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Introducing Quantum and Statistical Physics in the Footsteps of Einstein: A Proposal^{ †}

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## Abstract

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I have thought a hundred times more about quantum problems than I have about general relativity.(A. Einstein to O. Stern)

## 1. Introduction

## 2. Setting the Stage. Cavity Radiation and Planck’s Law

## 3. Part 1. Einstein 1905: Light Quanta

#### 3.1. Entropy of Thermal Radiation

#### 3.2. Some Applications of Light Quanta

## 4. Part 2. Einstein 1909: Wave-Particle Duality for Light

a theory of light which can be interpreted as a kind of fusion of the wave and the emission [that is, the corpuscular] theory

the wave structure and the quantum structure [...] are not to be considered mutually incompatible.

## 5. Part 3. Einstein 1916: Probability

## 6. Tools from Statistical Physics

#### 6.1. Boltzmann’s Formula and the Entropy of an Ideal Gas

#### 6.2. The Maxwell-Boltzmann Distribution

#### 6.3. The Gibbs Distribution and Energy Fluctuations

#### 6.4. Thermal Stability of the Ideal Gas

## 7. Implementation and Preliminary Results

## 8. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1. | Interestingly, Einstein himself adopted a point of view concerning the conceptual use of history in teaching similar to that expressed by the present authors in the wonderful book [7]. |

2. | It has even been proposed that it should be the starting point for teaching quantum physics [10] (this is also the teaching strategy of [3], but only in newer editions), since it avoids the subtleties of statistical physics. We may agree with this claim if the teaching–learning sequence is severely time constrained, but we think that students can benefit from a suitable treatment of statistical physics, and it is our point here that such a treatment can indeed help in grasping some quantum concepts. Moreover, thermal radiation is a very important topic in its own right, with many applications. |

3. | This result was later extended by him to the case of matter along the same lines [14]. Thus Einstein introduced wave–particle duality indipendently of De Broglie, and his arguments rest on a completely different basis. The two approaches can be regarded as complementary, since Einstein’s one is statistical in nature, while De Broglie’s one applies to individual (light or matter) quanta. |

4. | It is fair to recall that Einstein himself, unlike other founders of quantum mechanics, did not regard this introduction of probability as a fundamental one, but rather due to our ignorance of the true internal working of atoms. Hence, for him, this probabilistic aspect was not fundamentally different from those emerging in statistical physics. He was very disturbed by the lack of causality (“I, at any rate, am convinced that He is not playing at dice.” Letter from A. Einstein to M. Born, 4 December 1926, emphasis in original), and famously kept this viewpoint for all his life. |

5. | We point out in passing that for a better grasp of the topics of the path, and of elementary quantum theory in general, at least the qualitative idea that accelerated electric charge radiate electromagnetic waves has to be introduced. Because of the complexity of a full treatment, this topic is usually omitted from high school courses. However, this idea underlies thermal radiation and it was the main motivation for the inadequacy of Rutherford’s atom and its substitution with Bohr’s model. Hence, when presenting the material to students, we typically include this topic as a preliminary, with a brief discussion of Larmor’s formula, without derivation of course. |

6. | A word about notation: in this paper we deal with thermodynamic systems made up of a large number of elementary constituents, hence different notions of energy come into play. We denote by E the total energy of the system, by u the spectral (i.e., per unit frequency) energy density, and by $\epsilon $ the energy of an elementary constituent, such as a gas molecule or a light quantum. |

7. | Students should have the notion of the volumic energy density of an electromagnetic field. |

8. | In fact, before explaining it with energy quantization, Planck wrote it down by fitting the experimental curve. |

9. | |

10. | To avoid discussing Taylor series this can be easily explained by saying that the exponential function is approximated by its tangent in the point $x=0$ in a neighborhood of that point. |

11. | In fact this law (not to be confused with Wien’s displacement law) was obtained before Planck’s one by a semi empirical reasoning based on the Maxwell distribution, which indeed has a very similar mathematical form (cf. Section 6.2). |

12. | |

13. | The crucial idea is that, unlike the emission of a spherical electromagnetic wave, the emission of a light quantum is intrinsically directional, hence the atom recoils after the emission. |

14. | At this level, there is no way of computing the A and B coefficients. The computation of these coefficients by first principles was achieved in 1927 by Dirac, when quantum mechanics was fully developed and applied to radiation. |

15. | Another thing to notice is that, since there are many possible states in which the emitted quantum can go, the process of spontaneous emission is irreversible in a statistical sense. Actually, by engineering appropriate cavities, it is actually possible to change the number of states, which through Equation (27) allows us to actually manipulate the spontaneous emission probability, which can be suppressed or enhanced. For example, by considering a cavity which is so small that, so to speak, there is no room for the waves to be into it (i.e., if the dimensions of the cavity are smaller than the wavelength), the number of possible states can be greatly reduced, and this in fact can suppress spontaneous emission. In extreme cases, the number of possible states can be reduced to one, thus making the process of spontaneous emission reversible. Radiation stays in the cavity long enough that it can be reabsorbed by the atom before being dissipated, and this generates oscillations between lower and higher atomic states. Such phenomena are the object of a very active research area nowadays (see e.g., [37] for a very clear review). |

