Thermodynamics of Radiation and Momentum Masud Mansuripur† and Pin Han‡ † College of Optical Sciences, the University of Arizona, Tucson, Arizona, USA ‡ Graduate Institute of Precision Engineering, National Chung Hsing University, Taichung, Taiwan [Published in Optical Trapping and Optical Micromanipulation XIV, edited by K. Dholakia and G.C. Spalding, Proceedings of SPIE Vol. 10347, 103471Y, pp1-20 (2017). doi: 10.1117/12.2274589] Abstract. Theoretical analyses of and photon momentum in the past 150 years have focused almost exclusively on classical and/or quantum theories of electrodynamics. In these analyses, Maxwell’s equations, the properties of polarizable and/or magnetizable material media, and the stress tensors of Maxwell, Abraham, Minkowski, Chu, and Einstein-Laub have typically played prominent roles [1-9]. Each stress tensor has subsequently been manipulated to yield its own expressions for the electromagnetic (EM) force, torque, , and linear as well as angular momentum densities of the EM field. This paper presents an alternative view of radiation pressure from the perspective of thermal physics, invoking the properties of blackbody radiation in conjunction with empty as well as -filled cavities that contain EM energy in thermal equilibrium with the container’s walls. In this type of analysis, Planck’s quantum hypothesis, the spectral distribution of the trapped radiation, the of the photon gas, and Einstein’s and coefficients play central roles.

1. Introduction. At a fundamental level, many properties𝐴𝐴 of 𝐵𝐵the electromagnetic (EM) radiation are intimately tied to the basic tenets of thermodynamics, which is essentially the science that seeks answers to the question: “What must the laws of nature be like so that it is impossible to construct a perpetual motion machine of either the first or the second kind? ” This question refers to the empirical fact that perpetual motion machines have never been observed that could violate the first principle (i.e., energy cannot be added or destroyed in an isolated system), or contradict the second principle (i.e., entropy always increases for spontaneous processes in an isolated system) [10]. According to Albert Einstein (1879-1955), “the science of thermodynamics seeks by analytical means to deduce necessary conditions, which separate events have to satisfy, from the universally experienced fact that perpetual motion is impossible.” In his monumental work on EM radiation, Einstein, who was well-versed in thermal physics, drew profound original conclusions from the principles of statistical mechanics and thermodynamics. Einstein’s enthusiasm and admiration for this branch of physical science is palpable in the following reflection on his early career: “A law is more impressive the greater the simplicity of its premises, the more different are the kinds of things it relates, and the more extended its range of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content, which I am convinced, that within the framework of applicability of its basic concepts will never be overthrown.” The goal of the present paper is to review the physics of blackbody radiation and to draw a few conclusions concerning radiation pressure from the principles of thermodynamics and statistical physics. Section 2 provides a detailed derivation of Planck’s blackbody radiation formula from a combination of classical and quantum electrodynamics. In Sec.3 we relate Planck’s formula to the Stefan-Boltzmann law of blackbody radiation. Section 4 describes the radiation pressure inside an evacuated chamber whose walls are maintained at a constant . In Sec.5 we show that the Stefan-Boltzmann law is unaffected when the blackbody chamber is filled with a gas of refractive index ( ). Section 6 explains why the EM momentum density inside a gas-filled𝑇𝑇 chamber at temperature is given by the Abraham formula, despite the fact that the radiation pressure on the walls of the chamber 𝑛𝑛appears𝜔𝜔 to conform with the Minkowski formulation. Fluctuations of thermal radiation are the subject𝑇𝑇 of Sec.7, while different approaches to deriving an expression for the entropy of the ‘photon gas’ form the subject of Sec.8. In Sec.9 we return to Planck’s formula and derive it from a different perspective, this time from a combination of classical electrodynamics and the quantum theory of material media acting as a collection of harmonic oscillators. Section 10 addresses the intimate connection between Einstein’s and coefficients and Planck’s blackbody radiation formula. Certain implications of the second law of 𝐴𝐴 𝐵𝐵

1

thermodynamics for radiation pressure on various surfaces are the subject of Sec.11. In Sec.12 we reconstruct an argument from Einstein’s early papers to show that quantization of radiation is necessary if a hypothetical reflective foil floating inside a radiation-filled (but otherwise empty) chamber at temperature is to behave in accordance with the equipartition theorem of statistical physics. The final section is devoted to a summary of the results and a few concluding remarks. 𝑇𝑇 2. Blackbody radiation. With reference to Fig.1(a), consider a hollow rectangular parallelepiped box of dimensions × × , with perfectly reflecting interior walls, filled with EM radiation of all admissible frequencies. The boundary conditions allow only certain -vectors (and, considering that 𝑥𝑥 𝑦𝑦 𝑧𝑧 | | = , only𝐿𝐿 certain𝐿𝐿 𝐿𝐿 frequencies) in the form of 𝑘𝑘 𝒌𝒌 𝜔𝜔⁄𝑐𝑐 = ( ) + ( ) + ( ) . (1) In the above equation,𝒌𝒌 ( ,𝜋𝜋𝑛𝑛𝑥𝑥, ⁄𝐿𝐿)𝑥𝑥 are𝒙𝒙� arbitrary𝜋𝜋𝑛𝑛𝑦𝑦⁄𝐿𝐿 𝑦𝑦integers,𝒚𝒚� 𝜋𝜋 𝑛𝑛which𝑧𝑧⁄𝐿𝐿𝑧𝑧 may𝒛𝒛� be positive or negative but not zero. For every choice of these three integers, we construct a standing wave within the confines of the 𝑥𝑥 𝑦𝑦 𝑧𝑧 hollow cavity by superposing𝑛𝑛 the𝑛𝑛 eight𝑛𝑛 plane-waves associated with the combinations (± , ± , ± ).

𝑛𝑛𝑥𝑥 𝑛𝑛𝑦𝑦 𝑛𝑛𝑧𝑧 (a) (b) kz

Lz Ly

Lx z π/L ky y z

kx x π/Lx

Fig.1. (a) Rectangular parallelepiped chamber having dimensions × × . The internal walls of the cavity are perfectly reflecting, thus allowing standing EM waves of certain frequencies to survive indefinitely within 𝑥𝑥 𝑦𝑦 𝑧𝑧 the cavity. (b) Allowed -vectors of the resonant modes of the chamber𝐿𝐿 𝐿𝐿 form𝐿𝐿 a discrete mesh in the -space, with each volume element, having dimensions ( ) × ( ) × ( ), containing exactly one -vector. 𝑘𝑘 𝑘𝑘 Note that the constraint = 0 imposed𝜋𝜋⁄𝐿𝐿𝑥𝑥 by𝜋𝜋 Maxwell’s⁄𝐿𝐿𝑦𝑦 𝜋𝜋⁄ 𝐿𝐿1𝑧𝑧st equation must be reflected𝑘𝑘 in the choice of signs for the various -field components. Denoting the -field components by = 0 | | exp(i ), = | | exp𝒌𝒌 ∙ (𝑬𝑬i ), and = | | exp(i ), we write 𝐸𝐸 𝐸𝐸 𝐸𝐸𝑥𝑥0 𝑥𝑥0 𝑥𝑥0 𝑦𝑦0 𝑦𝑦0 𝑦𝑦0 𝑧𝑧0 𝑧𝑧0 𝑧𝑧0 ( ) ( ) 𝐸𝐸( , ) = Real𝜙𝜙 8 𝐸𝐸 exp𝐸𝐸{i[ 𝜙𝜙 ]} 𝐸𝐸 𝐸𝐸 𝜙𝜙 𝑚𝑚 𝑚𝑚 0 𝑬𝑬 𝒓𝒓 𝑡𝑡 � 𝑬𝑬 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝜔𝜔 = Re ( 𝑚𝑚=1+ + ) exp[i( + + )] ( + ) exp[i( + )]

(� 𝐸𝐸𝑥𝑥0𝒙𝒙� 𝐸𝐸𝑦𝑦0𝒚𝒚�+ 𝐸𝐸𝑧𝑧0)𝒛𝒛�exp[i( 𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦+ 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔)𝜔𝜔] +−( 𝐸𝐸𝑥𝑥0𝒙𝒙� 𝐸𝐸𝑦𝑦0𝒚𝒚� − 𝐸𝐸𝑧𝑧0)𝒛𝒛�exp[i( 𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 − 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔)𝜔𝜔] +−(𝐸𝐸𝑥𝑥0𝒙𝒙� − 𝐸𝐸𝑦𝑦0𝒚𝒚� 𝐸𝐸𝑧𝑧0𝒛𝒛�) exp[i(𝑘𝑘𝑥𝑥𝑥𝑥 −+𝑘𝑘𝑦𝑦𝑦𝑦 +𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔 )] 𝐸𝐸(𝑥𝑥0𝒙𝒙� − 𝐸𝐸𝑦𝑦0𝒚𝒚� −+𝐸𝐸𝑧𝑧0𝒛𝒛� ) exp[𝑘𝑘i(𝑥𝑥𝑥𝑥 − 𝑘𝑘+𝑦𝑦𝑦𝑦 − 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔 )] (𝐸𝐸𝑥𝑥0𝒙𝒙� −+ 𝐸𝐸𝑦𝑦0𝒚𝒚� − 𝐸𝐸𝑧𝑧0𝒛𝒛�) exp[i(−𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 + 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔)] −+ (𝐸𝐸𝑥𝑥0𝒙𝒙� −+ 𝐸𝐸𝑦𝑦0𝒚𝒚� + 𝐸𝐸𝑧𝑧0𝒛𝒛�) exp[i(−𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 − 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔)]

𝑥𝑥0 𝑦𝑦0 𝑧𝑧0 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥0 𝑦𝑦0 𝑧𝑧0 𝑥𝑥 𝑦𝑦 𝑧𝑧 = −8𝐸𝐸| 𝒙𝒙�| cos𝐸𝐸 ( 𝒚𝒚� −)𝐸𝐸sin𝒛𝒛� sin−𝑘𝑘( 𝑥𝑥 −) cos𝑘𝑘 𝑦𝑦( 𝑘𝑘 𝑧𝑧 − 𝜔𝜔)𝜔𝜔 8 𝐸𝐸 𝒙𝒙� sin𝐸𝐸( 𝒚𝒚� ) cos𝐸𝐸 𝒛𝒛� sin−(𝑘𝑘 𝑥𝑥)−cos𝑘𝑘 𝑦𝑦 − 𝑘𝑘 𝑧𝑧 − 𝜔𝜔𝜔𝜔

−8|𝐸𝐸𝑥𝑥0| sin( 𝑘𝑘𝑥𝑥𝑥𝑥) sin �𝑘𝑘𝑦𝑦𝑦𝑦�cos(𝑘𝑘𝑧𝑧𝑧𝑧) cos(𝜔𝜔𝜔𝜔 − 𝜙𝜙𝑥𝑥0) 𝒙𝒙�. − �𝐸𝐸𝑦𝑦0� 𝑘𝑘𝑥𝑥𝑥𝑥 �𝑘𝑘𝑦𝑦𝑦𝑦� 𝑘𝑘𝑧𝑧𝑧𝑧 �𝜔𝜔𝜔𝜔 − 𝜙𝜙𝑦𝑦0�(2)𝒚𝒚� It is− clear𝐸𝐸𝑧𝑧0 that 𝑘𝑘the𝑥𝑥𝑥𝑥 -field�𝑘𝑘𝑦𝑦𝑦𝑦 �components𝑘𝑘𝑧𝑧𝑧𝑧 𝜔𝜔parallel𝜔𝜔 − 𝜙𝜙𝑧𝑧0 to𝒛𝒛� the six walls of the chamber vanish at each wall. Where the -field is perpendicular to a surface, it does not vanish at the surface, because the relevant boundary condition is satisfied𝐸𝐸 by the presence of electric charges at these surfaces. 𝐸𝐸 2

