Thermodynamics of Radiation and Momentum (Part 2) Masud Mansuripur College of Optical Sciences, The University of Arizona, Tucson, Arizona, USA [Published in Complex Light and Optical Forces XII, edited by E. J. Galvez, D. L. Andrews, and J. Glückstad, Proceedings of SPIE Vol. 10549, 105490X~1-13 (2018). DOI: 10.1117/12.2286300]

Abstract. We derive some of the properties of blackbody radiation using thermodynamic identities. A few of the results reported earlier (in Part 1 of the present paper) will be re-derived here from a different perspective. We argue that the fluctuations of thermal radiation can be expressed as the sum of two contributions: one resulting from classical considerations in the limit when Planck’s constant goes to zero, and a second one that is rooted in the discrete nature of electromagnetic particles (). Johnson noise and the Nyquist theorem will be another topic of discussion, where the role played by blackbodyℏ radiation in generating noise within an electronic circuit will be emphasized. In the context of thermal fluctuations, we also analyze the various sources of noise in photodetection, relating the statistics of photo- electron counts to -density fluctuations associated with blackbody radiation. The remaining sections of the paper are devoted to an analysis of the thermodynamic properties of a monatomic under conditions of thermal equilibrium. This latter part of the paper aims to provide a basis for comparisons between a photon gas and a rarefied gas of identical rigid particles of matter. 1. Introduction. In a previous paper [1] we discussed the properties of blackbody radiation in general terms, emphasizing those aspects of the theory that pertain specifically to and photon momentum. In the present paper, we elaborate some of the ideas discussed in [1], derive some of the previously reported results from a different perspective, and compare the properties of a photon gas in thermal equilibrium (i.e., with the walls of an otherwise empty cavity) with the thermodynamic properties of a monatomic gas in thermal equilibrium with a reservoir at a given . In Sec.2 we derive the relation between the and the partition function for a closed system containing a fixed number of particles. Section 3 is devoted to a derivation of some of the fundamental𝑇𝑇 properties of thermodynamic systems under thermal equilibrium conditions. The thermodynamic identities obtained in these early parts of the paper will be needed in subsequent sections. Section 4 provides a brief derivation of the frequency at which the blackbody radiation density reaches its peak value. In Sec.5 we analyze the fluctuations of blackbody radiation in the classical limit when Planck’s constant goes to zero; the remaining term in the expression of the variance of blackbody radiation — that due to the discrete nature of photons — is examined in Sec.6. Sections 7 and 8 are devoted to a discussion of Johnsonℏ noise and the Nyquist theorem, where the prominent role of blackbody radiation in generating thermal noise within electronic circuits is emphasized. In Sec.9 we return to the topic of photon-number variance and examine the fluctuations of blackbody radiation in the context of idealized photodetection, relating the fluctuations of thermal light to the statistics of photodetection in the presence of chaotic illumination. The remaining sections of the paper are devoted to the thermodynamics of a simple monatomic gas under conditions of thermal equilibrium. We derive the Sackur-Tetrode expression for the of a monatomic gas, and proceed to discuss the various thermodynamic properties that can be extracted from the Sackur-Tetrode equation. The goal of the latter part of the paper is to provide a basis for comparisons between the thermo- dynamics of the photon gas and those of a rarefied gas consisting of rigid, identical, material particles. 2. The concept of free energy in thermodynamics. The Helmholtz free energy of an -particle system having, in equilibrium, , entropy , absolute temperature , pressure , and volume is defined as = [2− 4]. An important application of the concept𝐹𝐹 of free 𝑁𝑁energy is found in reversible isothermal expansion (or contraction)𝑈𝑈 processes.𝑆𝑆 Here the heat 𝑇𝑇 is given 𝑝𝑝to the system while𝑉𝑉 its volume 𝐹𝐹expands𝑈𝑈 − by𝑇𝑇𝑇𝑇 , delivering an amount of work = to the outside world. The change of entropy in this process is = , while the internal energy of the ∆system𝑄𝑄 changes by = . The pressure may ∆thus𝑉𝑉 be written as = ( ∆𝑊𝑊) 𝑝𝑝,∆ or,𝑉𝑉 considering that the temperature is kept constant, as = ( ∆𝑆𝑆 ∆)𝑄𝑄, ⁄.𝑇𝑇 The pressure in a reversible isothermal expansion (or ∆𝑈𝑈 ∆𝑄𝑄 − 𝑝𝑝∆𝑉𝑉 𝑝𝑝 − ∆𝑈𝑈 − 𝑇𝑇∆𝑆𝑆 ⁄∆𝑉𝑉 𝑝𝑝 − ∆𝐹𝐹⁄∆𝑉𝑉 𝑇𝑇 𝑁𝑁 1

contraction) is, therefore, given by = ( ) , . This expression should be contrasted with the formula = ( ) , , which is the pressure of a system undergoing a reversible adiabatic 𝑇𝑇 𝑁𝑁 expansion (or contraction). 𝑝𝑝 − 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑆𝑆 𝑁𝑁 An intimate𝑝𝑝 − 𝜕𝜕 connection𝜕𝜕⁄𝜕𝜕𝜕𝜕 exists between the partition function ( , , ) of an -particle thermo- dynamic system and its free energy . The following argument shows that = exp( ): 𝑍𝑍 𝑇𝑇 𝑁𝑁 𝑉𝑉 𝑁𝑁 ( ) 𝐵𝐵 1 = ln = ln = (1 ) + ln[ ⁄( , , )] 𝐹𝐹 ( , , ) 𝑍𝑍 −𝐹𝐹 𝑘𝑘 𝑇𝑇 exp −ℰ𝑚𝑚⁄𝑘𝑘𝐵𝐵𝑇𝑇 𝐵𝐵 𝐵𝐵 𝐵𝐵 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑆𝑆 −𝑘𝑘 ∑ 𝑝𝑝 𝑝𝑝 −𝑘𝑘 �𝑚𝑚 𝑝𝑝 � 𝑍𝑍 𝑇𝑇 𝑁𝑁 𝑉𝑉 � ⁄𝑇𝑇 ∑ 𝑝𝑝 ℰ 𝑘𝑘 𝑍𝑍 𝑇𝑇 𝑁𝑁 𝑉𝑉 ∑ 𝑝𝑝 = + ln[ ( , , )] ( , , ) = exp[ ( ) ]. (1)

𝐵𝐵 𝐵𝐵 Note →that the𝑇𝑇𝑇𝑇 above𝑈𝑈 arguments𝑘𝑘 𝑇𝑇 𝑍𝑍 are𝑇𝑇 𝑁𝑁predicated𝑉𝑉 →on the 𝑍𝑍assumpt𝑇𝑇 𝑁𝑁 𝑉𝑉ion that the− total𝑈𝑈 − number𝑇𝑇𝑇𝑇 ⁄𝑘𝑘 𝑇𝑇 of particles in the system is constant. 𝑁𝑁 3. Useful thermodynamic identities. The following thermodynamic identities are useful when dealing with a system of fixed volume and fixed number of particles , which is in thermal equilibrium while in thermal contact with a reservoir at temperature . Some of these identities were used by Einstein in his original papers on thermal radiation𝑉𝑉 [5−10]. 𝑁𝑁 𝑇𝑇 Partition function: ( , , ) = exp( ) = exp . (2) 𝜕𝜕𝜕𝜕 1 𝜀𝜀𝑚𝑚 𝑚𝑚 𝑚𝑚 𝐵𝐵 2 𝑚𝑚 𝐵𝐵 𝐵𝐵 Average energy: 𝑍𝑍(𝑇𝑇, 𝑉𝑉, 𝑁𝑁) = ∑ exp−𝜀𝜀( ⁄𝑘𝑘 𝑇𝑇 →) 𝜕𝜕 𝜕𝜕 𝑘𝑘 𝑇𝑇 �𝑚𝑚 𝜀𝜀 �− 𝑘𝑘 𝑇𝑇�

𝑚𝑚 𝑚𝑚 𝑚𝑚 𝐵𝐵 ( ) ℰ 𝑇𝑇=𝑉𝑉 𝑁𝑁 ∑ 𝜀𝜀 exp−𝜀𝜀 ⁄𝑘𝑘 𝑇𝑇 ⁄𝑍𝑍 exp

𝜕𝜕ℰ 1 2 𝜀𝜀𝑚𝑚 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝜀𝜀𝑚𝑚 2 𝑚𝑚 2 𝑚𝑚 𝐵𝐵 𝐵𝐵 𝐵𝐵 → 𝜕𝜕𝜕𝜕 𝑘𝑘 𝑇𝑇 �𝑚𝑚 𝜀𝜀 ( ) �− 𝑘𝑘 𝑇𝑇�⁄𝑍𝑍 − 𝑍𝑍 �𝑚𝑚 𝜀𝜀 �− 𝑘𝑘 𝑇𝑇� = = (3) 2 2 · 𝜕𝜕ℰ 〈𝜀𝜀 〉 − 〈𝜀𝜀〉 Var ℰ 2 2 Entropy: → 𝜕𝜕𝜕𝜕( , , 𝑘𝑘𝐵𝐵)𝑇𝑇= 𝑘𝑘𝐵𝐵𝑇𝑇 ln

𝐵𝐵 𝑚𝑚 𝑆𝑆 𝑇𝑇 𝑉𝑉 𝑁𝑁 = −𝑘𝑘 ∑ 𝑝𝑝𝑚𝑚 exp𝑝𝑝𝑚𝑚( ) [ ( ) + ln ] 𝐵𝐵 −1 𝐵𝐵 𝐵𝐵 = 𝑘𝑘( �)𝑚𝑚+𝑍𝑍 ln .− 𝜀𝜀𝑚𝑚⁄𝑘𝑘 𝑇𝑇 𝜀𝜀𝑚𝑚⁄ 𝑘𝑘 𝑇𝑇 𝑍𝑍 (4)

𝐵𝐵 ( ) = ℰ+⁄𝑇𝑇 ( 𝑘𝑘 𝑍𝑍) = = · (5) 𝜕𝜕𝜕𝜕 𝜕𝜕ℰ⁄𝜕𝜕𝜕𝜕 ℰ 𝜕𝜕ℰ⁄𝜕𝜕𝜕𝜕 Var ℰ 2 𝐵𝐵 3 𝜕𝜕𝜕𝜕 𝑇𝑇 𝑇𝑇 ⁄ ⁄ 𝑇𝑇 𝑘𝑘𝐵𝐵𝑇𝑇 → = −= · 𝑘𝑘 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑍𝑍 (6) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 1 → 𝜕𝜕ℰ 𝜕𝜕ℰ⁄𝜕𝜕𝜕𝜕 𝑇𝑇 = = = = = 2 ( ) ( ) 𝜕𝜕 𝑆𝑆 𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕 1 𝜕𝜕𝜕𝜕⁄𝜕𝜕ℰ 1 𝑘𝑘𝐵𝐵 2 2 2 → 𝜕𝜕ℰ 𝜕𝜕ℰ �𝜕𝜕ℰ� 𝜕𝜕ℰ �𝑇𝑇� − 𝑇𝑇 − 𝑇𝑇 𝜕𝜕ℰ⁄𝜕𝜕𝜕𝜕 − Var ℰ Var( ) = . (7) 2 −1 𝜕𝜕 𝑆𝑆 𝐵𝐵 2 In the case of→ a photonℰ gas −trapped𝑘𝑘 �𝜕𝜕 ℰwithin� (and in equilibrium with the walls of) an otherwise empty chamber at temperature , the above identities may be verified as follows (see Ref.[1], Eq.(35)):

( , ) = 𝑇𝑇 Var( ) = = . (8) 2 4 2 5 𝜋𝜋 𝑘𝑘𝐵𝐵 4 2 𝜕𝜕ℰ 4𝜋𝜋 𝑘𝑘𝐵𝐵 5 3 3 3 3 ℰ 𝑇𝑇 𝑉𝑉 �15𝑐𝑐 ℏ � 𝑉𝑉𝑇𝑇 → ℰ 𝑘𝑘𝐵𝐵𝑇𝑇 �𝜕𝜕𝜕𝜕� �15𝑐𝑐 ℏ � 𝑉𝑉𝑇𝑇 ( , ) = Var( ) = = . (9) 2 4 2 5 4𝜋𝜋 𝑘𝑘𝐵𝐵 3 3 𝜕𝜕𝜕𝜕 4𝜋𝜋 𝑘𝑘𝐵𝐵 5 3 3 3 3 𝑆𝑆 𝑇𝑇 𝑉𝑉 �45𝑐𝑐 ℏ � 𝑉𝑉𝑇𝑇 → ℰ 𝑘𝑘𝐵𝐵𝑇𝑇 �𝜕𝜕𝜕𝜕� �15𝑐𝑐 ℏ � 𝑉𝑉𝑇𝑇 2