16. | The other two processes depend on a well defined cause, that is the molecule being hit by a light quantum, so in that case the probability is related to the occurrence of that event. Hence the probabilistic nature of spontaneous emission is on a very different footing with respect to that of absorption and stimulated emission. |

17. | True, the same law appeared even earlier in radioactive decay, but for many years nobody could tell whether the nucleus obeyed the same quantum laws which hold at the atomic scales. |

18. | This model of course is good for describing the equilibrium state, not the process of relaxation to it, for which it is necessary to consider collisions between molecules; the equilibrium state is then characterized as non-changing under collisions. Here we only consider equilibrium states, so we do not need to concern about this complication. |

19. | This is a simple generalization of the argument by which we say a priori that in tossing a coin we get heads with probability $1/2$. |

20. | |

21. | Usually the Maxwell distribution is expressed in terms of the number of molecules with velocity comprised in that interval, which is $dN\left(\mathbf{v}\right)=NdP\left(\mathbf{v}\right)$, where N is the total number of molecules. |

22. | Typically high school students are exposed to the Maxwell distribution without proof. In that case, instructors may consider devoting some time to a simple proof of it. A very nice and instructive one which can be considered is that given by Maxwell himself in 1860 [40]. This proof uses the fact that an ideal gas is isotropic, and the statistical independence of the probabilities associated with each component of the velocity. The latter observation implies that we may write the function f as a product of functions, each one expressing the probability associated with that component; by isotropy, the distribution of the velocities along the three directions must be the same, hence these three functions must be equal. Hence we may write: $f\left(\mathbf{v}\right)=g\left({v}_{x}\right)g\left({v}_{y}\right)g\left({v}_{z}\right)$, for some function g. However, isotropy also means that $f\left(\mathbf{v}\right)$ can depend on $\mathbf{v}$ only through its modulus v. These two conditions are satisfied by the function (36), since $A{e}^{-\frac{m{v}^{2}}{2{k}_{B}T}}=A{e}^{-\frac{m({v}_{x}^{2}+{v}_{y}^{2}+{v}_{z}^{2})}{2{k}_{B}T}}=A{e}^{-\frac{m{v}_{x}^{2}}{2{k}_{B}T}}{e}^{-\frac{m{v}_{y}^{2}}{2{k}_{B}T}}{e}^{-\frac{m{v}_{z}^{2}}{2{k}_{B}T}}$, hence $g\left(x\right)=\sqrt[3]{A}{e}^{-\frac{m{x}^{2}}{2{k}_{B}T}}$. |

23. | It can be computed by changing variables from v to $\epsilon $ in the differential, since it is defined by $4\pi {v}^{2}dv=\omega \left(\epsilon \right)d\epsilon $; however its precise form is of no interest to us. In fact $\omega \left(\epsilon \right)\sim \sqrt{\epsilon}$. |

24. | This is the canonical Gibbs distribution, which is the appropriate one to use if the system is exchanging only energy with the environment. This is the only case we consider here. |

25. | Einstein gave a derivation of this formula also in his 1909 paper [12], however that derivation involves Taylor series, and therefore it is not suited for high school students. It can nevertheless be employed when proposing this material to a more advanced audience. The 1909 derivation is actually quite interesting also because it again involves a reversal of Boltzmann’s principle, analogous to that he used in 1905, which we saw in Section 3 [42]. |

26. | We notice that, again because of the fact that the Maxwell-Boltzmann and the Gibbs distribution have the same form, a formally identical equation can be derived to describe the fluctuations of the energies $\epsilon $ of the molecules of the gas around the average value $\overline{\epsilon}$. |

27. | It is a so-called quantum ideal (Bose) gas, which like the usual (classical) ideal gas is made up by many identical particles. However, in this case the latter are not independent far from the Wien limit, because of quantum correlations (cf. the discussion at the end of Section 4). |

28. | It is in fact safe to assert that fluctuations constitute the Leitmotiv of most of Einstein’s work in statistical physics [8]. |

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**MDPI and ACS Style**

Di Mauro, M.; Esposito, S.; Naddeo, A.
Introducing Quantum and Statistical Physics in the Footsteps of Einstein: A Proposal. *Universe* **2021**, *7*, 184.
https://doi.org/10.3390/universe7060184

**AMA Style**

Di Mauro M, Esposito S, Naddeo A.
Introducing Quantum and Statistical Physics in the Footsteps of Einstein: A Proposal. *Universe*. 2021; 7(6):184.
https://doi.org/10.3390/universe7060184

**Chicago/Turabian Style**

Di Mauro, Marco, Salvatore Esposito, and Adele Naddeo.
2021. "Introducing Quantum and Statistical Physics in the Footsteps of Einstein: A Proposal" *Universe* 7, no. 6: 184.
https://doi.org/10.3390/universe7060184