Next we calculate the distribution of the -field inside the chamber. The relationship between the components of , and is given by Maxwell’s 3rd equation as × = [2,11]. Thus, changes in the signs of the various components𝐻𝐻 of and must be reflected in the signs of the 0 0 0 0 0 components of 𝑬𝑬 for𝑯𝑯 each 𝒌𝒌plane-wave. Maxwell’s 4th equation requires𝒌𝒌 that𝑬𝑬 𝜇𝜇 𝜔𝜔𝑯𝑯= 0, but this is 0 automatically satisfied when is derived from 𝑬𝑬Maxwell’s𝒌𝒌 3rd equation. Denoting the -field 0 0 components by 𝑯𝑯 = | | exp(i ), etc., we will have 𝒌𝒌 ∙ 𝑯𝑯 𝑯𝑯0 𝐻𝐻 𝑥𝑥0( ) 𝑥𝑥0 ( ) 𝑥𝑥0 ( , ) = Real 8𝐻𝐻 exp𝐻𝐻 {i[ 𝜑𝜑 ]} 𝑚𝑚 𝑚𝑚 0 𝑯𝑯 𝒓𝒓 𝑡𝑡 � 𝑯𝑯 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝜔𝜔 = Re ( 𝑚𝑚=+1 + ) exp[i( + + )] + ( + ) exp[i( + )]

+(� 𝐻𝐻𝑥𝑥0𝒙𝒙� 𝐻𝐻𝑦𝑦0𝒚𝒚�+ 𝐻𝐻𝑧𝑧0)𝒛𝒛�exp[i( 𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦+ 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔)𝜔𝜔] + ( 𝐻𝐻𝑥𝑥0𝒙𝒙� 𝐻𝐻𝑦𝑦0𝒚𝒚� − 𝐻𝐻𝑧𝑧0)𝒛𝒛�exp[i( 𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 − 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔)𝜔𝜔] (𝐻𝐻𝑥𝑥0𝒙𝒙� − 𝐻𝐻𝑦𝑦0𝒚𝒚� 𝐻𝐻𝑧𝑧0𝒛𝒛�) exp[i(𝑘𝑘𝑥𝑥𝑥𝑥 −+𝑘𝑘𝑦𝑦𝑦𝑦 +𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔 )] 𝐻𝐻(𝑥𝑥0𝒙𝒙� − 𝐻𝐻𝑦𝑦0𝒚𝒚� −+𝐻𝐻𝑧𝑧0𝒛𝒛� ) exp[𝑘𝑘i(𝑥𝑥𝑥𝑥 − 𝑘𝑘+𝑦𝑦𝑦𝑦 − 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔 )] −(𝐻𝐻𝑥𝑥0𝒙𝒙� −+ 𝐻𝐻𝑦𝑦0𝒚𝒚� − 𝐻𝐻𝑧𝑧0𝒛𝒛�) exp[i(−𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 + 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔)] − (𝐻𝐻𝑥𝑥0𝒙𝒙� −+ 𝐻𝐻𝑦𝑦0𝒚𝒚� + 𝐻𝐻𝑧𝑧0𝒛𝒛�) exp[i(−𝑘𝑘𝑥𝑥𝑥𝑥 𝑘𝑘𝑦𝑦𝑦𝑦 − 𝑘𝑘𝑧𝑧𝑧𝑧 − 𝜔𝜔𝜔𝜔)]

𝑥𝑥0 𝑦𝑦0 𝑧𝑧0 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥0 𝑦𝑦0 𝑧𝑧0 𝑥𝑥 𝑦𝑦 𝑧𝑧 = − 8|𝐻𝐻 𝒙𝒙�| sin𝐻𝐻( 𝒚𝒚�)−cos𝐻𝐻 𝒛𝒛� cos−( 𝑘𝑘 𝑥𝑥)−sin𝑘𝑘(𝑦𝑦 𝑘𝑘 𝑧𝑧 −)𝜔𝜔𝜔𝜔+ 8− 𝐻𝐻 cos𝒙𝒙� (𝐻𝐻 𝒚𝒚�) sin𝐻𝐻 𝒛𝒛� cos(−𝑘𝑘 )𝑥𝑥sin− 𝑘𝑘 𝑦𝑦 − 𝑘𝑘 𝑧𝑧 − 𝜔𝜔 𝜔𝜔

+8𝐻𝐻| 𝑥𝑥0 | cos𝑘𝑘(𝑥𝑥𝑥𝑥 ) cos�𝑘𝑘𝑦𝑦𝑦𝑦� sin(𝑘𝑘𝑧𝑧𝑧𝑧 ) sin(𝜔𝜔𝜔𝜔 − 𝜑𝜑𝑥𝑥0 )𝒙𝒙� . �𝐻𝐻𝑦𝑦0� 𝑘𝑘𝑥𝑥𝑥𝑥 �𝑘𝑘𝑦𝑦𝑦𝑦� 𝑘𝑘𝑧𝑧𝑧𝑧 �𝜔𝜔𝜔𝜔 − 𝜑𝜑𝑦𝑦0� 𝒚𝒚�(3)

The 𝐻𝐻-𝑧𝑧field0 components𝑘𝑘𝑥𝑥𝑥𝑥 �𝑘𝑘𝑦𝑦𝑦𝑦 that� are𝑘𝑘𝑧𝑧𝑧𝑧 perpendicular𝜔𝜔𝜔𝜔 − 𝜑𝜑𝑧𝑧0 to𝒛𝒛� a given surface are seen to vanish at that surface, thus satisfying the boundary condition for the perpendicular -field. The tangential -fields, however, do not have to𝐻𝐻 vanish at various surfaces, as they are matched by the induced surface currents [2,11]. Each of the above standing waves constitutes a mode of𝐵𝐵 the resonator (or cavity)𝐻𝐻 . In fact, for each triplet of positive integers ( , , ), there exist two resonator modes, because each standing wave can have two independent states of polarization. This is due to the fact that the -field amplitudes 𝑥𝑥 𝑦𝑦 𝑧𝑧 ( , , ) are constrained𝑛𝑛 𝑛𝑛by 𝑛𝑛Maxwell’s 1st equation to satisfy = 0 and, therefore, the initial choice of represents two (rather than three) degrees of freedom for a plane-wave𝐸𝐸 whose -vector is 𝑥𝑥0 𝑦𝑦0 𝑧𝑧0 0 already𝐸𝐸 𝐸𝐸 fixed𝐸𝐸 by the choice of ( , , ). 𝒌𝒌 ∙ 𝑬𝑬 0 The volume𝑬𝑬 occupied by each cavity mode in the -space is = ( ); see Eq.𝑘𝑘(1). Using 𝑛𝑛𝑥𝑥 𝑛𝑛𝑦𝑦 𝑛𝑛𝑧𝑧 the dispersion relation + + = ( ) for plane-waves in 3vacuum3 , we compute the number of 𝑥𝑥 𝑦𝑦 𝑧𝑧 modes in the -space confined2 2 to a thin2 spherical2 shell of𝑘𝑘 radius =𝛿𝛿 𝜋𝜋 and� 𝐿𝐿thickness𝐿𝐿 𝐿𝐿 d = d to be 𝑥𝑥 𝑦𝑦 𝑧𝑧 4 d = 4 𝑘𝑘 d𝑘𝑘 ( 𝑘𝑘 ). Only𝜔𝜔⁄𝑐𝑐 one octant of the spherical shell, however, corresponds to positive2 values3 𝑘𝑘 of ( , ,2 ), which2 3 represent the various standing𝑘𝑘 waves𝜔𝜔⁄𝑐𝑐 . Therefore, the𝑘𝑘 above𝜔𝜔 ⁄number𝑐𝑐 𝑥𝑥 𝑦𝑦 𝑧𝑧 of𝜋𝜋 𝑘𝑘modes𝑘𝑘⁄ 𝛿𝛿must be𝐿𝐿 divided𝐿𝐿 𝐿𝐿 𝜔𝜔 by𝜔𝜔 ⁄8, 𝜋𝜋then𝑐𝑐 multiplied by 2 to account for the two polarization states of each 𝑥𝑥 𝑦𝑦 𝑧𝑧 standing wave. The𝑛𝑛 number𝑛𝑛 𝑛𝑛 of modes thus obtained may be further normalized by the volume of the cavity, as we are primarily interested in the energy-density (i.e., energy per unit volume) of the 𝑥𝑥 𝑦𝑦 𝑧𝑧 resonant EM waves. The number-density of all allowed modes in the frequency interval ( , 𝐿𝐿+𝐿𝐿d𝐿𝐿 ) is thus given by d ( ). Each resonant2 cavity2 mode3 associated with frequency may be occupied by identical𝜔𝜔 𝜔𝜔 𝜔𝜔 of energy with𝜔𝜔 𝜔𝜔a ⁄probability𝜋𝜋 𝑐𝑐 proportional to exp( ). Here is an arbitrary integer 𝜔𝜔 𝑚𝑚 (0, 1, 2, ), = 1.054572 × 10 is Planck’s reduced𝐵𝐵 constant, = 1.38065 × 10 °ℏ𝜔𝜔 is Boltzmann’s constant−34, and is the absolute−𝑚𝑚 temperatureℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇 of the𝑚𝑚 cavity. (The cavity walls−23, being perfect⋯ ℏ reflectors, cannot emit 𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽any radiation∙ 𝑠𝑠𝑠𝑠𝑠𝑠 into the chamber. One must 𝑘𝑘assume𝐵𝐵 that a material body𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝑒𝑒 ⁄of 𝐾𝐾temperature has resided in the𝑇𝑇 cavity for a sufficiently long time to have established thermal equilibrium with the EM field, before being removed without disturbing the radiation state.) The occupation probability𝑇𝑇 of a given mode of frequency at temperature is exp( ) normalized by the so-called partition function ( , ), which is defined as 𝐵𝐵 𝜔𝜔 𝑇𝑇 −𝑚𝑚ℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇 ( , ) = exp( ) = 𝑍𝑍 𝜔𝜔 𝑇𝑇 ( )· (4) 1 𝑚𝑚∞=0 𝐵𝐵 𝑍𝑍 𝜔𝜔 𝑇𝑇 ∑ −𝑚𝑚ℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇 1 − exp −ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 3