( , ) = ¾ = ¼ = · (10) 4 2 4 4 2 4 4 𝜋𝜋 𝑘𝑘𝐵𝐵𝑉𝑉 𝜕𝜕𝜕𝜕 𝜋𝜋 𝑘𝑘𝐵𝐵𝑉𝑉 − 1 3 3 3 3 𝑆𝑆 ℰ 𝑉𝑉 3 ��15𝑐𝑐 ℏ � ℰ → 𝜕𝜕ℰ ��15𝑐𝑐 ℏ � ℰ 𝑇𝑇 = = 2 4 2 4 2 4 5 −1 𝜕𝜕 𝑆𝑆 1 𝜋𝜋 𝑘𝑘𝐵𝐵𝑉𝑉 −5⁄4 4𝜋𝜋 𝑘𝑘𝐵𝐵𝑉𝑉𝑇𝑇 2 3 3 3 3 → 𝜕𝜕ℰ − 4 ��15𝑐𝑐 ℏ � ℰ − � 15𝑐𝑐 ℏ � Var( ) = = . (11) 2 −1 2 5 𝜕𝜕 𝑆𝑆 4𝜋𝜋 𝑘𝑘𝐵𝐵 5 𝐵𝐵 2 3 3 4. The peak frequency→ of blackbodyℰ −ra𝑘𝑘diation�𝜕𝜕ℰ �. The radia�15𝑐𝑐ntℏ energy� 𝑉𝑉𝑇𝑇 per unit area per unit time per unit angular frequency emanating from a blackbody at temperature was given in Ref.[1], Eq.(10) as follows:

( , ) = 𝑇𝑇 · (12) [ ( 3 ) ] ℏ𝜔𝜔 2 2 Defining = , the𝐼𝐼 above𝜔𝜔 𝑇𝑇 expression4𝜋𝜋 𝑐𝑐 exp mayℏ𝜔𝜔 be⁄𝑘𝑘𝐵𝐵 written𝑇𝑇 − 1 [( ) (2 ) ] [exp( ) 1]. ( ) ( ) The derivative of this expression𝐵𝐵 with respect to vanishes when exp𝐵𝐵 3 = 1 2 33 . The solutions of this equation,𝑥𝑥 whichℏ𝜔𝜔⁄ 𝑘𝑘can𝑇𝑇 be found graphically, are = 0 and 𝑘𝑘2𝑇𝑇.82144⁄ .𝜋𝜋𝜋𝜋 Theℏ peak𝑥𝑥 ⁄ intensity𝑥𝑥 − thus occurs at 2.82144 , where ( 𝑥𝑥, ) 1.4214( ) −(2𝑥𝑥 ) .− 𝑥𝑥⁄ 𝑥𝑥 𝑥𝑥 ≅ peak 𝐵𝐵 peak 𝐵𝐵 3 2 5. Blackbody𝜔𝜔 radiation≅ fluctuations𝑘𝑘 𝑇𝑇⁄ℏ in 𝐼𝐼the𝜔𝜔 classical𝑇𝑇 ≅ limit when𝑘𝑘 𝑇𝑇 ⁄ goes𝜋𝜋𝜋𝜋 ℏto zero. According to the Rayleigh-Jeans theory, each EM mode inside a blackbody cavity has of energy (on average), which ℏ is divided equally between its constituent electric and magnetic field component𝐵𝐵 s. The Maxwell-Lorentz electrodynamics estimates the number of modes per unit volume in the𝑘𝑘 𝑇𝑇frequency range ( , + d ) to be ( )d . Let us denote by the -field amplitude of a single mode having frequency in 𝜔𝜔 𝜔𝜔 𝜔𝜔 thermal2 equilibrium2 3 with the cavity walls0 . The total energy of the mode integrated over the volume of the cavity𝜔𝜔 ⁄𝜋𝜋 is𝑐𝑐 thus𝜔𝜔 given by = ½ 𝐸𝐸 . In𝐸𝐸 general, the single-mode energy is a function of𝜔𝜔 the 𝑉𝑉 temperature , which could also depend0 02 on frequency (e.g., the Planck mode), but it does not depend on the propagation direction 𝔼𝔼 inside𝜀𝜀 the𝐸𝐸 𝑉𝑉cavity, nor does it depend on the polarization𝔼𝔼 state of the mode. Most significantly,𝑇𝑇 the mode energy does not depend𝜔𝜔 on the volume of the cavity. If the cavity becomes larger, the -field amplitude𝒌𝒌 must become correspondingly smaller, so that the mode energy 𝔼𝔼 𝑉𝑉 would remain constant. 0 Consider a small𝐸𝐸 volume , 𝐸𝐸where = 2 is the wavelength associated with the frequency 𝔼𝔼 of the mode. Given that (on average)3 the total energy of the modes in the frequency range ( , + d ) throughout the entire volume 𝑣𝑣 ≪at any𝜆𝜆 given 𝜆𝜆instant𝜋𝜋𝜋𝜋 of⁄ 𝜔𝜔time will be ( )d , multiplication by 𝜔𝜔 yields the corresponding (average) energy within the small volume2 as follows:2 3 𝜔𝜔 𝜔𝜔 𝜔𝜔 𝑉𝑉 𝜔𝜔 𝑉𝑉𝑉𝑉⁄𝜋𝜋 𝑐𝑐 𝜔𝜔 Ave[ ( , , )] = ( )(½� )d = d = ( , ) d 𝑣𝑣 ⁄𝑉𝑉 𝐸𝐸 2 𝑣𝑣 . (13) 2 2 3 2 𝜔𝜔 𝔼𝔼 0 0 2 3 Next, we examine𝐸𝐸� 𝜔𝜔 𝑇𝑇 energy𝑣𝑣 fluctuations𝜔𝜔 𝑣𝑣⁄𝜋𝜋 𝑐𝑐 within𝜀𝜀 a𝐸𝐸 small𝑉𝑉 𝜔𝜔 volume�𝜋𝜋 𝑐𝑐 .� Taking𝑣𝑣 𝜔𝜔 aℰ single𝜔𝜔 𝑇𝑇 𝑣𝑣mode𝜔𝜔 of energy as a micro-canonical ensemble, statistical mechanics requires that Var( ) = ( ); see Eq.(3). In 𝑣𝑣 𝔼𝔼 the Rayleigh-Jeans regime, where = , we find Var( ) = ( ) = .𝐵𝐵 Since2 the various modes are 𝔼𝔼 𝑘𝑘 𝑇𝑇 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 independent of each other, the total variance𝐵𝐵 of energy within the 𝐵𝐵volume2 2 of the cavity in the frequency range ( , + d ) is readily seen𝔼𝔼 to𝑘𝑘 be𝑇𝑇 ( 𝔼𝔼 )d .𝑘𝑘 If𝑇𝑇 we now𝔼𝔼 assume that, within non- overlapping small volumes , energy fluctuations2 occur2 2 independently3 of𝑉𝑉 each other, we must multiply the above𝜔𝜔 total𝜔𝜔 variance𝜔𝜔 by to arrive at 𝜔𝜔 𝑉𝑉𝔼𝔼 ⁄𝜋𝜋 𝑐𝑐 𝜔𝜔 𝑣𝑣 Var[ ( , 𝑣𝑣⁄, 𝑉𝑉)] = ( )d = [ ( , ) ] d . (14) 2 2 2 3 2 3 2 A comparison with𝐸𝐸� Eq.(33)𝜔𝜔 𝑇𝑇 𝑣𝑣 in Ref.[1]𝜔𝜔 𝔼𝔼 now𝑣𝑣⁄𝜋𝜋 reveals𝑐𝑐 𝜔𝜔 that𝜋𝜋 these𝑐𝑐 ℰ classical𝜔𝜔 𝑇𝑇 ⁄𝜔𝜔 fluctuations𝑣𝑣 𝜔𝜔 are consistent with Planck’s blackbody radiation formula in the limit when 0.

ℏ → 3

6. Blackbody radiation fluctuations traceable to the discrete nature of photons. Consider identical, non-interacting particles distributed randomly and independently within an otherwise empty box of volume . The probability that of these particles end up together in a smaller volume inside𝑁𝑁 the box is

𝑉𝑉 𝑛𝑛 = 1 . 𝑣𝑣 (15) 𝑣𝑣 𝑛𝑛 𝑣𝑣 𝑁𝑁−𝑛𝑛 𝑁𝑁 𝑝𝑝𝑛𝑛 � � �𝑉𝑉� � − 𝑉𝑉� The binomial expansion formula ( + 𝑛𝑛) = confirms that = 1. Setting = 1 , we proceed to compute the 1st and 2𝑁𝑁nd derivatives𝑁𝑁 𝑁𝑁 with𝑛𝑛 𝑁𝑁respect−𝑛𝑛 to of the following𝑁𝑁 identity: 𝑥𝑥 𝑦𝑦 ∑𝑛𝑛=0 � � 𝑥𝑥 𝑦𝑦 ∑𝑛𝑛=0 𝑝𝑝𝑛𝑛 𝑛𝑛 𝑦𝑦 − 𝑥𝑥 ( ) = (1 ) = 1. 𝑥𝑥 (16) 𝑁𝑁 𝑁𝑁 𝑛𝑛 𝑁𝑁−𝑛𝑛 st 𝑆𝑆 𝑥𝑥 � � � 𝑥𝑥 − 𝑥𝑥 1 derivative: ( ) = [ (1 𝑛𝑛=0) 𝑛𝑛 ( ) (1 ) ] = 0 ′ 𝑁𝑁 𝑛𝑛−1 𝑁𝑁−𝑛𝑛 𝑛𝑛 𝑁𝑁−𝑛𝑛−1 𝑆𝑆 𝑥𝑥 � � � 𝑛𝑛𝑥𝑥 − 𝑥𝑥 − 𝑁𝑁 − 𝑛𝑛 𝑥𝑥 − 𝑥𝑥 + 𝑛𝑛 (1 ) (1 ) = 0

