, p.73, l.-6].
2. The nonadiabatic case: We give the definition of heat based on the principle of the conservation of energy [Zem, 75, l.9-l.16].
Remark 1. In order to make the procedure of axiomization specific, the content of the first law must be divided into two cases. In the adiabatic case, the evidence is stronger and more tangible. Both [Hua1, p.7, l.2-l.7] and [Kit, p.49, l.4-l.6] fail to emphasize these two subtle points.
Remark 2. Even before he properly defines heat [Reif, p.73 l.16], Reif unnecessarily gives an extra case: thermal interaction [Reif, pp.66-68, §2.6]. His poor logical analysis also complicates his discussion of the first law: Reif unnecessarily repeats the definition of heat twice [Reif, p.67, l.-10].

18. Fano, Chu, and Adler's views on axioms and definitions [Fan, p.11, l.-16-p.12, l.3].