The derivative with respect to of the natural logarithm of the partition function yields the average photon number occupancy for the mode under consideration, as follows: 𝜔𝜔 [ ( , )] ( ) = = ( ) = ( ) ∞ ( ) . (5) 𝜕𝜕 ln 𝑍𝑍 𝜔𝜔 𝑇𝑇 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 ∑𝑚𝑚=0 𝑚𝑚 exp −𝑚𝑚ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 𝐵𝐵 ∞ 𝐵𝐵 The desired derivati𝜕𝜕𝜕𝜕 ve is readily𝑍𝑍 found− ℏfrom⁄𝑘𝑘 𝑇𝑇Eq.(∑4)𝑚𝑚 to=0 beexp −𝑚𝑚ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 −ℏ〈𝑚𝑚〉⁄ 𝑘𝑘 𝑇𝑇 [ ( , )] [ ( )] ( ) = = ( ) ( ) · (6) 𝜕𝜕 ln 𝑍𝑍 𝜔𝜔 𝑇𝑇 𝜕𝜕 1 − exp −ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 ⁄𝜕𝜕𝜕𝜕 ℏ⁄ 𝑘𝑘𝐵𝐵𝑇𝑇 Comparing Eq.(5) with 𝜕𝜕Eq.(𝜕𝜕 6), we− find 1 − exp −ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 − exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 − 1 = ( ) . (7) 1 As expected, the occupancy of 〈modes𝑚𝑚〉 expwhoseℏ𝜔𝜔 ⁄𝑘𝑘𝐵𝐵𝑇𝑇 is− below1 or comparable to is large, whereas modes whose photon energy far exceeds are sparsely populated. The energy𝐵𝐵 density of black- ℏ𝜔𝜔 𝑘𝑘 𝑇𝑇 body radiation in the frequency interval ( , 𝐵𝐵+ d ) may now be obtained by multiplying the mode number density d ( ℏ) 𝜔𝜔with the average𝑘𝑘 𝑇𝑇 photon energy per mode, . The energy density (per unit volume per2 unit angular2 3 frequency)𝜔𝜔 is𝜔𝜔 thus found𝜔𝜔 to be [11-13] 𝜔𝜔 𝜔𝜔⁄ 𝜋𝜋 𝑐𝑐 〈𝑚𝑚〉ℏ𝜔𝜔 ( , ) = · (8) [ ( 3 ) ] ℏ𝜔𝜔 2 3 This is the well-known blackℰbody𝜔𝜔 𝑇𝑇 radiation𝜋𝜋 𝑐𝑐 energyexp ℏ𝜔𝜔-⁄density𝑘𝑘𝐵𝐵𝑇𝑇 − 1(energy per unit volume per unit angular frequency) that has been associated with the name of Max Planck. Integrating ( , ) over all frequencies ranging from 0 to , yields a total energy-density (per unit volume) proportional to , in ℰ 𝜔𝜔 𝑇𝑇 agreement with the Stefan-Boltzmann law. Gradshteyn & Ryzhik 3.411-17 [26]4 𝜔𝜔 ∞ 𝑇𝑇 ( ) = ( , )d = d = = . ∞ [ ( 3 ) ] 4 4 ∞ (3 ) 2 4 ∞ ℏ𝜔𝜔 𝑘𝑘𝐵𝐵𝑇𝑇 𝑥𝑥 d𝑥𝑥 𝜋𝜋 𝑘𝑘𝐵𝐵 4 2 3 2 3 3 3 3 0 𝜋𝜋 𝑐𝑐 exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 − 1 𝜋𝜋 𝑐𝑐 ℏ exp 𝑥𝑥 − 1 15𝑐𝑐 ℏ (9) 𝑢𝑢 𝑇𝑇 ∫ ℰ 𝜔𝜔 𝑇𝑇 𝜔𝜔 �0 𝜔𝜔 � � �0 � � 𝑇𝑇 At low frequencies or high , the exponential factor in the denominator on the right- hand-side of Eq.(8) may be approximated as 1 + ( ), thus reducing the energy-density formula to ( , ) ( ). This is the Rayleigh-Jeans 𝐵𝐵 UV Visible

Infrared ) formula for the2 energy2 density3 of blackbody radiation,ℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇

𝐵𝐵 1 ℰ 𝜔𝜔 𝑇𝑇 ≅ 𝜔𝜔 𝑘𝑘 𝑇𝑇⁄ 𝜋𝜋 𝑐𝑐 − derived under the assumption that each cavity mode, in 𝑚𝑚 5000 K 𝑛𝑛 ∙ accordance with the equi-partition theorem [11-14], 2 partakes equally of the available energy in the amount − Classical Theory (5000 K) 𝑚𝑚 ∙ of ½ per available degree of freedom. Since each 1 − 𝑟𝑟 𝑠𝑠 cavity 𝐵𝐵mode was deemed to behave as a harmonic ∙ oscillator𝑘𝑘 𝑇𝑇 holding ½ of kinetic energy and ½ 𝑘𝑘𝑘𝑘 of potential energy (i.e., electric and magnetic field ( 𝐵𝐵 4000 K of the mode𝑘𝑘 ),𝑇𝑇 a total energy of 𝑘𝑘was𝐵𝐵𝑇𝑇

assigned to each mode. 𝐵𝐵 𝑘𝑘 𝑇𝑇 3000 K Fig.2. Plots of the blackbody energy density versus the 0 2 4 6 8 10 12 14 12 10 8 6 4 2 0 Spectral radiance radiance Spectral wavelength for three different temperatures. Also shown is a 0 0.5 1.0 1.5 2.0 2.5 3.0 plot of the Rayleigh-Jeans energy density at = 5000K. Wavelength ( )

The Rayleigh-Jeans theory gave the𝑇𝑇 correct explanation for the low-frequency end𝜇𝜇 of𝜇𝜇 the blackbody spectrum, but it hugely over-estimated the high-energy (i.e., short-wavelength) part of the spectrum. This major disagreement between the prevailing theory of the day and experimental facts came to be known as the ultraviolet catastrophe, which was resolved only after Max Planck advanced his quantum hypothesis in the closing days of the 19th century.

4

3. Energy emanating from the surface of a blackbody. With reference to Fig.3, consider a small aperture of area on a wall of the blackbody examined in the preceding section. During a short time interval , a ray (i.e., -vector) emerging at an angle relative to the -axis brings out ½ cos times the energy-𝛿𝛿density𝛿𝛿 associated with that -vector. The reason𝛿𝛿𝛿𝛿 for the ½ factor𝑘𝑘 is that for every -vector 𝜃𝜃 𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 with a positive that leaves the box, there exists𝑘𝑘 one with a negative , which would not come out𝑘𝑘 of the 𝑥𝑥 aperture. The remaining𝑘𝑘 factor, cos , is the θ 𝑥𝑥 x volume of a parallelepiped𝑘𝑘 with base and height along the emergent ray. Now, 𝑐𝑐the𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 average𝜃𝜃 value of cos among all the -vectors in the first𝛿𝛿𝛿𝛿 octant of the 𝑐𝑐𝑐𝑐𝑐𝑐 -space is cos sin d = ½. The energy 𝜃𝜃 ⁄𝑘𝑘 emanating from the𝜋𝜋 aperture2 per unit area per unit time Fig.3. Radiation emerging from a small aperture 𝑥𝑥 𝑦𝑦 𝑧𝑧 0 per𝑘𝑘 𝑘𝑘 unit𝑘𝑘 angular frequency∫ 𝜃𝜃is thus𝜃𝜃 given𝜃𝜃 by of area on the surface of a blackbody box.

( , ) = ( , ) = 𝛿𝛿𝛿𝛿 · (10) [ ( 3 ) ] 1 ℏ𝜔𝜔 2 2 When integrated over all𝐼𝐼 frequencies,𝜔𝜔 𝑇𝑇 4 𝑐𝑐theℰ 𝜔𝜔above𝑇𝑇 formula4𝜋𝜋 𝑐𝑐 yieldsexp ℏ𝜔𝜔 ⁄𝑘𝑘𝐵𝐵𝑇𝑇 − 1 ( , )d = = 2 4 . (11) 𝐵𝐵 ∞ 𝜋𝜋 𝑘𝑘 4 4 2 4 The numerical value of , the Stefan-Boltzmann 2constant,3 is 5.670373(21) × 10 J/(s·m ·K ). A ∫0 𝐼𝐼 𝜔𝜔 𝑇𝑇 𝜔𝜔 �60𝑐𝑐 ℏ � 𝑇𝑇 𝜎𝜎𝑇𝑇 direct derivation of the Stefan-Boltzmann law from thermodynamics principles is given −in8 Appendix A. 𝜎𝜎 4. Radiation pressure inside the blackbody chamber. The momentum-density of a plane-wave propagating in free space along a given -vector is equal to the energy-density divided by the speed of light, . The direction of the momentum, of course, is aligned with the -vector. Consider a small volume in the shape of a parallelepiped adjacent𝑘𝑘 to a wall inside the blackbody of Fig.1. The volume of the parallelepiped𝑐𝑐 is cos , where is the base area, is a short𝑘𝑘 time interval, and is the angle between the surface-normal and the particular -vector under consideration. For each -vector pointing toward the wall, there𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 is an 𝜃𝜃identical 𝛿𝛿-𝛿𝛿vector pointing away,𝛿𝛿𝛿𝛿 so only half the momentum 𝜃𝜃content of the parallelepiped bounces off its adjacent wall. The𝑘𝑘 change of momentum upon reflection𝑘𝑘 from the wall involves multiplication by the factor𝑘𝑘 2 cos , because the perpendicular component of the incident momentum changes sign upon reflection, whereas the tangential component remains unchanged. Thus, the net momentum transferred to the wall during𝜃𝜃 the time interval is ½(2 cos ) times the momentum density. Dividing this transferred momentum by yields the force, and further2 division by yields the pressure on the wall, so the pressure exerted by each mode𝛿𝛿𝛿𝛿 (corresponding𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 to𝜃𝜃 a given - vector) is the energy density of the mode multiplied by cos 𝛿𝛿𝛿𝛿. Since the average cos over all the - 𝛿𝛿𝛿𝛿 𝑘𝑘 vectors in the first octant of the -space is cos 2 sin d = , the radiation2 pressure inside 𝜃𝜃 𝜃𝜃 𝑘𝑘 the chamber will be 𝜋𝜋⁄2 2 𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦𝑘𝑘𝑧𝑧 0 𝜃𝜃 𝜃𝜃 𝜃𝜃 ⅓ ( )∫= ( ). (12) 1 An alternative determination of the pressure inside the chamber involves calculating the EM force 𝑝𝑝 𝑇𝑇 3 𝑢𝑢 𝑇𝑇 exerted on each mirror surface. With reference to Fig.1, consider the × surface located at = . The perpendicular component of the -field acting on this surface is given in Eq.(2), namely, 𝐿𝐿𝑦𝑦 𝐿𝐿𝑧𝑧 𝑥𝑥 𝐿𝐿𝑥𝑥 ( = , , , ) = 8| 𝐸𝐸 | cos( ) sin( ) sin( ) cos( ). (13) The induced𝐸𝐸𝑥𝑥 𝑥𝑥 𝐿𝐿surface𝑥𝑥 𝑦𝑦 𝑧𝑧 -𝑡𝑡charge−-density𝐸𝐸𝑥𝑥0 on 𝜋𝜋the𝑛𝑛𝑥𝑥 mirror𝑘𝑘 𝑦𝑦at𝑦𝑦 = 𝑘𝑘𝑧𝑧𝑧𝑧 will thus𝜔𝜔𝜔𝜔 −be𝜙𝜙 𝑥𝑥0 = , yielding a force per unit area = ½ = ½ = 32 | | sin ( ) sin ( ) cos ( 0 ). The 𝑥𝑥 𝐿𝐿𝑥𝑥 𝜎𝜎𝑠𝑠 −𝜀𝜀 𝐸𝐸𝑥𝑥 0 2 0 2 2 2 2 𝐹𝐹𝑥𝑥 𝜎𝜎𝑠𝑠𝐸𝐸𝑥𝑥 − 𝜀𝜀 𝐸𝐸𝑥𝑥 − 𝜀𝜀 𝐸𝐸𝑥𝑥0 𝑘𝑘𝑦𝑦𝑦𝑦 𝑘𝑘𝑧𝑧𝑧𝑧 𝜔𝜔𝜔𝜔 − 𝜙𝜙𝑥𝑥0 5

factor ½ in the above equation arises from the fact that the average -field acting on the surface charge is only half the -field at the mirror surface, as the -field inside the perfectly conducting wall is zero. Time averaging as well as averaging over the and coordinates then 𝐸𝐸yields = 4 | | . In similar fashion, we find𝐸𝐸 the force of the magnetic field on𝐸𝐸 the mirror located at = by calculating2 first the - 𝑥𝑥 0 𝑥𝑥0 field from Eq.(3) followed by the induced 𝑦𝑦surface𝑧𝑧 current-density , as follows:〈𝐹𝐹 〉 − 𝜀𝜀 𝐸𝐸 𝑥𝑥 𝐿𝐿𝑥𝑥 𝐻𝐻 ( = , , , ) = 8 cos( ) | | sin( ) cos(𝑱𝑱𝑠𝑠 ) sin( ) | 0 | ( ) ( ) ( ) 𝑯𝑯 𝑥𝑥 𝐿𝐿𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑡𝑡 𝜋𝜋𝑛𝑛𝑥𝑥 �+𝐻𝐻𝑦𝑦 cos𝑘𝑘𝑦𝑦𝑦𝑦 sin𝑘𝑘𝑧𝑧𝑧𝑧 sin𝜔𝜔𝜔𝜔 − 𝜑𝜑𝑦𝑦0 𝒚𝒚� . (14) 0 ( = , , , ) = 8 cos( ) | 𝐻𝐻𝑧𝑧| cos( 𝑘𝑘𝑦𝑦)𝑦𝑦sin( 𝑘𝑘𝑧𝑧)𝑧𝑧sin( 𝜔𝜔𝜔𝜔 − 𝜑𝜑𝑧𝑧)0 𝒛𝒛��

| 0 | sin( ) cos( ) sin( ) 𝑱𝑱𝑠𝑠 𝑥𝑥 𝐿𝐿𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑡𝑡 𝜋𝜋𝑛𝑛𝑥𝑥 � 𝐻𝐻𝑧𝑧 𝑘𝑘𝑦𝑦𝑦𝑦 𝑘𝑘𝑧𝑧𝑧𝑧 𝜔𝜔𝜔𝜔 − 𝜑𝜑𝑧𝑧0 𝒚𝒚� . (15) The Lorentz force-density (i.e., force per− unit𝐻𝐻𝑦𝑦0 area)𝑘𝑘 due𝑦𝑦𝑦𝑦 to the𝑘𝑘𝑧𝑧 𝑧𝑧action 𝜔𝜔of𝜔𝜔 the− 𝜑𝜑 𝑦𝑦0-field𝒛𝒛�� on the above surface current is thus seen to be 𝐵𝐵 = ½ × = 32 | | sin ( ) cos ( ) sin ( )