1 1 𝑁𝑁 𝑁𝑁 𝑛𝑛 𝑁𝑁−𝑛𝑛 𝑁𝑁 𝑛𝑛 𝑁𝑁−𝑛𝑛 → �𝑥𝑥 1−𝑥𝑥 � � 𝑛𝑛 � � 𝑥𝑥 − 𝑥𝑥 − 1−𝑥𝑥 � � � 𝑥𝑥 − 𝑥𝑥 𝑛𝑛 𝑛𝑛 (1 ) = . (17) 𝑁𝑁 𝑁𝑁 𝑛𝑛 𝑁𝑁−𝑛𝑛 → � 𝑛𝑛 � � 𝑥𝑥 − 𝑥𝑥 𝑁𝑁𝑁𝑁 A comparison of Eq.(1𝑛𝑛5=)0 with𝑛𝑛 Eq.(17) reveals that = . As for the variance of , we need to differentiate Eq.(17) with respect to . We will have 〈𝑛𝑛〉 𝑁𝑁𝑁𝑁⁄𝑉𝑉 𝑛𝑛 2nd derivative: [ (1 𝑥𝑥 ) ( ) (1 ) ] = 𝑁𝑁 𝑛𝑛−1 𝑁𝑁−𝑛𝑛 𝑛𝑛 𝑁𝑁−𝑛𝑛−1 � 𝑛𝑛 � � 𝑛𝑛𝑥𝑥 − 𝑥𝑥 − 𝑁𝑁 − 𝑛𝑛 𝑥𝑥 − 𝑥𝑥 𝑁𝑁 𝑛𝑛 + (1 ) (1 ) = 1 1 𝑁𝑁 2 𝑁𝑁 𝑛𝑛 𝑁𝑁−𝑛𝑛 𝑁𝑁 𝑛𝑛 𝑁𝑁−𝑛𝑛 → �𝑥𝑥 1−𝑥𝑥� � 𝑛𝑛 � � 𝑥𝑥 − 𝑥𝑥 − 1−𝑥𝑥 � 𝑛𝑛 � � 𝑥𝑥 − 𝑥𝑥 𝑁𝑁 (1 𝑛𝑛 ) = + (1 ). 𝑛𝑛 (18) 2 𝑁𝑁 𝑛𝑛 𝑁𝑁−𝑛𝑛 2 2 → � 𝑛𝑛 � � 𝑥𝑥 − 𝑥𝑥 𝑁𝑁 𝑥𝑥 𝑁𝑁𝑁𝑁 − 𝑥𝑥 Comparing Eq.(15) with Eq.(18𝑛𝑛), we find the variance of , as follows: Var( ) = = ( ) ( 𝑛𝑛 ) = [1 ( )] . (19) 2 2 2 If , then Var𝑛𝑛 ( )〈𝑛𝑛 〉 − .〈 𝑛𝑛Suppose〉 𝑁𝑁 now𝑣𝑣⁄𝑉𝑉 that− 𝑁𝑁 the𝑣𝑣 ⁄particles𝑉𝑉 are− known𝑣𝑣⁄𝑉𝑉 to〈𝑛𝑛 have〉 a fixed energy, say, they are identical photons of energy . Then = while Var( ) ( ) = . As pointed𝑣𝑣 out≪ 𝑉𝑉by Einstein 𝑛𝑛[6,7≅],〈 𝑛𝑛this〉 constitutes the first term in the variance of the energy2 -density of blackbody radiation appearing on the rightℏ𝜔𝜔-hand-side〈ℰ〉 of Eq.(33)〈𝑛𝑛〉ℏ𝜔𝜔 in Ref. [1]. ℰ ≅ 〈𝑛𝑛〉 ℏ𝜔𝜔 ℏ𝜔𝜔〈ℰ〉 In contrast, if the chamber is filled with a simple (monatomic) gas, then the average energy of the gas occupying volume will be = 3 2, but computing the variance of energy will require

accounting for fluctuations in the energy of each𝐵𝐵 particle in addition to those of the number of particles. 𝑣𝑣 〈ℰ〉 〈𝑛𝑛〉𝑘𝑘 𝑇𝑇⁄ 7. Johnson noise and the Nyquist theorem. In the circuit shown in the 𝑛𝑛 ( ) figure, the -turn inductor has length , cross-sectional area , and inductance = = = . Here is the𝑅𝑅𝑅𝑅𝑅𝑅 total magnetic flux, and 𝒾𝒾 𝑡𝑡 + = 𝑛𝑛= is the magnetic2 fieldℓ inside the inductor𝒜𝒜— both produced 0 0 0 0 0 0 𝐿𝐿 Φ ⁄𝐼𝐼 𝑛𝑛𝑛𝑛𝜇𝜇 𝐻𝐻⁄𝐼𝐼 𝜇𝜇 𝑛𝑛 𝒜𝒜⁄ℓ Φ ( ) by the current0 0. Assuming the thermal noise that accompanies the resistance 𝑅𝑅 𝐻𝐻 is modelled𝐵𝐵⁄𝜇𝜇 𝑛𝑛 𝐼𝐼as⁄ aℓ separate voltage source, ( ), and denoting the resulting 𝑠𝑠 0 𝑣𝑣 𝑡𝑡 𝑆𝑆 𝐿𝐿 current by ( )𝐼𝐼, the current ( ) exp( i ) and voltage ( ) exp( i ) at 𝑠𝑠 − 𝑅𝑅the frequency will be related by 𝑣𝑣 𝑡𝑡 ∼ 𝐶𝐶 𝒾𝒾 𝑡𝑡 𝐼𝐼 𝜔𝜔 − 𝜔𝜔𝜔𝜔 𝑉𝑉𝑠𝑠 𝜔𝜔 − 𝜔𝜔𝜔𝜔 𝜔𝜔 4

( ) = + i ( ). (20) 1 At the resonance frequency 𝑉𝑉𝑠𝑠 𝜔𝜔, the impedance�𝑅𝑅 �𝐿𝐿𝐿𝐿 of− the𝐶𝐶𝐶𝐶 �circuit� 𝐼𝐼 𝜔𝜔 is at a minimum and, therefore, =

1. If the circuit happens to have a0 large quality-factor , the excited frequencies will be confined to02 a narrow band centered at . The 𝜔𝜔magnetic field energy stored in the inductor is the product of the𝐿𝐿𝐿𝐿 𝜔𝜔-field 𝑄𝑄 energy-density and the volume0 of the inductor, that is, 𝜔𝜔 𝐻𝐻 ( ) = ½ = ½ ( ) = ½ | ( )| . (21)

0 2 0 2 2 Considering that ℰthe𝜔𝜔 current𝜇𝜇 is𝐻𝐻 sinusoidal,𝒜𝒜ℓ 𝜇𝜇 namely,𝑛𝑛𝑛𝑛⁄ℓ 𝒜𝒜| (ℓ )| sin𝐿𝐿(𝐼𝐼 𝜔𝜔+ ), the time-averaged energy stored in the inductor is only one-half of ( ) given by Eq.(21). However, since we are interested in the total energy of the system (that is, the sum of the stored𝐼𝐼 𝜔𝜔 in the𝜔𝜔𝜔𝜔 inductor𝜑𝜑 and capacitor), the time- independent ( ) of Eq.(21) is a good estimateℰ 𝜔𝜔 for the total energy. From Eq.(20) we find

( ) ( ) ( ) ( ) | ( )| = ℰ 𝜔𝜔 = = · 2 2 2 2 ( ) 𝑠𝑠 ( 𝑠𝑠 0) ( 𝑠𝑠 0 ) 𝑠𝑠 0 2 𝑉𝑉 𝜔𝜔 𝑉𝑉 𝜔𝜔 + 𝛿𝛿𝛿𝛿( ) 𝑉𝑉 𝜔𝜔 + 𝛿𝛿𝛿𝛿 𝑉𝑉 𝜔𝜔 + 𝛿𝛿𝛿𝛿 1 2 1 2 2 2 2 2 2 2 2 𝜔𝜔0 − 𝛿𝛿𝛿𝛿 𝑅𝑅 + 4𝐿𝐿 𝛿𝛿𝛿𝛿 𝐼𝐼 𝜔𝜔 𝑅𝑅 +�𝐿𝐿𝜔𝜔 − 𝐶𝐶𝐶𝐶� 𝑅𝑅 +�𝐿𝐿 𝜔𝜔0+ 𝛿𝛿𝛿𝛿 − 𝐶𝐶 𝜔𝜔 + 𝛿𝛿𝛿𝛿 � ≅ 𝑅𝑅 + 𝐿𝐿 𝜔𝜔0+ 𝛿𝛿𝛿𝛿 − 2 (22) 0 � 𝐶𝐶𝜔𝜔0 � Since the assumed circuit has a narrow bandwidth, it is reasonable to suppose that ( ) over this bandwidth is fairly uniform, in which case, the integral of the squared current becomes 𝑅𝑅𝑅𝑅𝑅𝑅 𝑉𝑉𝑠𝑠 𝜔𝜔 ( ) ( ) ( ) ( ) | ( )| d d = = · (23) ∞ 2 2 ∞ ( ) 2 ∞ 2 ∞ 2 𝑉𝑉𝑠𝑠 𝜔𝜔0+ 𝜁𝜁 𝑉𝑉𝑠𝑠 𝜔𝜔0 d𝜁𝜁 𝑉𝑉𝑠𝑠 𝜔𝜔0 d𝑥𝑥 𝜋𝜋𝑉𝑉𝑠𝑠 𝜔𝜔0 2 2 2 2 2 2 𝜔𝜔=0 𝑅𝑅 + 4𝐿𝐿 𝜁𝜁 𝑅𝑅 1 + 2𝐿𝐿𝐿𝐿⁄𝑅𝑅 2𝑅𝑅𝑅𝑅 1+𝑥𝑥 2𝑅𝑅𝑅𝑅 The∫ average𝐼𝐼 𝜔𝜔 electromagnetic𝜔𝜔 ≅ �−∞ energy𝜁𝜁 stored≅ in the�−∞ form of magnetic field� in−∞ the inductance and also in the form of electric field in the capacitance is thus given by Eqs.(21) and (23), as follows: 𝐿𝐿 ( ) ( ) = ½ | 𝐶𝐶( )| d = = ½ . (24) 2 ∞ 𝜋𝜋𝑉𝑉𝑠𝑠 𝜔𝜔0 0 2 𝐵𝐵 In the above equation,〈ℰ 𝜔𝜔we 〉have invoked𝐿𝐿 ∫0 𝐼𝐼 𝜔𝜔the equipartition𝜔𝜔 4𝑅𝑅 theorem 𝑘𝑘of𝑇𝑇 statistical mechanics [2−4], which states that each degree of freedom of a system in thermal equilibrium at temperature has, on average, the energy ½ , where is the Boltzmann constant. The noise power-density per unit 𝑇𝑇 (angular) frequency associated𝐵𝐵 with the𝐵𝐵 resistance is thus given by 𝑘𝑘 𝑇𝑇 𝑘𝑘 ( ) =𝑅𝑅 (2 ) . (25) 2 𝐵𝐵 Note that the noise power-density𝑉𝑉𝑠𝑠 per𝜔𝜔 unit⁄𝑅𝑅 frequency⁄𝜋𝜋 𝑘𝑘 is𝑇𝑇 independent of the frequency ; this is characteristic of white noise. The noise power within a frequency range d = 2 d is thus given by ( )d = (2 ) d = 4 d . This result is often referred to as the Johnson-Nyquist theorem.𝜔𝜔 𝜔𝜔 𝜋𝜋 𝑓𝑓 𝐵𝐵 𝐵𝐵 𝑃𝑃8. 𝜔𝜔An 𝜔𝜔alternative⁄𝜋𝜋 𝑘𝑘 derivation𝑇𝑇 𝜔𝜔 𝑘𝑘 of𝑇𝑇 the𝑓𝑓 Johnson-Nyquist theorem. Let the resistor be connected to another resistor having the same resistance via a perfectly matched transmission line. The length of the transmission line is , its characteristic impedance is , and 𝑅𝑅

the propagation velocity of the EM fields along the line is . Transmission Line If we model the noiseℓ produced by the resistor at temperature𝑅𝑅 ( ) 𝑉𝑉 as a voltage source connected in series to the resistor, then the current flowing in the circuit will be ( ) = 𝑠𝑠 𝑇𝑇 ( ) 2 , which transmits𝑣𝑣 𝑡𝑡 an average electrical power ℓ 𝑅𝑅 𝑅𝑅 ( ) = ( ) 4 to the second resistor. Here𝒾𝒾 𝑡𝑡 the 𝑠𝑠 𝑣𝑣angled𝑡𝑡2 ⁄ brackets𝑅𝑅 2 represent statistical averaging. 〈𝑅𝑅𝒾𝒾 The𝑡𝑡 〉 EM〈𝑣𝑣 fields𝑠𝑠 𝑡𝑡 〉 ⁄inside𝑅𝑅 the transmission line may now be considered as blackbody radiation in a one- dimensional box. The allowed frequencies inside the box must have wavelengths such that the length of the box is an integer multiple of 2, that is, = 2 = / . Consequently, 𝜔𝜔𝑛𝑛 𝜆𝜆𝑛𝑛 ℓ 𝜆𝜆𝑛𝑛⁄ ℓ 𝑛𝑛𝜆𝜆𝑛𝑛⁄ 𝑛𝑛𝑛𝑛𝑛𝑛 𝜔𝜔𝑛𝑛 5

= / . The allowed frequencies are thus seen to be separated from each other by = / , indicating that, within a range of frequencies d = 2 d , the number of allowed modes inside the 𝑛𝑛 transmission𝜔𝜔 𝑛𝑛𝑛𝑛𝑛𝑛 ℓ line is d = ( / )d . 𝛿𝛿𝛿𝛿 𝜋𝜋𝜋𝜋 ℓ In accordance with the Planck distribution, 𝜔𝜔the average𝜋𝜋 𝑓𝑓 energy content of each mode is given by = [exp( 𝜔𝜔⁄𝛿𝛿𝛿𝛿) 1]2ℓ, which,𝑉𝑉 𝑓𝑓 in the classical limit (i.e., ), reduces to .