0 0 2 2 2 2 𝑭𝑭 𝑱𝑱𝑠𝑠 𝜇𝜇 𝑯𝑯 𝜇𝜇 �+𝐻𝐻|𝑦𝑦0 | cos𝑘𝑘(𝑦𝑦𝑦𝑦 ) sin 𝑘𝑘(𝑧𝑧𝑧𝑧 ) sin 𝜔𝜔(𝜔𝜔 − 𝜑𝜑𝑦𝑦0 ) . (16) 2 2 2 2 Once again, the factor ½ in Eq.(16)𝐻𝐻𝑧𝑧0 arises from𝑘𝑘𝑦𝑦𝑦𝑦 the fact𝑘𝑘𝑧𝑧 𝑧𝑧that the 𝜔𝜔effective𝜔𝜔 − 𝜑𝜑𝑧𝑧0 �𝒙𝒙�-field acting on the surface current is only half the -field at the mirror surface, as the -field inside the perfectly conducting wall is zero. Time averaging as well as averaging over the and coordinates𝐵𝐵 yields = 4 (| | + | | ). Next, we𝐵𝐵 sum the average force densities exerted𝐵𝐵 on the three walls of the box that 𝑦𝑦 𝑧𝑧 〈𝐹𝐹𝑥𝑥〉 are0 perpendicular2 to2 the , , and axes to obtain 𝜇𝜇 𝐻𝐻𝑦𝑦0 𝐻𝐻𝑧𝑧0 + + 𝑥𝑥 𝑦𝑦= 4 𝑧𝑧 | | + | | + | | + 8 | | + | | + | | 0 2 2 2 0 2 2 2 〈𝐹𝐹𝑥𝑥〉 〈𝐹𝐹𝑦𝑦〉 〈𝐹𝐹𝑧𝑧〉 = −4 𝜀𝜀 |� 𝐸𝐸𝑥𝑥|0 + | 𝐸𝐸𝑦𝑦|0 + | 𝐸𝐸𝑧𝑧|0 .� 𝜇𝜇 � 𝐻𝐻𝑥𝑥0 𝐻𝐻𝑦𝑦0 𝐻𝐻𝑧𝑧0 � (17) 0 2 2 2 Here we have used the fact𝜀𝜀 � that,𝐸𝐸𝑥𝑥0 for plane𝐸𝐸𝑦𝑦0 waves,𝐸𝐸𝑧𝑧0 � = (see Appendix B). The right-hand-side of Eq.(17) is the average energy-density of the0 mode, namely,0 𝜀𝜀 𝑬𝑬0 ∙ 𝑬𝑬0 𝜇𝜇 𝑯𝑯0 ∙ 𝑯𝑯0 ½ + ½ = = 4 , (18)

0 0 0 0 as can be readily confirmed by〈 inspecting𝜀𝜀 𝑬𝑬 ∙ 𝑬𝑬 Eq𝜇𝜇.(2).𝑯𝑯 ∙ 𝑯𝑯Since〉 𝜀𝜀Eq〈𝑬𝑬.(17)∙ 𝑬𝑬〉 is valid𝜀𝜀 𝑬𝑬 for0 ∙ 𝑬𝑬each0 and every mode, its sum over all admissible modes leads once again to Eq.(12), which relates the overall pressure ( ) to the total energy-density ( ). Here we have invoked the symmetry of the problem, which ensures that the pressure is the same on all three walls, thus allowing one to replace the left-hand-side 𝑝𝑝of 𝑇𝑇Eq.(17) with 3 ( ), albeit after𝑢𝑢 𝑇𝑇adding up the contributions of all admissible modes.

5.𝑝𝑝 Blackbody𝑇𝑇 radiation emerging from a gas-filled chamber. Returning to Fig.1, we now let the cuboid chamber of dimensions × × with perfectly-electrically-conducting (PEC) walls be filled with a homogeneous and transparent medium (e.g., a dilute gas) of refractive index ( ). As before, we assume 𝑥𝑥 𝑦𝑦 𝑧𝑧 that the walls of the box 𝐿𝐿at temperature𝐿𝐿 𝐿𝐿 are in thermal equilibrium with the EM radiation inside the box. Moreover, we assume that the transparent medium that has uniformly filled the𝑛𝑛 chamber𝜔𝜔 is also in thermal equilibrium with the walls. Now, the wavelength𝑇𝑇 of each mode is reduced by the factor ( ), which causes the mode-density inside the chamber to be that of an empty chamber multiplied by ( ). The spectral energy-density ( , ) of the trapped radiation is thus greater, by the cube of the𝑛𝑛 𝜔𝜔 3refractive index, than that in an otherwise empty chamber. 𝑛𝑛 𝜔𝜔 Despite the increasedℰ intensity𝜔𝜔 𝑇𝑇 of trapped radiation, the Stefan-Boltzmann law will not be affected by the presence of the medium that occupies the chamber. This is because the rays attempting to escape the cavity through the small aperture depicted in Fig.3 will suffer total internal reflection when their propagation angle relative to the -axis exceeds the critical angle = sin [1 ( )]. (In the

crit −1 𝑥𝑥 𝜃𝜃 ⁄𝑛𝑛 𝜔𝜔 6

following discussion we shall ignore the Fresnel reflection losses at the interface between the host medium of refractive index ( ) and the free-space region outside the box.) For a ray propagating inside the box at an angle relative to the -axis, the volume of a small parallelepiped with base and height ( ) along the incident ray𝑛𝑛 𝜔𝜔 will be ( ) cos . As mentioned earlier, half the EM energy inside this volume will exit𝜃𝜃 through the aperture𝑥𝑥 during the time provided that < . Now,𝛿𝛿𝛿𝛿 the average 𝑐𝑐⁄𝑛𝑛 𝛿𝛿𝛿𝛿 𝑐𝑐⁄𝑛𝑛 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 𝜃𝜃 value of cos for rays in the first octant of the -space that leave the chambercrit is given by 𝛿𝛿𝛿𝛿 𝜃𝜃 𝜃𝜃

𝜃𝜃 cos sin 𝑘𝑘=𝑥𝑥𝑘𝑘½𝑦𝑦𝑘𝑘sin𝑧𝑧 = 1 (2 ), (19) 𝜃𝜃crit 2 crit 2 which is the value of ½ obtained∫0 earlier𝜃𝜃 𝜃𝜃for𝑑𝑑 𝑑𝑑an empty chamber𝜃𝜃 divided⁄ 𝑛𝑛 by ( ). Another division by ( ) is occasioned by the aforementioned fact that the speed of light inside2 the chamber has dropped from to ( ). The net result is that the emitted energy (per unit area per𝑛𝑛 unit𝜔𝜔 time per unit angular frequency)𝑛𝑛 𝜔𝜔 must now be divided by ( ), which cancels out the corresponding increase in the mode- density𝑐𝑐 within𝑐𝑐⁄𝑛𝑛 the𝜔𝜔 chamber. The Stefan3-Boltzmann emission law given by Eqs.(10) and (11) thus remains unchanged, in spite of the ( ) enhancement𝑛𝑛 𝜔𝜔 of the EM energy-density within the material medium that has uniformly filled the chamber.3 𝑛𝑛 𝜔𝜔 6. Electromagnetic momentum inside a Fabry-Perot cavity. Figure 4 shows a Fabry-Perot cavity formed between two perfectly-electrically-conducting (PEC) flat mirrors located at = 0 and = . The medium filling the gap between the mirrors has a refractive index , and the EM wave trapped between the mirrors is the superposition of a pair of plane-waves propagating along the𝑧𝑧 -axis, each𝑧𝑧 having𝐿𝐿 angular frequency , -vector = ±( ) , and wavelength 𝑛𝑛= 2 ( ). The gap between the mirrors is an integer-multiple of half the wavelength, i.e., = 2 = ( ), where𝑧𝑧 = 1, 2, 3, is an arbitrary positive𝜔𝜔 𝑘𝑘integer. Assuming𝒌𝒌 𝑛𝑛𝑛𝑛 the⁄𝑐𝑐 𝒛𝒛�-field is linearly polarized𝜆𝜆 𝜋𝜋 along𝜋𝜋⁄ 𝑛𝑛𝑛𝑛 the -axis, the - and - field distributions within the cavity are 𝐿𝐿 𝑚𝑚𝑚𝑚⁄ 𝑚𝑚𝑚𝑚𝑚𝑚⁄ 𝑛𝑛𝑛𝑛 𝑚𝑚 ⋯ 𝐸𝐸 𝑥𝑥 𝐸𝐸 𝐻𝐻 ( , ) = [cos( ) cos( + )] = 2 sin( ) sin( ) . (20a)