Needless to say, this is𝐵𝐵 twice the thermal energy content per mode, ½ ,𝐵𝐵 associated with a classical𝐵𝐵 〈ℰ〉 ℏ𝜔𝜔⁄ ℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇 − ℏ𝜔𝜔 ≪ 𝑘𝑘 𝑇𝑇 〈ℰ〉 ≅ 𝑘𝑘 𝑇𝑇 oscillator in one dimension. The factor of 2 arises from the fact that each𝐵𝐵 mode of the transmission line could have two independent polarization states. All in all, the blackbody 𝑘𝑘radiation𝑇𝑇 inside the transmission line has energy (2 / )d within the frequency range d . Dividing this entity by yields the energy per unit length, a further𝐵𝐵 division by 2 yields the energy density that flows either to the right or the left along the transmission𝑘𝑘 𝑇𝑇ℓ line,𝑉𝑉 and,𝑓𝑓 finally, multiplication by the𝑓𝑓 wave velocity yieldsℓ the rate at which EM energy is transferred to each resistor at either end of the line, namely, d . A comparison with our 𝑉𝑉 earlier result now shows that ( ) = 4 d . This completes our alternative𝐵𝐵 proof of the Johnson- 𝑘𝑘 𝑇𝑇 𝑓𝑓 Nyquist theorem. 2 𝐵𝐵 〈𝑣𝑣𝑠𝑠 𝑡𝑡 〉 𝑘𝑘 𝑇𝑇𝑇𝑇 𝑓𝑓 9. Photodetection statistics. The following argument follows largely in the footsteps of E.M. Purcell [11]. Let the cycle-averaged (also known as low-pass-filtered) intensity of quasi-monochromatic radiation arriving at an ideal, small-area photodetector be denoted by ( ). Within a short time-interval ( , + ), the probability that a single photo-electron is released is ( ) , where is a characteristic parameter of the photodetector for the incident radiation, which is centered𝐼𝐼 𝑡𝑡at the frequency and has linewidth𝑡𝑡 𝑡𝑡 ∆𝑡𝑡 . 𝛼𝛼𝛼𝛼 𝑡𝑡 ∆𝑡𝑡 𝛼𝛼 [ ] We denote by ( , ) the probability that photo-electrons are released in the 0time interval 0, . For = 0 we will have 𝜔𝜔 ∆𝜔𝜔 𝑝𝑝 𝑛𝑛 𝑡𝑡 𝑛𝑛 ( , ) 𝑡𝑡 𝑛𝑛 (0, + ) = (0, )[1 ( ) ] = ( ) (0, ) d𝑝𝑝 0 𝑡𝑡 𝑝𝑝 𝑡𝑡 ∆𝑡𝑡 𝑝𝑝 𝑡𝑡 (−0,𝛼𝛼𝛼𝛼) =𝑡𝑡 ∆exp𝑡𝑡 → ( ) d𝑡𝑡 . −𝛼𝛼𝛼𝛼 𝑡𝑡 𝑝𝑝 𝑡𝑡 (26) 𝑡𝑡 ′ ′ Similarly, the probability→ that 𝑝𝑝 photo𝑡𝑡 -electrons�−𝛼𝛼 ∫are0 𝐼𝐼 𝑡𝑡released𝑑𝑑𝑡𝑡 � between = 0 and can be derived from the following first-order difference-differential equation: 𝑛𝑛 𝑡𝑡 𝑡𝑡 ( + 1, + ) = ( + 1, )[1 ( ) ] + ( , ) ( )

( , ) 𝑝𝑝 𝑛𝑛 𝑡𝑡 ∆𝑡𝑡 𝑝𝑝 =𝑛𝑛 𝑡𝑡( )[ −( 𝛼𝛼+𝛼𝛼 𝑡𝑡1,∆)𝑡𝑡 𝑝𝑝( 𝑛𝑛, 𝑡𝑡)].𝛼𝛼 𝛼𝛼 𝑡𝑡 ∆𝑡𝑡 (27) d𝑝𝑝 𝑛𝑛+1 𝑡𝑡 Using the method of proof→ by inductiond𝑡𝑡 and−𝛼𝛼 with𝛼𝛼 𝑡𝑡 the𝑝𝑝 𝑛𝑛 aid of𝑡𝑡 Eq.(− 26𝑝𝑝 ),𝑛𝑛 we𝑡𝑡 arrive at

( , ) = ( )d exp ( )d . (28) ! 𝑛𝑛 1 𝑡𝑡 ′ ′ 𝑡𝑡 ′ ′ 0 0 The probability distribution𝑝𝑝 𝑛𝑛 𝑡𝑡 function𝑛𝑛 �𝛼𝛼 ∫ 𝐼𝐼in𝑡𝑡 Eq.(𝑡𝑡28� ) is known�−𝛼𝛼 ∫ as𝐼𝐼 𝑡𝑡the Poisson𝑡𝑡 � distribution. Note that, for 0, we have ( , ) 0 for = 0, 1, 2, , and that ( , ) = 1, as expected from a probability distribution function. In a photon-counting experiment during∞ the fixed time interval [0, ], we may 𝑛𝑛=0 𝑡𝑡define≥ the parameter𝑝𝑝 𝑛𝑛 𝑡𝑡 ≥= 𝑛𝑛( )d and write⋯ ( , )∑= exp𝑝𝑝 (𝑛𝑛 𝑡𝑡 ) !. The average number of 𝑇𝑇 counted photo-electrons can then𝑇𝑇 be computed as follows: 𝑛𝑛 𝜂𝜂 𝛼𝛼 ∫0 𝐼𝐼 𝑡𝑡 𝑡𝑡 𝑝𝑝 𝑛𝑛 𝑇𝑇 −𝜂𝜂 𝜂𝜂 ⁄𝑛𝑛 〈𝑛𝑛〉 = ( , ) = exp( ) ! = exp( ) ( 1)! = . (29) ∞ ∞ 𝑛𝑛 ∞ 𝑛𝑛−1 Similarly,〈𝑛𝑛〉 ∑,𝑛𝑛 the=0 𝑛𝑛average𝑛𝑛 𝑛𝑛 𝑇𝑇 squared −number𝜂𝜂 ∑𝑛𝑛= of0 𝑛𝑛 photo𝜂𝜂 ⁄𝑛𝑛-electrons,𝜂𝜂 is− 𝜂𝜂found∑𝑛𝑛= to1 𝜂𝜂 be ⁄ 𝑛𝑛 − 𝜂𝜂 2 〈𝑛𝑛 〉= ( , ) = exp( ) ! = exp( ) ( 1)! 2 ∞ 2 ∞ 2 𝑛𝑛 ∞ 𝑛𝑛 〈𝑛𝑛 〉 = ∑exp𝑛𝑛=(0 𝑛𝑛 )𝑝𝑝 𝑛𝑛 𝑇𝑇 ( 1 +−1𝜂𝜂) ∑𝑛𝑛=( 0 𝑛𝑛 𝜂𝜂1)!⁄ 𝑛𝑛 −𝜂𝜂 ∑𝑛𝑛=1 𝑛𝑛𝜂𝜂 ⁄ 𝑛𝑛 − ∞ 𝑛𝑛 −𝜂𝜂 ∑𝑛𝑛=1 𝑛𝑛 − 𝜂𝜂 ⁄ 𝑛𝑛 − 6

= exp( ) [ ( 2)! + ( 1)!] ∞ 𝑛𝑛 ∞ 𝑛𝑛 = exp(−𝜂𝜂) [ ∑𝑛𝑛=2 𝜂𝜂 (⁄ 𝑛𝑛 −!) + ∑𝑛𝑛=1(𝜂𝜂 ⁄ 𝑛𝑛!)−] = + . (30) 2 ∞ 𝑛𝑛 ∞ 𝑛𝑛 2 The variance of the number−𝜂𝜂 𝜂𝜂 of∑ photo𝑛𝑛=0 𝜂𝜂-electrons⁄𝑛𝑛 𝜂𝜂relea∑𝑛𝑛=sed0 𝜂𝜂 in⁄ the𝑛𝑛 interval𝜂𝜂 [0𝜂𝜂, ] is thus given by Var( ) = Ave( ) = = . 𝑇𝑇 (31) 2 2 2 So far, our assumption has𝑛𝑛 been that𝑛𝑛 the− 〈 𝑛𝑛incident〉 〈 intensity𝑛𝑛 〉 − 〈𝑛𝑛 〉 ( ) is𝜂𝜂 a predetermined function of time. If ( ) happens to be a stationary random process having ensemble averages ( ) = and ( ) 𝐼𝐼 𝑡𝑡 ( ) ( + ) ( ) = ( ), then the ensemble averages of and will become 0 𝐼𝐼 𝑡𝑡 2 2 2 〈𝐼𝐼 𝑡𝑡 〉 𝐼𝐼 〈𝐼𝐼 𝑡𝑡 𝐼𝐼 𝑡𝑡 𝜏𝜏 〉⁄〈𝐼𝐼 𝑡𝑡 〉 𝑔𝑔 𝜏𝜏 = ( ) d = . 𝜂𝜂 𝜂𝜂 (32) 𝑇𝑇 0 〈𝜂𝜂〉 𝛼𝛼 ∫0 〈𝐼𝐼 𝑡𝑡 〉 𝑡𝑡 𝛼𝛼𝐼𝐼 𝑇𝑇 = ( ) ( )d d 𝑇𝑇 𝑇𝑇 2 2 ′ ′ ′ 〈𝜂𝜂 〉 = 𝛼𝛼 〈∫𝑡𝑡=0 ∫𝑡𝑡 =0 𝐼𝐼(𝑡𝑡)𝐼𝐼(𝑡𝑡 + 𝑡𝑡)d𝑡𝑡 〉d + ( ) ( )d d 2 𝑇𝑇 𝑇𝑇−𝜏𝜏 𝑇𝑇 𝑇𝑇 = 𝛼𝛼 〈∫(𝜏𝜏=)0 ∫𝑡𝑡=0 𝐼𝐼(𝑡𝑡 𝐼𝐼 𝑡𝑡| |)𝜏𝜏 ( 𝑡𝑡)( 𝜏𝜏)d ∫ 𝜏𝜏=0 ∫𝑡𝑡=𝜏𝜏 𝐼𝐼 𝑡𝑡 𝐼𝐼 𝑡𝑡 − 𝜏𝜏 𝑡𝑡 𝜏𝜏〉 𝑇𝑇 2 2 2 𝜏𝜏=−𝑇𝑇 = 𝛼𝛼 〈𝐼𝐼 𝑡𝑡 〉 ∫[1 (𝑇𝑇| −| 𝜏𝜏)]𝑔𝑔( )(𝜏𝜏)d𝜏𝜏 The width of ( )( ) 1, which 𝑇𝑇 2 02 2 is inversely proportional2 to , −𝑇𝑇 ⁄ ( ) (𝛼𝛼 𝐼𝐼 𝑇𝑇) � 1 +− 𝜏𝜏 𝑇𝑇 𝑔𝑔 ( )𝜏𝜏 1𝜏𝜏 d . (33) is assumed𝑔𝑔 to𝜏𝜏 be− . 𝑇𝑇 ∆𝜔𝜔 0 2 −1 2 −𝑇𝑇 The ensemble-averaged≪ 𝑇𝑇 variance of≅ the𝛼𝛼𝐼𝐼 number𝑇𝑇 � of 𝑇𝑇photo�-electrons�𝑔𝑔 𝜏𝜏 −released� 𝜏𝜏� in [0, ] is given by ( ) Var( ) = = + = 1 + ( 𝑇𝑇) 1 d . (34) 𝑇𝑇 2 2 2 2 0 0 2 Noting that 𝑛𝑛= 〈𝑛𝑛 and〉 − 〈d𝑛𝑛enoting〉 〈𝜂𝜂 the〉 area〈𝜂𝜂 〉under− 〈𝜂𝜂〉 ( )(𝛼𝛼𝐼𝐼) 𝑇𝑇 �1 by 𝛼𝛼𝐼𝐼, Eq.(�−𝑇𝑇�𝑔𝑔34) may𝜏𝜏 − be �written𝜏𝜏� as

0 2 0 〈𝑛𝑛〉 𝛼𝛼𝐼𝐼 𝑇𝑇 Var( ) = [1 +𝑔𝑔( 𝜏𝜏 − )]. 𝜏𝜏 (35) 0 If the incident light happens to 𝑛𝑛be fully〈𝑛𝑛〉 unpolarized,〈𝑛𝑛〉𝜏𝜏 ⁄𝑇𝑇 then the two orthogonal components of polarization, say, and , will have similar intensities while remaining uncorrelated. They will thus give rise to equal average photo-electron numbers, = , and also equal variances of the photo-electron numbers, Var( )𝑝𝑝= Var𝑠𝑠( ). One can then use Eq.(35) to write 〈𝑛𝑛𝑝𝑝〉 〈𝑛𝑛𝑠𝑠〉 Var(𝑛𝑛𝑝𝑝 ) = Var𝑛𝑛𝑠𝑠( ) + Var( ) = [1 + ( )] + [1 + ( )]

total 0 0 𝑛𝑛 𝑛𝑛𝑝𝑝 𝑛𝑛𝑠𝑠 = 〈𝑛𝑛𝑝𝑝〉 [1 +〈𝑛𝑛½𝑝𝑝〉(𝜏𝜏 ⁄𝑇𝑇 〈𝑛𝑛)𝑠𝑠]〉. 〈𝑛𝑛𝑠𝑠〉𝜏𝜏 ⁄𝑇𝑇 (36) total total 0 In a somewhat modified experiment, one〈𝑛𝑛 may〉 use a 〈50/50𝑛𝑛 〉 𝜏𝜏beam⁄𝑇𝑇 -splitter to divide a linearly polarized, quasi-monochromatic incident beam between two identical small-area photodetectors. For these two detector, = and Var( ) = Var( ), where the individual averages and variances satisfy Eq.(35). If we now add the two photo-electron counts together to form = + , we will have 1 2 1 2 = + and〈𝑛𝑛 〉 〈𝑛𝑛=〉 + 𝑛𝑛+ 2 𝑛𝑛. Consequently, Var( ) = Var( ) + Var( ) + 1 2 2( ). Substitution2 2 into Eq.(2 35) now yields 𝑛𝑛 𝑛𝑛 𝑛𝑛 〈𝑛𝑛〉 〈𝑛𝑛1〉 〈𝑛𝑛2〉 〈𝑛𝑛 〉 〈𝑛𝑛1〉 〈𝑛𝑛2〉 〈𝑛𝑛1𝑛𝑛2〉 𝑛𝑛 𝑛𝑛1 𝑛𝑛2 = [1 + ( )] 〈𝑛𝑛1𝑛𝑛2〉 − 〈𝑛𝑛1〉〈𝑛𝑛2〉 . (37) 0 This positive cross-correlation〈𝑛𝑛1 𝑛𝑛between2〉 〈𝑛𝑛 1the〉〈𝑛𝑛 2〉photo-electron𝜏𝜏 ⁄𝑇𝑇 counts of the two detectors is an alternative expression of the result obtained in the Hanbury Brown-Twiss experiment [12].

7

For a single-mode, quasi-monochromatic laser beam, whose cycle-averaged intensity is nearly constant, we will have ( )( ) 1, resulting in pure Poisson statistics for the number of released photo- electrons, where Var( ) =2 = . In this case the excess noise, i.e., the second term on the right- 𝑔𝑔 𝜏𝜏 ≅ hand-side of Eq.(35), is absent. 0 In the case of blackbody𝑛𝑛 〈𝑛𝑛 〉radiation𝛼𝛼𝐼𝐼 𝑇𝑇 (at fixed temperature ) from a small aperture of area , center frequency , linewidth , and integration time , Eq.(32) of Ref.[1] gives the single-mode photon number variance as Var( ) = exp( ) [exp( )𝑇𝑇� 1] = + . This variance𝐴𝐴 must be multiplied𝜔𝜔 by the density∆𝜔𝜔 (per unit volume) of modes,𝑇𝑇 ( 2 ), by the relevant2 mode volume 𝐵𝐵 � 𝐵𝐵 � , by a factor ½ to account𝑚𝑚 for theℏ fraction𝜔𝜔⁄𝑘𝑘 𝑇𝑇 of⁄ photonsℏ𝜔𝜔 ⁄that𝑘𝑘 𝑇𝑇 2exit− the 2blackbody3〈𝑚𝑚〉 〈 through𝑚𝑚〉 the aperture, and by another factor of ½ to account for the average obliquity 𝜔𝜔factor∆𝜔𝜔 ⁄cos𝜋𝜋 𝑐𝑐 . The first term in the expression 𝑐𝑐𝑐𝑐of 𝑇𝑇the photon-number variance, ( 2 ) , thus corresponds to the first term = on 〈 𝜃𝜃〉 the right-hand-side of Eq.(36). Noting that 2 = 2 , we may write = ( ) total. The second0 𝐴𝐴 𝜔𝜔⁄ 𝜋𝜋𝜋𝜋 〈𝑚𝑚〉𝑇𝑇∆𝜔𝜔 2 〈𝑛𝑛 〉 𝛼𝛼𝐼𝐼 𝑇𝑇 term in the expression of the photon-number variance for blackbody0 radiation then becomes ( ) . Agreement with the second𝜆𝜆 term𝜋𝜋𝜋𝜋 ⁄on𝜔𝜔 the right-hand-side𝛼𝛼𝐼𝐼 of Eq.(𝐴𝐴⁄𝜆𝜆36)〈 𝑚𝑚above〉∆𝜔𝜔 requires that ~ 2 and 2 = [ ( )( ) 1]d ~( ) . The first requirement is satisfied because the detector must 𝐴𝐴⁄𝜆𝜆 〈𝑚𝑚〉 𝑇𝑇∆𝜔𝜔𝑇𝑇 2 0 2 −1 act as a point-detector−𝑇𝑇 for the blackbody radiation emerging from the small aperture. The second requirement𝐴𝐴 𝜆𝜆 𝜏𝜏 is generally� 𝑔𝑔 𝜏𝜏satisfied− 𝜏𝜏 for∆ chaotic𝜔𝜔 (or thermal) light [13]. The excess noise of blackbody radiation thus complies with Eq.(36). 10. States of a single particle in a rigid box. Consider a rectangular parallelepiped box of dimensions × × . The evacuated box, held at a constant temperature , contains a single point-particle of mass , velocity , momentum = , and kinetic energy = ½ = (2 ). A plane-wave 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝐿𝐿solution𝐿𝐿 of𝐿𝐿 Schrödinger’s equation ( 2 ) ( , ) = i (𝑇𝑇 , ) 2 in 2free space is given by ( , 𝑚𝑚) = exp[i(𝒗𝒗 )], where𝒑𝒑 𝑚𝑚 𝒗𝒗2= / and2 = / , withℰ 𝑚𝑚being𝑣𝑣 Planck’s𝑝𝑝 ⁄ 𝑚𝑚 reduced constant. − ℏ ⁄ 𝑚𝑚 𝛻𝛻 𝜓𝜓 𝒓𝒓 𝑡𝑡 ℏ 𝜕𝜕𝜕𝜕 𝒓𝒓 𝑡𝑡 ⁄𝜕𝜕𝜕𝜕 To ensure that0 the wave-function vanishes at the interior walls of the box, we need a superposition of eight𝜓𝜓 𝒓𝒓 𝑡𝑡plane𝜓𝜓-waves having𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔-vectors𝜔𝜔 =𝒌𝒌±( 𝒑𝒑 ℏ ) 𝜔𝜔± ( ℰ ℏ ) ±ℏ ( ) , where , , are positive integers 1, 2, 3, . The combined wave-function will be 𝑘𝑘 𝒌𝒌 𝜋𝜋𝑛𝑛𝑥𝑥⁄𝐿𝐿𝑥𝑥 𝒙𝒙� 𝜋𝜋𝑛𝑛𝑦𝑦⁄𝐿𝐿𝑦𝑦 𝒚𝒚� 𝜋𝜋𝑛𝑛𝑧𝑧⁄𝐿𝐿𝑧𝑧 𝒛𝒛� 𝑛𝑛𝑥𝑥 𝑛𝑛𝑦𝑦 𝑛𝑛𝑧𝑧 ( , ) = exp i( ⋯+ + ) exp i( + ) exp i( + ) 0 𝜓𝜓 𝒓𝒓 𝑡𝑡 𝜓𝜓 �+ exp� i𝑘𝑘(𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 𝑘𝑘𝑧𝑧𝑧𝑧 �)− exp� i𝑘𝑘(𝑥𝑥𝑥𝑥 𝑘𝑘+𝑦𝑦𝑦𝑦 − 𝑘𝑘+𝑧𝑧𝑧𝑧 � −) + exp� 𝑘𝑘𝑥𝑥i(𝑥𝑥 − 𝑘𝑘𝑦𝑦𝑦𝑦+ 𝑘𝑘𝑧𝑧𝑧𝑧 � ) + exp�i(𝑘𝑘𝑥𝑥𝑥𝑥 − 𝑘𝑘𝑦𝑦𝑦𝑦 −+𝑘𝑘𝑧𝑧𝑧𝑧 �)− exp� −i(𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 𝑘𝑘𝑧𝑧𝑧𝑧 �) exp�( −i 𝑘𝑘𝑥𝑥)𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 − 𝑘𝑘𝑧𝑧𝑧𝑧 � = 2i exp[i(� −𝑘𝑘+𝑥𝑥𝑥𝑥 − 𝑘𝑘)]𝑦𝑦𝑦𝑦 exp𝑘𝑘𝑧𝑧[𝑧𝑧i( � − � −)]𝑘𝑘𝑥𝑥𝑥𝑥exp− 𝑘𝑘[i𝑦𝑦(𝑦𝑦 − 𝑘𝑘𝑧𝑧+𝑧𝑧 �� )] +−exp𝜔𝜔𝜔𝜔[i( )] 0 𝜓𝜓 �× sin(𝑘𝑘𝑥𝑥𝑥𝑥) exp𝑘𝑘𝑦𝑦(𝑦𝑦 i −) 𝑘𝑘𝑥𝑥𝑥𝑥 − 𝑘𝑘𝑦𝑦𝑦𝑦 − −𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 −𝑘𝑘𝑥𝑥𝑥𝑥 − 𝑘𝑘𝑦𝑦𝑦𝑦 � = 4 [exp(i ) exp( i )] sin( ) sin( ) exp( i ) 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔 0 = −8i𝜓𝜓 sin( 𝑘𝑘𝑥𝑥𝑥𝑥) sin−( )−sin𝑘𝑘𝑥𝑥(𝑥𝑥 ) exp𝑘𝑘(𝑦𝑦𝑦𝑦i ). 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔 (38) 0 The− amplitude𝜓𝜓 𝑘𝑘𝑥𝑥𝑥𝑥 may be𝑘𝑘 𝑦𝑦determined𝑦𝑦 𝑘𝑘𝑧𝑧𝑧𝑧 (aside− from𝜔𝜔𝜔𝜔 an arbitrary phase-factor) by requiring the integral of | ( , )| over the 0volume of the box to be unity. Defining = , we will have | | = 1/ 8 . 𝜓𝜓 The desired2 wave-function is thus given by 0 𝜓𝜓 𝒓𝒓 𝑡𝑡 𝑉𝑉 𝐿𝐿𝑥𝑥𝐿𝐿𝑦𝑦𝐿𝐿𝑧𝑧 𝜓𝜓 √ 𝑉𝑉 ( , ) = 2 2 sin( ) sin( ) sin( ) exp[ i( )]. (39) The above wave-function with ( , , ) = ( , , )0, where ( , , ) are 𝜓𝜓 𝒓𝒓 𝑡𝑡 � ⁄𝑉𝑉 𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔 − 𝜑𝜑 positive integers, represents all possible states of a single point-particle inside a rigid box of dimensions 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑧𝑧 𝑧𝑧 𝑥𝑥 𝑦𝑦 𝑧𝑧 × × and volume . The particle’s𝑘𝑘 𝑘𝑘 linear𝑘𝑘 momentum𝜋𝜋𝑛𝑛 ⁄𝐿𝐿 and𝜋𝜋𝑛𝑛 kinetic⁄𝐿𝐿 𝜋𝜋 energy𝑛𝑛 ⁄𝐿𝐿 are readily𝑛𝑛 found𝑛𝑛 𝑛𝑛to be