( 𝑬𝑬, )𝑧𝑧=𝑡𝑡 [cos𝐸𝐸0 ( 𝑘𝑘𝑘𝑘 −)𝜔𝜔+𝜔𝜔 cos− ( +𝑘𝑘𝑘𝑘 )𝜔𝜔] 𝜔𝜔 =𝒙𝒙�2( 𝐸𝐸0 ) cos𝑘𝑘𝑘𝑘 ( )𝜔𝜔cos𝜔𝜔 (𝒙𝒙� ) . (20b) Here 𝑯𝑯 𝑧𝑧=𝑡𝑡 𝐻𝐻0 is the𝑘𝑘𝑘𝑘 −impedance𝜔𝜔𝜔𝜔 of𝑘𝑘 𝑘𝑘free𝜔𝜔 space,𝜔𝜔 𝒚𝒚� while𝑛𝑛𝑛𝑛 0⁄=𝑍𝑍01 𝑘𝑘𝑘𝑘 is the𝜔𝜔 𝜔𝜔speed𝒚𝒚� of light in vacuum. In the system of units adopted here, the 0 0 0 x 0 0 permeability𝑍𝑍 and� permittivity𝜇𝜇 ⁄𝜀𝜀 of free space are denoted by 𝑐𝑐 ⁄�𝜇𝜇 𝜀𝜀 and , respectively.𝑆𝑆𝑆𝑆 It is further assumed that the gap

medium0 0is non-magnetic, that is, its relative permeability is𝜇𝜇 ( )𝜀𝜀= 1.0, while its relative permittivity is given by ( ) = . The dielectric susceptibility of the gap 𝐸𝐸𝑥𝑥 medium𝜇𝜇 𝜔𝜔 is2 thus specified as ( ) = 1. z 𝜀𝜀 𝜔𝜔 𝑛𝑛 2 Fig.4. Fabry-Perot cavity formed𝜒𝜒 𝜔𝜔between𝑛𝑛 a −pair of parallel PEC mirrors. The gap medium has length and refractive index . 𝑦𝑦 The standing wave residing in the region between the mirrors 𝐻𝐻 = 2 𝐿𝐿 = 2 ( ) 𝑛𝑛 has angular frequency , wavelength , electric 𝐿𝐿 𝑚𝑚𝑚𝑚⁄ field , and magnetic field . PEC Mirror PEC Mirror 𝜔𝜔 𝜆𝜆 𝜋𝜋𝜋𝜋⁄ 𝑛𝑛𝑛𝑛 The energy𝐸𝐸𝑥𝑥 -density of the𝐻𝐻 EM𝑦𝑦 field may now be calculated as follows [2,11]: ( , ) = ½ + ½ = 2 0 sin ( )0 sin ( ) + 2 cos ( ) cos ( ) ℰ 𝑧𝑧 𝑡𝑡 𝜀𝜀 𝜀𝜀𝑬𝑬 ∙ 𝑬𝑬 𝜇𝜇 𝑯𝑯 ∙ 𝑯𝑯 2 2 2 2 2 2 2 0 {[ ( )] ( )0 [ ( )] ( )} = 𝜀𝜀 𝑛𝑛 𝐸𝐸0 1 cos𝑘𝑘𝑘𝑘 2 𝜔𝜔sin𝜔𝜔 𝜇𝜇 +𝐻𝐻01 + cos𝑘𝑘𝑘𝑘2 cos𝜔𝜔𝜔𝜔 2 2 2 2 0 [ ( ) ( )] = 𝑛𝑛 𝜀𝜀 𝐸𝐸0 1 +−cos 2 𝑘𝑘𝑘𝑘 cos 2 𝜔𝜔𝜔𝜔 . 𝑘𝑘𝑘𝑘 𝜔𝜔𝜔𝜔 (21) 2 0 2 𝑛𝑛 𝜀𝜀 𝐸𝐸0 𝑘𝑘𝑘𝑘 𝜔𝜔𝜔𝜔 7

Denoting the cross-sectional area of the cavity by , the total EM energy trapped in the cavity, , is seen to be constant in time. One may argue that, at any given instant, the number of photons propagating2 2 rightward (and also leftward) is ½ 𝐴𝐴 . Considering that the propagation velocity 0 0 is𝑛𝑛 𝜀𝜀 𝐸𝐸, 𝐴𝐴the𝐴𝐴 average number of photons per unit area2 per2 unit time that impinge on each mirror is seen to 0 0 be ½ . 𝑛𝑛 𝜀𝜀 𝐸𝐸 𝐴𝐴𝐴𝐴⁄ℏ𝜔𝜔 𝑐𝑐⁄𝑛𝑛 The0 radiation2 pressure (i.e., force per unit area) on the PEC mirror located at = is evaluated in accordance𝑛𝑛𝜀𝜀 𝐸𝐸0 𝑐𝑐with⁄ℏ𝜔𝜔 the Lorentz force law [2,7,11], as follows: 𝑧𝑧 𝐿𝐿 ( = , ) = ½ × = ( ) cos( ) cos( ) × 2 ( ) cos( ) cos( ) ( ) 0 𝑭𝑭 𝑧𝑧 𝐿𝐿 𝑡𝑡 𝑱𝑱𝑠𝑠 𝑩𝑩 = 2𝑛𝑛𝑛𝑛0⁄𝑍𝑍0 cos 𝑘𝑘𝑘𝑘 . 𝜔𝜔𝜔𝜔 𝒙𝒙� 𝜇𝜇 𝑛𝑛𝑛𝑛0⁄𝑍𝑍0 𝑘𝑘𝑘𝑘 𝜔𝜔𝜔𝜔 𝒚𝒚� (22) 2 2 2 = ( = 0 , ) × In the above equation, 𝑛𝑛 𝜀𝜀 𝐸𝐸0 𝜔𝜔 is𝜔𝜔 the𝒛𝒛� surface-current-density at the mirror surface, and = ( = , ) is the magnetic -field− immediately before the mirror. The factor ½ appearing in the 𝑱𝑱𝑠𝑠 𝑯𝑯 𝑧𝑧 𝐿𝐿 𝑡𝑡 𝒛𝒛� Lorentz0 force expression− accounts for the fact that the effective -field acting on the surface-current is the𝑩𝑩 average𝜇𝜇 𝑯𝑯 𝑧𝑧 of 𝐿𝐿the 𝑡𝑡 -fields at = 𝐵𝐵, where = 0, and = . (The pressure on the mirror at = 0 is 𝑠𝑠 the same as that given by Eq.(22), except+ that its direction is reversed.)−𝐵𝐵 Note that the number of photons𝑱𝑱 impinging on each𝐵𝐵 mirror per𝑧𝑧 unit𝐿𝐿 area per unit𝑩𝑩 time, which𝑧𝑧 𝐿𝐿we obtained earlier, must be multiplied𝑧𝑧 by 2 in order to arrive at the time-averaged pressure ( = , ) = given by Eq.(22). This fact has been interpreted by some authors in the past as supporting evidence 2that0 each2 photon residing in a dielectric𝑛𝑛ℏ𝜔𝜔⁄𝑐𝑐 medium of refractive index carries the Minkowski〈𝐹𝐹𝑧𝑧 𝑧𝑧 momentum𝐿𝐿 𝑡𝑡 〉 𝑛𝑛 𝜀𝜀 𝐸𝐸0 [3-6]. In what follows, we shall argue that the radiation pressure exerted on the mirrors in accordance with Eq.(22) is consistent with a nuanced interpretation of Abraham’s𝑛𝑛 photon momentum [7,8]. 𝑛𝑛ℏ𝜔𝜔⁄𝑐𝑐 Let us then compute the Abraham momentum-density ( , ) = × of the EM field that forms the standing wave inside the dielectric-filled cavity. Recall that, for2 a single plane-wave propagating in a dielectric host of refractive index , the Poynting𝓹𝓹𝐴𝐴 𝑧𝑧 vector𝑡𝑡 𝑬𝑬 ( 𝑯𝑯, ⁄)𝑐𝑐= × yields the EM energy that crosses a unit area per unit time [2,11]. The volume occupied by this energy is , which also contains a total Abraham momentum of ×𝑛𝑛 ( ). Normalizing𝑺𝑺 this𝒓𝒓 𝑡𝑡 EM𝑬𝑬 momentum𝑯𝑯 by the corresponding EM energy, and recalling that individual photons have energy , one concludes𝑐𝑐⁄𝑛𝑛 that the Abraham momentum per photon is ( ). In 𝑬𝑬the case𝑯𝑯⁄ 𝑛𝑛of𝑛𝑛 the standing wave specified by Eqs.(20a) and (20b), we find the EM momentum-density to be ℏ𝜔𝜔 ℏ𝜔𝜔⁄ 𝑛𝑛𝑛𝑛 ( , ) = ( ) sin(2 ) sin(2 ) . (23) 2 The dielectric slab of thickness = 0 2 thus appears as a contiguous series of 2 layers of 𝓹𝓹𝐴𝐴 𝑧𝑧 𝑡𝑡 𝑛𝑛𝜀𝜀 𝐸𝐸0 ⁄𝑐𝑐 𝑘𝑘𝑘𝑘 𝜔𝜔𝜔𝜔 𝒛𝒛� thickness 4 demarcated by the adjacent zeros of sin(2 ), similar to the one between = (2 1) 4 and = 2 = . The𝐿𝐿 EM𝑚𝑚 (Abraham)𝑚𝑚⁄ momentum per unit cross-sectional area𝑚𝑚 contained in each such𝜆𝜆⁄ layer is readily determined as follows: 𝑘𝑘𝑘𝑘 𝑧𝑧 𝑚𝑚 − 𝜆𝜆⁄ 𝑧𝑧 𝑚𝑚𝑚𝑚⁄ 𝐿𝐿 ( ) ( ) ( ) ( ) , d = sin 2 . (24) 𝑚𝑚𝑚𝑚⁄2 0 2 Note that the EM momentum∫ 2𝑚𝑚−1 𝜆𝜆 ⁄content4 𝓹𝓹𝐴𝐴 𝑧𝑧 of𝑡𝑡 each𝑧𝑧 layer− 𝜀𝜀 oscillates𝐸𝐸0 ⁄𝜔𝜔 between𝜔𝜔𝜔𝜔 ±𝒛𝒛�( ) with frequency 2 , even though, in accordance with Eq.(21), the EM energy content of each2 layer is fixed at 0 0 4 = (2 ). The time-rate-of-change of the momentum content𝜀𝜀 𝐸𝐸 of⁄𝜔𝜔 the𝒛𝒛� layer may now 𝜔𝜔 be2 determined0 2 by differentiating2 Eq.(24), as follows: 𝑛𝑛 𝜀𝜀 𝐸𝐸0 𝜆𝜆⁄ 𝜋𝜋𝑛𝑛𝐸𝐸0 ⁄ 𝑍𝑍0𝜔𝜔 ( ) ( ) ( ) , d = 2 cos 2 . (25) d 𝑚𝑚𝑚𝑚⁄2 0 2 Before using the aboved𝑡𝑡 ∫result2𝑚𝑚−,1 we𝜆𝜆⁄ 4need𝓹𝓹𝐴𝐴 an𝑧𝑧 𝑡𝑡expression𝑧𝑧 − for𝜀𝜀 𝐸𝐸 the0 time-𝜔𝜔rate𝜔𝜔 -𝒛𝒛of� -change of the mechanical momentum residing in each layer, which is the EM force exerted on the electric dipoles that constitute the dielectric material. The EM force-density acting on the polarization ( , ) = ( 1) ( , ) of the

dielectric medium within the Fabry-Perot cavity is given by [3-9] 0 𝑷𝑷 𝒓𝒓 𝑡𝑡 𝜀𝜀 𝜀𝜀 − 𝑬𝑬 𝒓𝒓 𝑡𝑡

8

0 ( , ) = ( ) + ( ) ×

0 𝑭𝑭 𝑧𝑧 𝑡𝑡 = 2𝑷𝑷 ∙(𝜵𝜵 𝑬𝑬1) 𝜕𝜕𝑷𝑷sin⁄𝜕𝜕𝜕𝜕( )𝜇𝜇cos𝑯𝑯( ) × 2 ( ) cos( ) cos( ) 0 0 = 2𝜀𝜀( 𝜀𝜀 −1) 𝜔𝜔𝐸𝐸0( 𝑘𝑘𝑘𝑘) sin(2𝜔𝜔𝜔𝜔)𝒙𝒙�cos (𝜇𝜇 )𝑛𝑛𝐸𝐸.0 ⁄𝑍𝑍0 𝑘𝑘𝑘𝑘 𝜔𝜔𝜔𝜔 𝒚𝒚� (26) 2 0 2 2 Integrating the above 𝑛𝑛force−-density𝜀𝜀 𝐸𝐸0 over𝑛𝑛𝑛𝑛 ⁄the𝑐𝑐 thickness𝑘𝑘𝑘𝑘 of a 4𝜔𝜔-thick𝜔𝜔 𝒛𝒛� layer, we find ( ) ( ) ( ) ( ) , d = 2 𝜆𝜆1⁄ cos . (27) 𝑚𝑚𝑚𝑚⁄2 2 0 2 2 The time-averaged∫ 2𝑚𝑚EM−1 force𝜆𝜆⁄4 𝑭𝑭 per𝑧𝑧 𝑡𝑡 unit𝑧𝑧 cross− -sectional𝑛𝑛 − 𝜀𝜀area𝐸𝐸0 of each𝜔𝜔 𝜔𝜔 𝒛𝒛�4-thick layer is given by ±( 1) . Adjacent layers, of course, experience equal but opposite forces and, given that the 𝜆𝜆⁄ total2 number0 22 of such layers is even, the overall EM force experienced by the dielectric slab of thickness𝑛𝑛 − 𝜀𝜀 is𝐸𝐸 0seen𝒛𝒛� to be zero at all times. Focusing our attention on the 4-thick dielectric layer that is in contact with 𝑚𝑚the mirror at = , we conclude that the time-rate-of-change of the total momentum content of𝐿𝐿 this layer is the sum of contributions from the EM momentum,𝜆𝜆 ⁄as given by Eq.(25), and from the mechanical momentum, as 𝑧𝑧given𝐿𝐿 by Eq.(27). In order to tie together the various results of the preceding discussion, we need to bring in one final ingredient of the classical theory of electrodynamics, namely, the EM stress tensor, which provides the rate of flow of the EM momentum per unit time per unit cross-sectional area along the , , and directions [2,7]. The Einstein-Laub stress tensor of the EM field is given by [8] 𝑥𝑥 𝑦𝑦 𝑧𝑧 ( , ) = ½( + ) . (28)