𝐿𝐿 𝑥𝑥 𝐿𝐿𝑦𝑦 𝐿𝐿𝑧𝑧 𝑉𝑉 = = ( , , ). (40) 𝒑𝒑 ℏ𝒌𝒌 𝜋𝜋ℏ 𝑛𝑛𝑥𝑥⁄𝐿𝐿𝑥𝑥 𝑛𝑛𝑦𝑦⁄𝐿𝐿𝑦𝑦 𝑛𝑛𝑧𝑧⁄𝐿𝐿𝑧𝑧 8

= = 2 = ( 2 ) ( ) + ( ) + ( ) . (41) 2 2 2 2 2 2 2 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑧𝑧 𝑧𝑧 11. Partitionℰ functionℏ𝜔𝜔 forℏ 𝑘𝑘particl⁄ 𝑚𝑚es in𝜋𝜋 a ℏbox⁄ and𝑚𝑚 � derivation𝑛𝑛 ⁄𝐿𝐿 of𝑛𝑛 the⁄𝐿𝐿 Sackur𝑛𝑛-Tetrode⁄𝐿𝐿 � equation. For the single particle in a box described in the preceding section, the partition function is the sum over all possible states of the exponential factor exp( ). We will have

𝐵𝐵 ( , = 1ℓ, ) = exp ( −2ℰℓ⁄𝑘𝑘 𝑇𝑇)[( ) + ( ) + ( ) ] , , 2 2 2 2 2 𝐵𝐵 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑧𝑧 𝑧𝑧 𝑍𝑍 𝑇𝑇 𝑁𝑁 𝑉𝑉 �𝑛𝑛𝑥𝑥 𝑛𝑛𝑦𝑦 𝑛𝑛𝑧𝑧 �− 𝜋𝜋 ℏ ⁄ 𝑚𝑚𝑘𝑘 𝑇𝑇 𝑛𝑛 ⁄𝐿𝐿 𝑛𝑛 ⁄𝐿𝐿 𝑛𝑛 ⁄𝐿𝐿 � ∞ exp ( 2 )[( ) + ( ) + ( ) ] d d d ∞ ∞ 2 2 𝐵𝐵 2 2 2 𝑛𝑛𝑧𝑧=0 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑧𝑧 𝑧𝑧 𝑥𝑥 𝑦𝑦 𝑧𝑧 ≅ � �𝑛𝑛𝑦𝑦=0 � �− 𝜋𝜋 ℏ ⁄ 𝑚𝑚𝑘𝑘 𝑇𝑇 𝑛𝑛 ⁄𝐿𝐿 𝑛𝑛 ⁄𝐿𝐿 𝑛𝑛 ⁄𝐿𝐿 � 𝑛𝑛 𝑛𝑛 𝑛𝑛 𝑛𝑛 =0 = (2𝑥𝑥 ) exp[ ( + + )] d d d exp( ) d = 2 ∞ ∞ 2 𝐵𝐵 2 2 3⁄2 2 2 2 = ( 𝑚𝑚𝑘𝑘 𝑇𝑇2⁄𝜋𝜋 ℏ) 𝐿𝐿. 𝑥𝑥𝐿𝐿𝑦𝑦𝐿𝐿𝑧𝑧 ∭0 − 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 𝑦𝑦 𝑧𝑧 ∫0 −𝑥𝑥 𝑥𝑥 √𝜋𝜋(⁄42) 𝐵𝐵 2 3⁄2 For a system𝑚𝑚𝑘𝑘 𝑇𝑇 ⁄of 𝜋𝜋ℏ identical𝑉𝑉 , non-interacting particles, ( , , ) = ( , 1, ) !. This is because exp[ ( + + + ) ] = [ exp( )] , and also because𝑁𝑁 the particles are 𝑁𝑁 𝑍𝑍 𝑇𝑇 𝑁𝑁 𝑉𝑉𝑁𝑁 𝑍𝑍 𝑇𝑇 𝑉𝑉 ⁄𝑁𝑁 indistinguishℓ1 ℓ2 ℓ𝑁𝑁able, whichℓ1 requiresℓ2 divisionℓ𝑁𝑁 𝐵𝐵 by ! ℓin orderℓ to𝐵𝐵 eliminate the contribution of particle arrangements∑ ∑ ⋯ ∑ that− areℰ mereℰ permutations⋯ ℰ ⁄ 𝑘𝑘of𝑇𝑇 each ∑other. −Tℰhe⁄ 𝑘𝑘Helmholtz𝑇𝑇 free energy of the -particle system (described in Section 2) is thus given by 𝑁𝑁 Stirling’s approximation:𝑁𝑁 = ln ( , , ) ln ( 2 ) ! ! 𝐵𝐵 𝐵𝐵 𝐵𝐵 2 3𝑁𝑁⁄2 𝑁𝑁 𝑁𝑁 −𝑁𝑁 𝐹𝐹 −𝑘𝑘 𝑇𝑇 ln𝑍𝑍( 𝑇𝑇 𝑁𝑁 𝑉𝑉2 ≅ −)𝑘𝑘 𝑇𝑇 � 𝑚𝑚𝑘𝑘 𝑇𝑇⁄ 𝜋𝜋ℏ 𝑉𝑉 ⁄𝑁𝑁 �Small compared𝑁𝑁 ≅ 𝑁𝑁 to𝑒𝑒 other√𝑁𝑁 terms𝑒𝑒 𝐵𝐵 𝐵𝐵 2 3𝑁𝑁⁄2 𝑁𝑁 𝑁𝑁 −𝑁𝑁 ≅= −𝑘𝑘 𝑇𝑇 �ln𝑚𝑚(𝑘𝑘 𝑇𝑇⁄ 𝜋𝜋2ℏ ) 𝑉𝑉( ��𝑁𝑁) 𝑒𝑒+ √𝑁𝑁𝑒𝑒½��ln 1 . (43)

𝐵𝐵 𝐵𝐵 2 3⁄2 Since = −𝑘𝑘 𝑇𝑇,� 𝑁𝑁and �since,𝑚𝑚𝑘𝑘 for𝑇𝑇⁄ the𝜋𝜋ℏ ,𝑉𝑉⁄ 𝑁𝑁 =� 3𝑁𝑁 − 2, the𝑁𝑁 − above� equation may be written

𝐵𝐵 𝐹𝐹 𝑈𝑈 − 𝑇𝑇𝑇𝑇 ( , , ) ln (2𝑈𝑈 𝑁𝑁𝑘𝑘 𝑇𝑇)⁄ ( ) + . (44) 5 𝐵𝐵 𝐵𝐵 2 3⁄2 This is the well-known𝑆𝑆 𝑇𝑇 𝑁𝑁 Sackur𝑉𝑉 ≅ -𝑁𝑁Tetrode𝑘𝑘 � � equation𝜋𝜋𝜋𝜋𝑘𝑘 𝑇𝑇 [2]⁄ℎ, which𝑉𝑉 gives⁄𝑁𝑁 � the2 entropy� of an ideal monatomic gas as a function of the temperature , the number of particles , and the volume . Here =

1.38065 × 10 Joule/deg is Boltzmann’s constant, while = 6.626068 × 10 Joule·sec is 𝐵𝐵the .−23 The formula breaks down𝑇𝑇 at low and/or𝑁𝑁 at high particle−34 densities,𝑉𝑉 partly𝑘𝑘 because of the neglect of the interaction among particles, andℎ partly because of the approximations involved in Eq.(42) in going from a sum to an integral. (The expression for the entropy cannot be valid for very low temperatures, because the entropy does not approach a constant value when 0 K.) Compare and contrast the entropy of a gas of monatomic particles with the entropy of the photon gas (i.e., blackbody radiation) given in Ref.[1], Eq.(37). 𝑇𝑇 →

12. Internal energy, pressure, and of the ideal monatomic gas. One may use the partition function ( , , ) to calculate the internal energy of the ideal monatomic gas in accordance with Eq.(2), as follows: 𝑍𝑍 𝑇𝑇 𝑁𝑁 𝑉𝑉 𝑈𝑈 = ln[ ( , , )] = ln ( 2 ) ! = . (45) 3 𝐵𝐵 2 𝐵𝐵 2 𝐵𝐵 2 3𝑁𝑁⁄2 𝑁𝑁 𝐵𝐵 Alternatively,𝑈𝑈 𝑘𝑘 𝑇𝑇 𝜕𝜕we can𝑍𝑍 𝑇𝑇 compute𝑁𝑁 𝑉𝑉 ⁄ 𝜕𝜕the𝜕𝜕 average𝑘𝑘 𝑇𝑇 𝜕𝜕 kinetic� 𝑚𝑚𝑘𝑘 energy𝑇𝑇⁄ 𝜋𝜋 ℏof a single𝑉𝑉 ⁄particle𝑁𝑁 �⁄𝜕𝜕𝜕𝜕 by direct2 𝑁𝑁𝑘𝑘 𝑇𝑇integration over all possible states of the particle, namely,

9

( , = 1, ) = , , ( 2 ) ( ) + ( ) + ( ) 2 2 2 2 2 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑈𝑈 𝑇𝑇 𝑁𝑁 𝑉𝑉 ∑𝑛𝑛×𝑛𝑛exp𝑛𝑛 𝜋𝜋(ℏ ⁄ 𝑚𝑚2 � 𝑛𝑛𝑥𝑥⁄)𝐿𝐿𝑥𝑥( 𝑛𝑛)𝑦𝑦⁄+𝐿𝐿(𝑦𝑦 𝑛𝑛)𝑧𝑧⁄+𝐿𝐿𝑧𝑧( � ) ( , = 1, )

2 2 𝐵𝐵 2 2 2 �− 𝜋𝜋 ℏ ⁄ 𝑚𝑚𝑘𝑘 𝑇𝑇 � 𝑛𝑛𝑥𝑥⁄𝐿𝐿𝑥𝑥 𝑛𝑛𝑦𝑦⁄𝐿𝐿𝑦𝑦 𝑛𝑛𝑧𝑧⁄𝐿𝐿𝑧𝑧 ��⁄𝑍𝑍 𝑇𝑇 𝑁𝑁 𝑉𝑉 ( 2 ) −1 𝐵𝐵 2 3⁄2 ≅ � 𝑚𝑚𝑘𝑘 𝑇𝑇⁄ 𝜋𝜋 ℏ 𝑉𝑉� × ∞ ( 2 )[( ) + ( ) + ( ) ] ∞ ∞ 2 2 2 2 2 𝑛𝑛𝑧𝑧=0 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑧𝑧 𝑧𝑧 � �𝑛𝑛𝑦𝑦=0 � 𝜋𝜋 ℏ ⁄ 𝑚𝑚 𝑛𝑛 ⁄𝐿𝐿 𝑛𝑛 ⁄𝐿𝐿 𝑛𝑛 ⁄𝐿𝐿 ×𝑛𝑛𝑥𝑥exp=0 ( 2 )[( ) + ( ) + ( ) ] d d d 2 2 2 2 2 𝐵𝐵 𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑦𝑦 𝑧𝑧 𝑧𝑧 𝑥𝑥 𝑦𝑦 𝑧𝑧 ( �− 𝜋𝜋) ℏ ⁄ 𝑚𝑚𝑘𝑘 𝑇𝑇 𝑛𝑛 ⁄𝐿𝐿 𝑛𝑛 ⁄𝐿𝐿 𝑛𝑛 ⁄𝐿𝐿 � 𝑛𝑛 𝑛𝑛 𝑛𝑛 = ( + + ) exp[ ( + + )] d d d ( 2) 2 3⁄2 ∞ 𝑘𝑘𝐵𝐵𝑇𝑇 2𝑚𝑚𝑘𝑘𝐵𝐵𝑇𝑇⁄𝜋𝜋 ℏ 𝑉𝑉 ∞ ∞ ⁄ 2 2 2 2 2 2 2 3 2 𝑦𝑦=0 𝑧𝑧=0 𝑚𝑚𝑘𝑘𝐵𝐵𝑇𝑇⁄2𝜋𝜋ℏ 𝑉𝑉 �𝑥𝑥=0 � 𝑥𝑥 𝑦𝑦 𝑧𝑧 − 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 𝑦𝑦 𝑧𝑧 = 8 ½ exp( ∫ ) d = . exp( ) d = 3 8 (46) ∞ 3 ∞ 4 2 3 4 2 𝐵𝐵 𝐵𝐵 0 Since the particles� 𝑘𝑘 𝑇𝑇 ⁄of√ the𝜋𝜋 � system∫0 𝜋𝜋 are𝑟𝑟 identical−𝑟𝑟 and𝑟𝑟 independent,2 𝑘𝑘 𝑇𝑇 ∫ the𝑥𝑥 total energy−𝑥𝑥 𝑥𝑥of the√ system𝜋𝜋⁄ will be = 3 2, as before; see Eq.(45). The pressure on each wall of the container may be derived 𝑁𝑁 from the kinetic𝐵𝐵 energies of the individual atoms, ½ ( + + ), by recalling that the component of momentum𝑈𝑈 𝑁𝑁 𝑘𝑘perpendicular𝑇𝑇⁄ to a given wall, say, , must2𝑝𝑝 2be mul2 tiplied by the number of particles 𝑥𝑥 𝑦𝑦 𝑧𝑧 ½( ) that strike a unit area of the wall per unit𝑚𝑚 𝑣𝑣time,𝑣𝑣 then 𝑣𝑣doubled (to account for the change of momentum from before to after collision). We thus𝑚𝑚 find𝑣𝑣𝑥𝑥 𝑁𝑁⁄𝑉𝑉 𝑣𝑣𝑥𝑥 = = . (47) 2𝑈𝑈 2 Here = is the energy-density of 𝑝𝑝the gas.3𝑉𝑉 From3 𝑢𝑢 Eqs.(46) and (47), one can readily obtain the ideal gas law, namely, = . Comparing Eq.(47) with Eq.(12) of Ref.[1], we note that the pressure 𝑢𝑢 𝑈𝑈⁄𝑉𝑉 of the ideal monatomic gas as 𝐵𝐵a function of its energy-density is twice that of the photon gas (i.e., blackbody radiation). 𝑝𝑝𝑝𝑝 𝑁𝑁𝑘𝑘 𝑇𝑇 The Sackur-Tetrode equation describing the entropy of an ideal monatomic gas consisting of identical particles of mass having (absolute) temperature and volume is given by Eq.(44). The fundamental thermodynamic identity [2-4] relates the various partial derivatives of the entropy ( , , 𝑁𝑁) to the temperature , chemical𝑚𝑚 potential , and pressure of the𝑇𝑇 system, as follows:𝑉𝑉 𝑆𝑆 𝑈𝑈 𝑁𝑁 𝑉𝑉 ( ) = 1 ; ( ) = ; ( ) = 𝑇𝑇 , 𝜇𝜇𝑐𝑐 , 𝑝𝑝 , . (48) Using Eq.(45𝜕𝜕𝜕𝜕),⁄ 𝜕𝜕we𝜕𝜕 rewrite𝑁𝑁 𝑉𝑉 the⁄𝑇𝑇 entropy in𝜕𝜕 𝜕𝜕Eq⁄𝜕𝜕.𝜕𝜕(44𝑈𝑈) 𝑉𝑉as a −function𝜇𝜇𝑐𝑐⁄𝑇𝑇 of , 𝜕𝜕 and𝜕𝜕⁄𝜕𝜕 𝜕𝜕, 𝑈𝑈namely,𝑁𝑁 𝑝𝑝⁄ 𝑇𝑇 ( , , ) = ln (4 3 ) ( 𝑈𝑈 )𝑁𝑁 + 𝑉𝑉· (49) 5 𝐵𝐵 2 3⁄2 5⁄2 In accordance with 𝑆𝑆Eqs𝑈𝑈.(4𝑁𝑁8𝑉𝑉), the 𝑁𝑁partial𝑘𝑘 � derivatives� 𝜋𝜋𝜋𝜋𝜋𝜋⁄ ofℎ the above𝑉𝑉⁄𝑁𝑁 expression� 2� for entropy now yield

( ) , = = 1 . (50) 3 𝑁𝑁 𝑉𝑉 𝐵𝐵 (𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕) , = 2 𝑁𝑁𝑘𝑘 ⁄𝑈𝑈= ⁄.𝑇𝑇 (51)

𝑈𝑈 𝑁𝑁 𝐵𝐵 𝜕𝜕𝜕𝜕=⁄𝜕𝜕𝜕𝜕 ( 𝑁𝑁)𝑘𝑘 ,⁄𝑉𝑉= 𝑝𝑝⁄𝑇𝑇 ln (4 3 ) ( )

𝐵𝐵 2 3⁄2 5⁄2 𝜇𝜇𝑐𝑐 −𝑇𝑇 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑈𝑈 𝑉𝑉 = −𝑘𝑘 𝑇𝑇 ln�(2𝜋𝜋𝜋𝜋𝜋𝜋⁄ ℎ ) 𝑉𝑉( ⁄𝑁𝑁 ) . � (52)

𝐵𝐵 𝐵𝐵 2 3⁄2 Considering that the proton mass is =−𝑘𝑘1.67262158𝑇𝑇 � 𝜋𝜋𝜋𝜋𝑘𝑘×𝑇𝑇10⁄ℎ kg, 𝑉𝑉an⁄ 𝑁𝑁order� -of-magnitude estimate for the chemical potential given by Eq.(52) is 13 . −This27 is based on assuming = , 𝑚𝑚𝑝𝑝 𝑐𝑐 𝐵𝐵 𝑝𝑝 𝜇𝜇10≅ − 𝑘𝑘 𝑇𝑇 𝑚𝑚 𝑚𝑚

= 100 , = 1.0 , and = 6.0221415 × 10 , i.e., Avogadro’s number. Reducing the temperature by a factor 3of 10 will change the chemical23 potential of the gas to 9.55 , whereas 𝑇𝑇 𝐾𝐾 𝑉𝑉 𝑚𝑚 𝑁𝑁 raising the density by a factor of 10 (while maintaining at 100 ) will change the𝐵𝐵 chemical potential to 10.7 . The more massive the atoms/molecules of the gas 𝜇𝜇are,𝑐𝑐 ≅ the− more𝑘𝑘 negative𝑇𝑇 its 𝑁𝑁⁄𝑉𝑉 𝑇𝑇 𝐾𝐾 chemical potential becomes.𝐵𝐵 𝜇𝜇𝑐𝑐 ≅ − 𝑘𝑘 𝑇𝑇 Example 1. Suppose an -particle monatomic gas at temperature and volume is heated to a new temperature without changing its volume or number of particles. The internal energy of the gas thus 1 increases by = (3 2)𝑁𝑁 ( ), requiring an amount of heat𝑇𝑇 = . The𝑉𝑉 change in entropy 𝑇𝑇2 may be calculated as follows:𝐵𝐵 ∆𝑈𝑈 ⁄ 𝑁𝑁𝑘𝑘 𝑇𝑇2 − 𝑇𝑇1 ∆𝑄𝑄 ∆𝑈𝑈 ( ) = = = ln( ) 𝑇𝑇2 𝑇𝑇2 . (53) d𝑄𝑄 3⁄2 𝑁𝑁𝑘𝑘𝐵𝐵d𝑇𝑇 3 𝐵𝐵 𝑇𝑇 𝑇𝑇 2 2⁄ 1 The above result is ∆seen𝑆𝑆 to� 𝑇𝑇be1 consistent�𝑇𝑇1 with the Sackur𝑁𝑁-𝑘𝑘Tetrode𝑇𝑇 expression𝑇𝑇 for the entropy given by Eq.(44).

Example 2. As another example, let an -particle monatomic gas expand reversibly from an initial volume to a final volume while remaining at a constant temperature . The internal energy of the gas will remain at = (3 2) , but the𝑁𝑁 pressure = will deliver an amount of work to 𝑉𝑉1 𝑉𝑉2 𝑇𝑇 the outside world that is given by𝐵𝐵 𝐵𝐵 𝑈𝑈 ⁄ 𝑁𝑁𝑘𝑘 𝑇𝑇 𝑝𝑝 𝑁𝑁𝑘𝑘 𝑇𝑇⁄𝑉𝑉 ∆𝑊𝑊 = d = d = ln( ) 𝑉𝑉2 . (54) 𝑉𝑉2 𝑁𝑁𝑘𝑘𝐵𝐵𝑇𝑇 𝐵𝐵 𝑉𝑉1 𝑉𝑉 2⁄ 1 Since the internal energy∆𝑊𝑊 ∫ of𝑝𝑝 the𝑉𝑉 gas� has𝑉𝑉1 remained𝑉𝑉 constant,𝑁𝑁𝑘𝑘 𝑇𝑇 the𝑉𝑉 above𝑉𝑉 work must be supplied by the heat bath, which maintains the gas at a fixed temperature. Consequently, = . The change in the entropy of the gas is, 𝑈𝑈therefore, given by ∆𝑊𝑊 ∆𝑄𝑄 ∆𝑊𝑊 ∆ 𝑆𝑆 = = = ln( ). (55) ∆𝑄𝑄 ∆𝑊𝑊 Once again, the above result∆ is𝑆𝑆 seen𝑇𝑇 to be𝑇𝑇 consistent𝑁𝑁𝑘𝑘𝐵𝐵 with𝑉𝑉2⁄ the𝑉𝑉1 Sackur-Tetrode expression for the entropy given by Eq.(44). Note that the entropy, being a property of a system in thermal and diffusional equilibrium, does not depend on the path taken by the system to arrive at its final state. Thus, in a modified version of the present example, shown in the figure below, the entropy of the system will increase by the same amount as given by Eq.(55) if the gas were to expand irreversibly and adiabatically from an initial volume to a final volume , Vacuum ∆without𝑆𝑆 adding or subtracting any amount of heat whatsoever. In Ideal gas 1 2 (V2 −V1) this system, the diaphragm shatters at 𝑉𝑉some instant of time, thus𝑉𝑉 (T, N, V1) allowing the gas to expand irreversibly into the evacuated part of the chamber. Since the gas molecules retain their initial kinetic Thermal energies, the internal energy and the temperature of the gas Diaphragm insulation will not change in consequence of the expansion. The pressure , however, drops in proportion 𝑈𝑈to the ratio , while𝑇𝑇 the entropy increases by = ln( ). 𝑝𝑝 𝑉𝑉1⁄𝑉𝑉2 𝐵𝐵 Example 3. ∆Suppose𝑆𝑆 𝑁𝑁𝑘𝑘 a single𝑉𝑉2⁄𝑉𝑉 1particle is added to a monatomic -particle gas at temperature and volume . The particle must have a kinetic energy of (3 2) , which raises the total energy of the 𝑁𝑁 𝑇𝑇 system from = (3 2) to = (3 2)( + 1) . If the𝐵𝐵 goal is to maintain the internal energy of the system𝑉𝑉 at its initial value of , then the final temperature⁄ 𝑘𝑘 must𝑇𝑇 be lowered from to ( + 1). 1 𝐵𝐵 2 𝐵𝐵 The change of𝑈𝑈 entropy⁄ at 𝑁𝑁constant𝑘𝑘 𝑇𝑇 𝑈𝑈 and may⁄ 𝑁𝑁thus be 𝑘𝑘obtained𝑇𝑇 from Eq.(49), as follows: 𝑈𝑈1 𝑇𝑇 𝑁𝑁𝑁𝑁⁄ 𝑁𝑁 𝑈𝑈 𝑉𝑉 11

0 = ( , + 1, ) ( , , ) ln (4 3 ) ( ) 5 𝐵𝐵 2 3⁄2 5⁄2 ∆𝑆𝑆 𝑆𝑆 𝑈𝑈 𝑁𝑁 𝑉𝑉 − 𝑆𝑆 𝑈𝑈 𝑁𝑁 𝑉𝑉 ≅ 𝑘𝑘 �ln �(2 𝜋𝜋𝜋𝜋𝜋𝜋⁄ ℎ ) (𝑉𝑉⁄𝑁𝑁 ) . � − 2𝑁𝑁 � (56)