0 0 Consequently, for the standing𝓣𝓣⃖�⃗ 𝒓𝒓 𝑡𝑡 wave 𝜀𝜀described𝑬𝑬 ∙ 𝑬𝑬 𝜇𝜇by𝑯𝑯 Eqs.(∙ 𝑯𝑯 20⃡𝐈𝐈 −a)𝑫𝑫 and𝑫𝑫 − (20𝑩𝑩𝑩𝑩b), the rate of flow of EM linear momentum along the -axis is determined as follows: ( , ) = ½ + ½ 𝑧𝑧 0 ( )0 ( ) ( ) ( ) 𝒯𝒯𝑧𝑧𝑧𝑧 𝑧𝑧 𝑡𝑡 = 2 𝜀𝜀 𝑬𝑬 ∙sin𝑬𝑬 𝜇𝜇sin𝑯𝑯 ∙ 𝑯𝑯 + 2 cos cos 2 2 2 2 2 2 2 0 [ ( )] ( ) 0 [ ( )] ( ) = 𝜀𝜀 𝐸𝐸01 cos𝑘𝑘𝑘𝑘2 sin𝜔𝜔𝜔𝜔 𝑛𝑛+𝜀𝜀 𝐸𝐸0 1 𝑘𝑘+𝑘𝑘cos 2 𝜔𝜔𝜔𝜔 cos 2 2 2 2 2 0 {[ ( ) ( )] [ 0 ( ) ( )] ( )} = 𝜀𝜀 𝐸𝐸0 1−+ 𝑘𝑘𝑘𝑘1 cos 𝜔𝜔𝜔𝜔 𝑛𝑛1𝜀𝜀 𝐸𝐸0 + 1 cos𝑘𝑘𝑘𝑘 cos𝜔𝜔2𝜔𝜔 . (29) 2 2 2 2 2 = 0 cos(2 ) = cos(2 ) = 1 At , where𝜀𝜀 𝐸𝐸 0 𝑛𝑛 − 𝜔𝜔𝜔𝜔 −, the− outflow𝑛𝑛 of the 𝜔𝜔EM𝜔𝜔 momentum𝑘𝑘𝑘𝑘 per unit cross- sectional area is 2 cos ( ), which is consistent with the pressure exerted on the mirror in accordance𝑧𝑧 with𝐿𝐿 Eq.(2 22).2 At𝑘𝑘 𝑘𝑘2 = ¼𝑚𝑚𝑚𝑚, however, cos(2 ) = 1, and the rate of flow of EM 0 0 momentum into the𝑛𝑛 𝜀𝜀4𝐸𝐸-thick layer𝜔𝜔𝜔𝜔 that is in contact with the mirror is 2 sin ( ). From this we 𝑧𝑧 𝐿𝐿 − 𝜆𝜆 𝑘𝑘𝑘𝑘 − must now subtract the time-rate-of-change of the total momentum content of0 the2 dielectric2 layer, i.e., the sum of the expressions𝜆𝜆⁄ given by Eqs.(25) and (27). We will have 𝜀𝜀 𝐸𝐸0 𝜔𝜔𝜔𝜔 ( ¼ , ) ( , )d ( , )d ¼ ¼ d 𝐿𝐿 𝐿𝐿 𝐿𝐿− 𝜆𝜆 𝒯𝒯𝑧𝑧𝑧𝑧 𝐿𝐿 − 𝜆𝜆 𝑡𝑡 𝒛𝒛� − d𝑡𝑡 ∫ = 2𝓹𝓹𝐴𝐴 𝑧𝑧 [𝑡𝑡sin𝑧𝑧(− ∫𝐿𝐿)−+ 𝜆𝜆cos𝑭𝑭 (𝑧𝑧2𝑡𝑡 )𝑧𝑧+ ( 1) cos ( )] 0 2 2 2 2 = 2𝜀𝜀 𝐸𝐸0 cos𝜔𝜔(𝜔𝜔 ) . 𝜔𝜔𝜔𝜔 𝑛𝑛 − 𝜔𝜔𝜔𝜔 𝒛𝒛� (30) 2 2 2 0 = The final expression in Eq.(30)𝑛𝑛 is 𝜀𝜀the𝐸𝐸0 rate of 𝜔𝜔outflow𝜔𝜔 𝒛𝒛� of EM momentum at , which is also the pressure exerted on the mirror at = ; see Eq.(22). We conclude that the force exerted on the end mirrors is consistent with the Abraham momentum-density of the EM field trapped𝑧𝑧 𝐿𝐿 within the Fabry- Perot cavity, as given by Eq.(23). In𝑧𝑧 the𝐿𝐿 process, the field also exerts a force on the dielectric slab, with the force-density given by Eq.(26). The argument presented here is thus seen to be consistent with the assertion that individual photons propagating along the + and directions inside the cavity possess the EM Abraham momentum ±( ) . 𝑧𝑧 −𝑧𝑧 ℏ𝜔𝜔⁄𝑛𝑛𝑛𝑛 𝒛𝒛� 9

7. Energy-density fluctuations. The square of the photon number contained in a given radiation mode trapped inside the cavity of Fig.1 is readily found to be 𝑚𝑚 = [1 exp( )] exp( )

2 𝐵𝐵 ∞ 2 𝐵𝐵 〈𝑚𝑚 〉 = [1− exp(−ℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇 )](∑𝑚𝑚=0 𝑚𝑚) −𝑚𝑚ℏ𝜔𝜔exp⁄𝑘𝑘( 𝑇𝑇 ) 2 2 d ∞ 𝐵𝐵 𝐵𝐵 2 𝑚𝑚=0 𝐵𝐵 = [1 − exp(−ℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇)](𝑘𝑘 𝑇𝑇⁄ℏ) d𝜔𝜔 [∑1 exp( −𝑚𝑚ℏ𝜔𝜔⁄𝑘𝑘)]𝑇𝑇 2 2 d −1 𝐵𝐵 𝐵𝐵 2 𝐵𝐵 ( ) = − −ℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇 𝑘𝑘 𝑇𝑇⁄ℏ d𝜔𝜔 − −ℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇 [ ( ) ] · (31) exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 + 1 2 Consequently, the photonexp numberℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵 𝑇𝑇variance− 1 is

( ) = [ ( ) ] · (32) 2 2 exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 2 The variance of energy per unit volume〈𝑚𝑚 〉 − per〈𝑚𝑚 unit〉 angularexp ℏ 𝜔𝜔frequency⁄𝑘𝑘𝐵𝐵𝑇𝑇 − 1 is thus found to be ( ) Var[ ( , )] = = ( , ) + [ ( , ) ] · (33) 2[ 4 ( ) ] ℏ 𝜔𝜔 exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 2 3 2 2 3 2 (As an aside,ℰ 𝜔𝜔note𝑇𝑇 that the𝜋𝜋 𝑐𝑐 secondexp ℏ 𝜔𝜔term⁄𝑘𝑘𝐵𝐵 𝑇𝑇on− the1 rightℏ-𝜔𝜔handℰ 𝜔𝜔-side𝑇𝑇 of Eq.(33)𝜋𝜋 𝑐𝑐 ℰ is𝜔𝜔 the𝑇𝑇 classical⁄𝜔𝜔 expression of the EM field-intensity fluctuations, which must be approached when 0.) If we now define the average-to-standard-deviation ratio for the energy-density of thermal radiation, Eqs.(8) and (33) yield ℏ → ( , ) ( ) = · (34) Average ℰ 𝜔𝜔 𝑇𝑇 𝜔𝜔 exp −ℏ𝜔𝜔⁄2𝑘𝑘𝐵𝐵𝑇𝑇 3⁄2 Note that the above ratio hasStandard the dimensionsDeviation of 𝜋𝜋𝑐𝑐, which indicates that the right-hand-side of Eq.(34) must be multiplied by d in order to yield the3 average-to-standard-deviation ratio for the radiation contained in a volume within the frequency�𝑠𝑠 range⁄𝑚𝑚 d . Considering that the various radiation modes are independent of each other,√𝑣𝑣 𝜔𝜔 the total variance (per unit volume) can be obtained by integrating Eq.(33) over all frequencies, as follows:𝑣𝑣 𝜔𝜔

( ) Var[ ( , )]d = d ∞ 2[ 4 ( ) ] ∞ ℏ 𝜔𝜔 exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 2 3 2 0 𝜋𝜋 𝑐𝑐 exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 − 1 ∫ ℰ 𝜔𝜔 𝑇𝑇 𝜔𝜔 �0 ( ) 𝜔𝜔 = d = [exp( ) 1] d 5 5 ∞ [ 4 ( ) ] 5 5 ∞ 𝑘𝑘𝐵𝐵𝑇𝑇 𝑥𝑥 exp 𝑥𝑥 𝑘𝑘𝐵𝐵𝑇𝑇 4 d −1 2 3 3 G&R2 3.411-17 [26] 2 3 3 𝜋𝜋 𝑐𝑐 ℏ exp 𝑥𝑥 − 1 𝜋𝜋 𝑐𝑐 ℏ d𝑥𝑥 �0 𝑥𝑥 − �0 𝑥𝑥 𝑥𝑥 − 𝑥𝑥 Integration by parts = d = = 16( ) . (35) 5 5 ∞ ( 3) 2 5 5 4𝑘𝑘𝐵𝐵𝑇𝑇 𝑥𝑥 4𝜋𝜋 𝑘𝑘𝐵𝐵𝑇𝑇 5 2 3 3 3 3 𝐵𝐵 𝜋𝜋 𝑐𝑐 ℏ exp 𝑥𝑥 − 1 15𝑐𝑐 ℏ The above result is in agreement�0 with the thermodynamic𝑥𝑥 identity𝑘𝑘 𝜎𝜎⁄𝑐𝑐 𝑇𝑇 = . Given that has the dimensions of ( ), the coefficient of in the preceding2 2equation𝐵𝐵 2 is seen to have the dimensions of . Dividing the 2average energy-density,5 (4 〈)𝜀𝜀 〉, −by〈 𝜀𝜀the〉 square𝑘𝑘 𝑇𝑇 root𝜕𝜕〈𝜀𝜀 〉of⁄ 𝜕𝜕the𝜕𝜕 variance given𝜎𝜎 by Eq.(325), we3 find𝐽𝐽 ⁄the𝑠𝑠 ∙average𝑚𝑚 -to-standard-deviation𝑇𝑇 ratio 4for the energy of thermal radiation contained in a 𝐽𝐽volume⁄𝑚𝑚 to be ( ) . 𝜎𝜎⁄𝑐𝑐 𝑇𝑇