𝐵𝐵 𝐵𝐵 2 3⁄2 According to Eq.(48), the above change≅ 𝑘𝑘in the� entropy𝜋𝜋𝜋𝜋𝑘𝑘 𝑇𝑇 of⁄ℎ the system𝑉𝑉⁄𝑁𝑁 must� be equal to , in agreement with Eq.(52). Note that the kinetic energy of the added particle, (3 2)[ ( + 1)] , is essentially the−𝜇𝜇 same𝑐𝑐⁄𝑇𝑇 as that of all the other particles in the system. This energy is very different from the chemical potential , which has a negative value in the present example. The⁄ significance𝑁𝑁⁄ 𝑁𝑁 of𝑘𝑘 𝐵𝐵the𝑇𝑇 chemical potential is in connection with the change in the internal energy, ( , , ), which may be obtained by comparison𝜇𝜇𝑐𝑐 with the fundamental thermodynamic identity in the following way: ∆𝑈𝑈 𝑆𝑆 𝑁𝑁 𝑉𝑉 = ( ) , + ( ) , + ( ) , = (1 ) ( ) + ( )

𝑁𝑁 𝑉𝑉 𝑈𝑈 𝑉𝑉 𝑈𝑈 𝑁𝑁 𝑐𝑐 ∆𝑆𝑆 𝜕𝜕𝜕𝜕⁄𝜕𝜕=𝜕𝜕( ∆𝑈𝑈 ) , 𝜕𝜕𝜕𝜕⁄+𝜕𝜕𝜕𝜕( ∆𝑁𝑁 ) , 𝜕𝜕𝜕𝜕⁄+𝜕𝜕𝜕𝜕( ∆𝑉𝑉 ) , ⁄𝑇𝑇=∆𝑈𝑈 − +𝜇𝜇 ⁄𝑇𝑇 ∆𝑁𝑁 𝑝𝑝.⁄ 𝑇𝑇 ∆𝑉𝑉(57)

The→ above∆𝑈𝑈 equation𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 should𝑁𝑁 𝑉𝑉∆𝑆𝑆 be interpreted𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑆𝑆 𝑉𝑉as∆ 𝑁𝑁follows.𝜕𝜕𝜕𝜕 When⁄𝜕𝜕𝜕𝜕 𝑆𝑆 the𝑁𝑁∆ 𝑉𝑉number𝑇𝑇∆ 𝑆𝑆of particles𝜇𝜇𝑐𝑐∆𝑁𝑁 − 𝑝𝑝 ∆is𝑉𝑉 constant, the change of the internal energy is equal to the amount of heat = added to the system minus the amount of work = delivered to the outside world. However, if the number of 𝑁𝑁particles changes by , the change of the entropy∆𝑈𝑈 is no longer equal to / ∆. 𝑄𝑄In this𝑇𝑇 ∆case,𝑆𝑆 we must correct by the addition of ∆ in𝑊𝑊 order𝑝𝑝∆ to𝑉𝑉 account for the change of the entropy in consequence of the addition of new particles∆𝑁𝑁 to the system.∆𝑆𝑆 ∆𝑄𝑄 𝑇𝑇 𝑇𝑇∆𝑆𝑆 𝑐𝑐 From Eq𝜇𝜇 .∆(𝑁𝑁57) it is seen that ( ) , = , in agreement with the first identity in Eq.(48). Also,

( ) , = and ( ) , = , 𝑁𝑁but𝑉𝑉 these latter thermodynamic relations cannot be derived from the fundamental thermodynamic𝜕𝜕𝜕𝜕 ⁄identity𝜕𝜕𝜕𝜕 by𝑇𝑇 a simple division and multiplication by on the left- 𝑆𝑆 𝑉𝑉 𝑐𝑐 𝑆𝑆 𝑁𝑁 hand𝜕𝜕𝜕𝜕⁄-𝜕𝜕side𝜕𝜕 of each𝜇𝜇 equation.𝜕𝜕𝜕𝜕⁄ In𝜕𝜕𝜕𝜕 other words,−𝑝𝑝 attention must be paid to what is being varied and what is being kept constant. For example, Eq.(49) shows that, when and are fixed, it is necessary𝜕𝜕𝜕𝜕 to have = constant. Differentiation with respect to then yields 3⁄2 𝑆𝑆 𝑁𝑁 𝑈𝑈 𝑉𝑉( ) , + = 0 ( 𝑉𝑉 ) , = ( ) = = . (58) 3 1⁄2 3⁄2 2 ( 𝑆𝑆 𝑁𝑁 ) 𝑆𝑆 𝑁𝑁 𝐵𝐵 To show2 𝜕𝜕𝜕𝜕 that⁄𝜕𝜕 𝜕𝜕 𝑈𝑈 𝑉𝑉, = 𝑈𝑈 , one may use→ Eq.(𝜕𝜕𝜕𝜕49⁄𝜕𝜕) to𝜕𝜕 solve the− 3following𝑈𝑈⁄𝑉𝑉 equation− 𝑁𝑁𝑘𝑘 for𝑇𝑇⁄ 𝑉𝑉 : −𝑝𝑝 𝑆𝑆 𝑉𝑉 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 ( )𝜇𝜇𝑐𝑐 = ( + , + 1, ) ( , , ) = 0. ∆𝑈𝑈 (59)

Considering that, in ∆the𝑆𝑆 above𝑉𝑉 𝑆𝑆 equation,𝑈𝑈 ∆𝑈𝑈 𝑁𝑁 = 1,𝑉𝑉 it is− clear𝑆𝑆 𝑈𝑈 that𝑁𝑁 𝑉𝑉 the solution must be = . An alternative, although ultimately equivalent, method of arriving at the same result is via partial differentiation, as follows: ∆𝑁𝑁 ∆𝑈𝑈 𝜇𝜇𝑐𝑐

( ) ( ) , + ( ) , = (1 ) ( ) = 0

𝑉𝑉 𝑁𝑁 𝑉𝑉 𝑈𝑈 𝑉𝑉 𝑐𝑐 ∆𝑆𝑆 ≅ 𝜕𝜕 𝜕𝜕 ⁄(𝜕𝜕𝜕𝜕 ∆)𝑈𝑈, 𝜕𝜕𝜕𝜕⁄ 𝜕𝜕 𝜕𝜕 ∆ 𝑁𝑁 ⁄=𝑇𝑇 (∆𝑈𝑈 − 𝜇𝜇)⁄,𝑇𝑇. ∆𝑁𝑁 (60) Example 4. Consider→ an ∆-particle𝑈𝑈⁄∆𝑁𝑁 ideal𝑆𝑆 𝑉𝑉 ≅ ga𝜇𝜇s𝑐𝑐 initially→ held at temperature𝜇𝜇𝑐𝑐 𝜕𝜕𝜕𝜕⁄ 𝜕𝜕 and𝜕𝜕 𝑆𝑆 volume𝑉𝑉 . In a reversible, adiabatic expansion, no heat is given to or taken from the gas, while the work delivered to the outside world by the pressure 𝑁𝑁of the expanding gas causes the internal energy𝑇𝑇 and, consequ𝑉𝑉 ently, the temperature to drop. Since the change of entropy in a reversible process at temperature is equal to , and since, in the present adiabatic process, is zero at all temperatures,𝑈𝑈 we conclude that the entropy has𝑇𝑇 remained constant. We write ∆𝑆𝑆 𝑇𝑇 ∆𝑄𝑄⁄𝑇𝑇 ∆𝑄𝑄 𝑆𝑆 = = ( ) = = = 2 d𝑈𝑈 2 d𝑉𝑉 d𝑈𝑈 2 d𝑉𝑉 ∆𝑊𝑊 𝑝𝑝∆𝑉𝑉 3 𝑈𝑈⁄𝑉𝑉 ∆𝑉𝑉 −∆𝑈𝑈 → 𝑈𝑈 − 3 𝑉𝑉 → � 𝑈𝑈 − 3 � 𝑉𝑉 12

ln( ) = ln( ) = ( ) 2 2⁄3 2 1 2 1 2 1 1 2 → 𝑈𝑈 ⁄=𝑈𝑈 ( − )3 𝑉𝑉 ⁄𝑉𝑉 → =𝑈𝑈constant⁄𝑈𝑈 . 𝑉𝑉 ⁄𝑉𝑉 (61) 2⁄3 2⁄3 This is consistent→ 𝑇𝑇 2with⁄𝑇𝑇1 the Sackur𝑉𝑉1⁄𝑉𝑉2-Tetrode→ equation,𝑇𝑇𝑉𝑉 Eq.(44), which shows that the entropy does not change when both and are kept constant. Another consequence of Eq.(61) is obtained by multiplying both sides of the equation2⁄3 with , then using the ideal gas law to replace with . 𝑁𝑁 𝑇𝑇𝑉𝑉 We will have 𝐵𝐵 𝐵𝐵 𝑁𝑁𝑘𝑘 𝑁𝑁𝑘𝑘 𝑇𝑇 𝑝𝑝𝑝𝑝 = constant. (62) 5⁄3 This is a well-known relation between𝑝𝑝𝑉𝑉 the pressure and volume of an ideal gas under adiabatic expansion (or compression). More generally, this relation is written as = constant, where, for imperfect , the exponent is below 5 3. The corresponding relation between𝛾𝛾 pressure and energy- density for a non-ideal gas is = ( 1) , indicating that only a fraction𝑝𝑝𝑉𝑉 of the total energy of the atoms/molecules belongs to the𝛾𝛾 kinetic energy⁄ of translational motion, which is directly associated with the pressure of the gas. 𝑝𝑝𝑝𝑝 𝛾𝛾 − 𝑈𝑈

Concluding remarks. Invoking several thermodynamic identities, we have derived and analyzed certain properties of blackbody radiation that were previously discussed, albeit from a different perspective, in an earlier paper [1]. We examined the sources of various contributions to energy fluctuations, the connection between blackbody radiation and the Johnson noise in electronic circuits, and the intimate relation between the statistics of photo-electron counts in photodetection and the random variations of the energy- density of thermal radiation. In the latter part of the paper we derived expressions for the entropy, energy- density, pressure, and chemical potential of an ideal monatomic gas under the conditions of thermal equi- librium. Our goal here has been to compare and contrast the thermodynamic properties of the photon gas (i.e., blackbody radiation) against those of a rarefied gas consisting of rigid, identical particles of matter.

References 1. M. Mansuripur and Pin Han, “Thermodynamics of Radiation Pressure and Photon Momentum,” Optical Trapping and Optical Micromanipulation XIV, edited by K. Dholakia and G.C. Spalding, Proceedings of SPIE Vol. 10347, 103471Y, pp1- 20 (2017). 2. C. Kittel and H. Kroemer, Thermal Physics, 2nd edition, W. H. Freeman and Company, New York (1980). 3. C. Kittel, Elementary Statistical Physics, Dover, New York (2016). 4. A. B. Pippard, The Elements of Classical Thermodynamics, Cambridge, United Kingdom (1964). 5. A. Einstein, “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt” (On a Heuristic Point of View about the Creation and Conversion of Light), Annalen der Physik 17, 132-148 (1905). 6. A. Einstein, “Zum gegenwärtigen Stande des Strahlungsproblems” (On the Present Status of the Radiation Problem), Physikalische Zeitschrift 10, 185–193 (1909). 7. A. Einstein, "Über die Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung" (On the Development of Our Views Concerning the Nature and Composition of Radiation), Physikalische Zeitschrift 10, 817-825 (1909). 8. A. Einstein, “Strahlungs-emission und -absorption nach der Quantentheorie” (Emission and Absorption of Radiation in Quantum Theory), Verhandlungen der Deutschen Physikalischen Gesellschaft 18, 318-323 (1916). 9. A. Einstein, “Quantentheorie der Strahlung” (Quantum Theory of Radiation), Mitteilungen der Physikalischen Gesellschaft, Zürich, 16, 47–62 (1916). 10. A. Einstein, “Zur Quantentheorie der Strahlung” (On the Quantum Theory of Radiation), Physikalische Zeitschrift 18, 121– 128 (1917). 11. E.M. Purcell, Nature 178 (4548), 1449-50 (1956). 12. R. Hanbury Brown and R.Q. Twiss, “Correlation between Photons in two Coherent Beams of Light,” Nature 177, 27 (1956). 13. M. Mansuripur, Classical Optics and Its Applications, 2nd edition, Cambridge University Press, United Kingdom (2009); see Chapter 9.

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