As for the total energy (per unit area per unit𝐵𝐵 time3 ) emerging from an aperture on the surface of the blackbody depicted in Fig.3, the𝑉𝑉 average� is𝜎𝜎 𝜎𝜎⁄𝑐𝑐𝑘𝑘 while𝑇𝑇 the variance is 4 , obtained by multiplying

the expression in Eq.(35) by 4. The average4-to-standard-deviation ratio𝐵𝐵 for5 the EM energy emerging from an aperture of area during a time interval𝜎𝜎𝑇𝑇 is thus ( 4 ) 𝑘𝑘. 𝜎𝜎𝑇𝑇 𝑐𝑐⁄ 𝐵𝐵 3 𝐴𝐴 𝜏𝜏 � 𝜎𝜎𝜎𝜎𝜎𝜎⁄ 𝑘𝑘 𝑇𝑇 10

8. Entropy of thermal radiation. Let an evacuated chamber of volume contain a photon gas in thermal equilibrium with the walls of the chamber at temperature . Each EM mode could have photons ( = 0, 1, 2, ) with probability = [1 exp( )] exp( 𝑉𝑉 ). The entropy of 𝑇𝑇 𝑚𝑚 each such mode is, therefore, given by mode 𝑚𝑚 ⋯ 𝑝𝑝𝑚𝑚 − −ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 −𝑚𝑚ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 𝑆𝑆 ( , ) = ln

mode 𝐵𝐵 ∞ 𝑆𝑆 𝜔𝜔 𝑇𝑇 = −𝑘𝑘 ∑𝑚𝑚=0 𝑝𝑝𝑚𝑚 {ln𝑝𝑝[𝑚𝑚1 exp( )] ( )} 𝐵𝐵 ∞ = −( 𝑘𝑘 ∑𝑚𝑚)=0 𝑝𝑝𝑚𝑚 ln[−1 exp−(ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 −)] 𝑚𝑚ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 = 𝐵𝐵 ln[1 exp( 𝐵𝐵 )] ℏ𝜔𝜔( ⁄𝑇𝑇 〈𝑚𝑚) 〉 − 𝑘𝑘 − −ℏ𝜔𝜔⁄𝑘𝑘 𝑇𝑇 . (36) ℏ𝜔𝜔⁄𝑇𝑇 𝐵𝐵 Since the various modesexp ℏ are𝜔𝜔⁄ 𝑘𝑘independent𝐵𝐵𝑇𝑇 − 1 − 𝑘𝑘 of each− other, −theℏ 𝜔𝜔total⁄𝑘𝑘 𝐵𝐵entropy𝑇𝑇 of the trapped photon gas is the sum of the of all available modes. Given that the mode density (per unit volume) is d ( ), the entropy of the photon gas is readily evaluated by integrating over all frequencies , as follows:2 2 3 𝜔𝜔 𝜔𝜔⁄ 𝜋𝜋 𝑐𝑐 𝜔𝜔 ( , ) = ( , )d ∞ 2 𝑉𝑉𝜔𝜔 2 3 mode 𝜋𝜋 𝑐𝑐 𝑆𝑆 𝑇𝑇 𝑉𝑉 �0 � � 𝑆𝑆 𝜔𝜔 𝑇𝑇 𝜔𝜔 = ln[1 exp( )] d ∞ ( 3 ) 𝑘𝑘𝐵𝐵𝑉𝑉 ℏ𝜔𝜔 ⁄𝑘𝑘𝐵𝐵𝑇𝑇 2 2 3 𝜋𝜋 𝑐𝑐 exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 − 1 𝐵𝐵 � � �0 � − 𝜔𝜔 Use integration− − ℏby𝜔𝜔 parts⁄𝑘𝑘 𝑇𝑇 � 𝜔𝜔 = d ∞ ( 3 ) 4𝑘𝑘𝐵𝐵𝑉𝑉 ℏ𝜔𝜔 ⁄𝑘𝑘𝐵𝐵𝑇𝑇 2 3 15 �3𝜋𝜋 𝑐𝑐 � � exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 − 1 𝜔𝜔 0 4 = 𝜋𝜋 ⁄= . (37) 3 ∞ (3 ) 2 4 4𝑘𝑘𝐵𝐵𝑉𝑉 𝑘𝑘𝐵𝐵𝑇𝑇 𝑥𝑥 d𝑥𝑥 4𝜋𝜋 𝑘𝑘𝐵𝐵 3 2 3 3 3 3𝜋𝜋 𝑐𝑐 ℏ exp 𝑥𝑥 − 1 45𝑐𝑐 ℏ An alternative derivation� � � of the� � above0 result assumes� that� 𝑉𝑉an𝑇𝑇 evacuated chamber of fixed volume is slowly heated from 0K to the temperature . The entropy of the photon gas is then evaluated by integrating with respect to the rising temperature from 0 to . Since the heat flows into the𝑉𝑉 chamber in increments of = = [ 𝑇𝑇(15 )] ( ), we will have ∆𝑄𝑄⁄𝑇𝑇 2 4 3 3 4 𝑇𝑇 𝐵𝐵 ( , ) = ∆𝑄𝑄= 𝑉𝑉∆𝑢𝑢 𝜋𝜋 𝑘𝑘 𝑉𝑉⁄ =𝑐𝑐 ℏ ∆ 𝑇𝑇 d = 𝑇𝑇 2 4 𝑇𝑇 4 2 4 2 4 . (38) d𝑄𝑄 𝜋𝜋 𝑘𝑘𝐵𝐵𝑉𝑉 d�𝑇𝑇 � 4𝜋𝜋 𝑘𝑘𝐵𝐵𝑉𝑉 𝑇𝑇 2 4𝜋𝜋 𝑘𝑘𝐵𝐵 3 3 3 3 3 3 3 𝑇𝑇 15𝑐𝑐 ℏ 𝑇𝑇 15𝑐𝑐 ℏ 0 45𝑐𝑐 ℏ Alternatively,𝑆𝑆 𝑇𝑇 𝑉𝑉 one�0 may imagine� an� evacuated�0 chamber� at� a∫ constant𝑇𝑇 𝑇𝑇 temperature� � 𝑉𝑉 𝑇𝑇, whose volume slowly (i.e., reversibly) rises from to . Considering that = + , and that the radiation pressure inside the chamber is = ( ) 3 while ( ) = (15 ), we will have𝑇𝑇 1 2 𝑉𝑉 𝑉𝑉 2 4 ∆4𝑄𝑄 ∆𝑈𝑈3 3 𝑃𝑃∆𝑉𝑉 𝐵𝐵 𝑃𝑃 𝑢𝑢 𝑇𝑇 ⁄ 𝑢𝑢 𝑇𝑇 (𝜋𝜋) 𝑘𝑘 𝑇𝑇( ⁄) 𝑐𝑐 ℏ ( , ) ( , ) = = d 𝑉𝑉2 𝑉𝑉2 d𝑄𝑄 𝑢𝑢 𝑇𝑇 +⅓𝑢𝑢 𝑇𝑇 2 1 𝑇𝑇 𝑇𝑇 𝑆𝑆 𝑇𝑇 𝑉𝑉 − 𝑆𝑆 𝑇𝑇 𝑉𝑉 �𝑉𝑉1 �𝑉𝑉1 𝑉𝑉 = d = ( ) 2 4 2 4 3 . (39) 2 4𝜋𝜋 𝑘𝑘𝐵𝐵 𝑉𝑉 3 4𝜋𝜋 𝑘𝑘𝐵𝐵𝑇𝑇 3 3 3 3 Clearly, all these different approaches�45𝑐𝑐 toℏ �computing∫𝑉𝑉1 𝑇𝑇 𝑉𝑉 the �entropy45𝑐𝑐 ℏ �of 𝑉𝑉the2 − photon𝑉𝑉1 gas in thermal equilibrium with its container of volume and temperature yield identical results. We note in passing that the energy fluctuations of thermal radiation derived earlier (see Eq.(35)) could also be obtained from either of the thermodynamic identities 𝑉𝑉 = ( 𝑇𝑇 ) or = ( ) .

2 2 𝐵𝐵 3 2 2 𝐵𝐵 2 2 −1 〈𝜀𝜀 〉 − 〈𝜀𝜀〉 𝑘𝑘 𝑇𝑇 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 〈𝜀𝜀 〉 − 〈𝜀𝜀〉 −𝑘𝑘 𝜕𝜕 𝑆𝑆⁄𝜕𝜕𝜀𝜀 11

9. Dielectric particle in thermal equilibrium with electromagnetic radiation. This section is devoted to an alternative derivation of Planck’s blackbody radiation formula, Eq.(8). Let a small spherical particle of volume and dielectric susceptibility ( ) — when subjected to an external -field cos( ) — be in thermal equilibrium with the radiation0 inside an otherwise empty box at temperature 0. The Lorentz oscillator model𝑣𝑣 for the particle yields the𝜀𝜀 following𝜒𝜒 𝜔𝜔 formula for its susceptibility at𝐸𝐸 frequency𝐸𝐸 𝒙𝒙� : 𝜔𝜔𝜔𝜔 𝑇𝑇 ( ) = (2 )( ) · 𝜔𝜔 (40) 𝜔𝜔𝑝𝑝 2 2 2 2 3 In the above equation, 𝜒𝜒= 𝜔𝜔 /(𝜔𝜔0 −)𝜔𝜔 is −thei�𝛾𝛾 𝛾𝛾plasma+ 4𝜋𝜋 𝜔𝜔frequency𝑝𝑝⁄3 𝑣𝑣⁄𝜆𝜆0 and� = the resonance frequency of the material, =2 is2 the damping coefficient associated with loss mechanisms other 𝑝𝑝 0 0 than radiation resistance (e.g.,𝜔𝜔 absorption𝑁𝑁𝑞𝑞 𝑚𝑚 𝜀𝜀losses), and = 2 is the vacuum𝜔𝜔 �𝛼𝛼 wavelength⁄𝑚𝑚 of the 𝛾𝛾 𝛽𝛽⁄𝑚𝑚 excitation field. Here is the number-density of the electric0 dipoles that comprise the spherical particle, and are the electric charge and mass of the oscillating𝜆𝜆 electron𝜋𝜋𝜋𝜋⁄𝜔𝜔 within individual dipoles, is the dipole’s spring constant,𝑁𝑁 is the dipole’s friction coefficient, and is the permittivity of free space. 𝑞𝑞 𝑚𝑚 𝛼𝛼 In what follows, we shall assume that the particle is transparent0 and that, therefore, can be set to zero. The only damping 𝛽𝛽mechanism will then be radiation resistance,𝜀𝜀 whose contribution appears as the second term inside the square brackets in Eq.(40). If we further assume that the particle𝛾𝛾 has a narrow resonance peak at = , we can define the effective damping coefficient = (6 ) =

(6 ), where 0 is the total charge and the total mass of oscillating electrons.eff 2 02 3 𝜔𝜔 𝜔𝜔 𝛾𝛾 𝑣𝑣𝜔𝜔𝑝𝑝𝜔𝜔 ⁄ 𝜋𝜋𝑐𝑐 0 2At02 thermal equilibrium, each vibrational degree of freedom of the spherical particle has, on average, 𝜇𝜇½𝑞𝑞� 𝜔𝜔 of⁄ energy.𝜋𝜋𝑚𝑚�𝑐𝑐 Given 𝑞𝑞�that harmonic oscillators𝑚𝑚� have two degrees of freedom (associated with their kinetic𝐵𝐵 and potential energies), the oscillators along the , , and axes will each have an average thermal𝑘𝑘 𝑇𝑇 energy of , for a total thermal energy of 3 . Now, the impulse response of the Lorentz ( ) 𝑥𝑥 𝑦𝑦 𝑧𝑧 ( ) oscillator diminishes𝐵𝐵 over time as exp 2 , indicating𝐵𝐵 that the radiated energy decays as exp . All in all, one can say𝑘𝑘 𝑇𝑇 that, in thermal equilibrium, the randomly𝑘𝑘 𝑇𝑇 vibrating particle radiates EM energy at an average rate of 3 Joule/sec [1−1].𝛾𝛾 𝛾𝛾⁄ −𝛾𝛾𝛾𝛾

The oscillators areeff 𝐵𝐵immersed in a bath of EM radiation. Let the -field acting on the particle be ( ) = ( ) exp𝛾𝛾( i𝑘𝑘 𝑇𝑇). Then the rate at which the particle extracts energy from the field will be 𝐸𝐸 0 𝑬𝑬 𝑡𝑡 𝐸𝐸 (𝜔𝜔) 𝒙𝒙�d ( )−d𝜔𝜔𝜔𝜔 = ½Re[ i ( ) | ( )| ] = ½ | ( )| ( ) 2 2 · (41) eff 𝑝𝑝 2 𝛾𝛾 𝜔𝜔 𝑣𝑣𝜔𝜔 2 0 0 2 2 2 2 2 0 0 Considering〈𝑬𝑬 𝑡𝑡 ∙ 𝒑𝒑 the𝑡𝑡 ⁄sharpness𝑡𝑡〉 of− the𝜔𝜔𝜀𝜀 resonance𝜒𝜒 𝜔𝜔 𝑣𝑣 𝐸𝐸 line𝜔𝜔 around �the𝜔𝜔0 −center𝜔𝜔 +-frequency𝛾𝛾eff 𝜔𝜔 � 𝜀𝜀 𝐸𝐸 =𝜔𝜔 , we may approximate the term with ( + )( ) 2 ( ), and ½ | (0 )| with 𝜔𝜔 𝜔𝜔 ½ | ( )| = ( , ), 02which2 is the total0 (i.e., electric0 plus 0magnetic)0 field energy0 -0density2 (i.e., 𝜔𝜔 − 𝜔𝜔 𝜔𝜔 𝜔𝜔 𝜔𝜔 − 𝜔𝜔 ≅ 𝜔𝜔 𝜔𝜔 − 𝜔𝜔 𝜀𝜀 𝐸𝐸 𝜔𝜔 energy0 0 per0 unit2 volume0 per unit frequency) at the resonance frequency of the oscillator at temperature . Integrating𝜀𝜀 𝐸𝐸 𝜔𝜔 over ℰ (in𝜔𝜔 order𝑇𝑇 to cover the entire linewidth) now yields 𝑇𝑇

(𝜔𝜔) d ( ) d d ( , ) d = ½ ( , ) ( ) 2 · (42) ∞ ∞ – eff 𝑝𝑝 𝛾𝛾 𝜔𝜔 𝑣𝑣 2 0 0 2 2 0 4 𝜔𝜔 𝜔𝜔0 + 𝛾𝛾eff 𝑝𝑝 At thermal∫ 〈𝑬𝑬 equilibrium,𝑡𝑡 ∙ 𝒑𝒑 𝑡𝑡 ⁄ 𝑡𝑡where〉 𝜔𝜔 ≅ theℰ 𝜔𝜔energy𝑇𝑇 � radiation−∞ and absorption𝜔𝜔 rates𝜋𝜋𝜔𝜔 computed𝑣𝑣ℰ 𝜔𝜔 𝑇𝑇 above are equal, we will have ½ ( , ) = 3 ( , ) = ( ). (43) 2 2 2 3 The above equation, 0the well-knowneff 𝐵𝐵 Rayleigh-Jeans formula for 𝐵𝐵the EM energy-density inside a 𝜋𝜋𝜔𝜔𝑝𝑝𝑣𝑣ℰ 𝜔𝜔 𝑇𝑇 𝛾𝛾 𝑘𝑘 𝑇𝑇 → ℰ 𝜔𝜔 𝑇𝑇 𝑘𝑘 𝑇𝑇𝜔𝜔 ⁄ 𝜋𝜋 𝑐𝑐 black box at temperature , is known to be wrong because it predicts, unrealistically, the existence of high EM energy-densities at high frequencies (in proportion to ). This failure of classical physics at the th end of the 19 century came𝑇𝑇 to be known as the ultraviolet catastrophe2 . The ultraviolet catastrophe was eventually resolved when Max Planck proposed his quantum𝜔𝜔 hypothesis, which allows a (harmonic) oscillator to possess only energies that are integer-multiples of . Here is the reduced while is the resonance frequency of the harmonic oscillator.0 The oscillator will thus have an ℏ𝜔𝜔 ℏ 𝜔𝜔0 12

energy with the probability = exp( ); here = 0, 1, 2, 3, is the integer-valued

quantum number0 that specifies the allowed energy levels0 of the oscillator. The coefficient is determined from the𝑛𝑛 ℏfact𝜔𝜔 that the sum of the probabilities𝑝𝑝𝑛𝑛 𝛼𝛼 must−𝑛𝑛 ℏequal𝜔𝜔 ⁄𝑘𝑘 1.0,𝐵𝐵𝑇𝑇 that is, 𝑛𝑛 ⋯ 𝛼𝛼 = exp( ) = = 1 ( ) 𝛼𝛼 ∞ ∞ 0 ∑𝑛𝑛=0 𝑝𝑝𝑛𝑛 𝛼𝛼 ∑𝑛𝑛= 0 = −1 𝑛𝑛ℏexp𝜔𝜔 ⁄( 𝑘𝑘𝐵𝐵𝑇𝑇 1 −). exp −ℏ𝜔𝜔0⁄𝑘𝑘𝐵𝐵𝑇𝑇 (44)

0 Consequently, the average energy→ of the𝛼𝛼 harmonic− osci−ℏllator𝜔𝜔 ⁄ 𝑘𝑘is𝐵𝐵 𝑇𝑇found to be ( ) = exp( ) ∞ ∞ 0 0 ( ) 0 ∑𝑛𝑛=0 𝑛𝑛ℏ𝜔𝜔 𝑝𝑝𝑛𝑛 = 𝛼𝛼ℏ𝜔𝜔 ∑𝑛𝑛=0 𝑛𝑛 −𝑛𝑛=ℏ𝜔𝜔 ⁄𝑘𝑘𝐵𝐵𝑇𝑇 [ ( )] ( ) · (45) 𝛼𝛼ℏ𝜔𝜔0 exp −ℏ𝜔𝜔0⁄𝑘𝑘𝐵𝐵𝑇𝑇 ℏ𝜔𝜔0 2 Only in the limit of large and/or small1 − exp − doesℏ𝜔𝜔0⁄ 𝑘𝑘the𝐵𝐵𝑇𝑇 averageexp energyℏ𝜔𝜔0⁄𝑘𝑘 𝐵𝐵(kinetic𝑇𝑇 − 1 plus potential) of the oscillator given by Eq.(45) approach the classical0 value of . The correct formula for the energy- 𝑇𝑇 𝜔𝜔 density of blackbody radiation is thus obtained by replacing 𝐵𝐵 in Eq.(43) with the average energy of the 𝑘𝑘 𝑇𝑇 oscillator given by Eq.(45). We will have 𝐵𝐵 𝑘𝑘 𝑇𝑇 ( , ) = · (46) [ ( 3 ) ] ℏ𝜔𝜔 2 3 This is the same equation, Eq.(8),ℰ 𝜔𝜔 derived𝑇𝑇 𝜋𝜋 earlier𝑐𝑐 exp ℏwhen𝜔𝜔⁄𝑘𝑘𝐵𝐵 𝑇𝑇we− 1invoked the quantum hypothesis in conjunction with the various modes of the EM field trapped inside an empty box in thermal equilibrium with the walls at temperature .

10. Einstein’s and coefficients𝑇𝑇 . In his famous 1916 and 1917 papers [15-18], Einstein suggested that an atom/molecule in a state |1 , when immersed in EM radiation having local energy-density ( , ) at frequency and𝑨𝑨 temperature𝑩𝑩 , will transition to the excited state |2 by absorbing a quantum of energy from the field at a rate proportional⟩ to ( , ) = ( , ). The energy difference between |ℰ1 𝜔𝜔 and𝑇𝑇 |2 was noted 𝜔𝜔to be = 𝑇𝑇, and the proportionality coefficient for⟩ the absorption-induced transition ℏ𝜔𝜔 𝐼𝐼 𝜔𝜔 𝑇𝑇 𝑐𝑐ℰ 𝜔𝜔 𝑇𝑇 ⟩ ⟩ was denoted by 2 . Einstein1 then suggested that the excited atom/molecule will have two pathways for returning to the lower𝜀𝜀 −-𝜀𝜀energyℏ 𝜔𝜔state: i) spontaneous emission, which occurs at a rate of, say, transitions 12 per second, and 𝐵𝐵ii) stimulated emission, whose rate is proportional to the local intensity ( , ), with a 21 proportionality coefficient . Given the energy difference = between |1 𝐴𝐴and |2 , the probabilities of being in |1 and |2 are in the ratio of = exp( ). In thermal𝐼𝐼 𝜔𝜔 equilibrium,𝑇𝑇 21 the rate of the |1 |2 transition𝐵𝐵 must equal that of the |2 |1 transition,∆𝜀𝜀 ℏ𝜔𝜔 that is, ⟩ ⟩ ⟩ ⟩ 𝑝𝑝2⁄𝑝𝑝1 −ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 ( , ) = ( , ) + ( , ) = ⟩ → ⟩ ⟩ → ⟩ ( ) · (47) 𝐴𝐴 A𝑝𝑝 comparison1𝐵𝐵12𝐼𝐼 𝜔𝜔 𝑇𝑇 with 𝑝𝑝Eq.(2𝐵𝐵2146𝐼𝐼) 𝜔𝜔reveals𝑇𝑇 that𝑝𝑝2 𝐴𝐴 =→ , that𝐼𝐼 𝜔𝜔 is,𝑇𝑇 absorption𝐵𝐵12 exp andℏ𝜔𝜔⁄ stimulated𝑘𝑘𝐵𝐵𝑇𝑇 − 𝐵𝐵21 emission have identical rates, and that = ( ). 12 21 3 2𝐵𝐵2 𝐵𝐵 11. Implications of the𝐴𝐴 second21⁄𝐵𝐵21 lawℏ of𝜔𝜔 thermodynamics⁄ 𝜋𝜋 𝑐𝑐 for radiation pressure. In this section we infer several facts about the pressure of thermal radiation using the second law. According to Feynman, “one cannot devise a process whose only result is to convert heat to work at a single temperature” [11]. An alternative statement of the second law, once again according to Feynman [11], is that “heat cannot be taken in at a certain temperature and converted into work with no other change in the system or the surroundings.” We apply the second law to a rotary machine placed within thermal radiation, because the rotary machine returns to its initial state after a full revolution. If mechanical work is performed after a full revolution of the device, and if no other changes are made in the system, then the second law will be violated. We have seen that the pressure of thermal radiation on a perfect reflector is ( ) = ( ) 3, where ( ) is the energy-density of the EM field at equilibrium with its container held at the temperature ; see 𝑝𝑝 𝑇𝑇 𝑢𝑢 𝑇𝑇 ⁄ 𝑢𝑢 𝑇𝑇 𝑇𝑇 13

Eq.(12). Does this result change if the reflector is not flat (e.g., it has a curved surface), or if the reflector’s surface is jagged (i.e., it is a scatterer)? What happens if the surface is not reflective at all, but rather an ideal absorber? What if the surface is a partial absorber (and partial reflector)? To answer these questions, consider a rotary machine whose vanes have different optical characteristics on their opposite facets; for instance, while the front side of each vane is a flat reflector, the back side might be a hemispherical reflector, as shown in Fig.5(a). The rotor is placed inside an empty box which contains EM radiation at the equilibrium temperature . If the pressure of radiation on one side of each vane happens to differ from the corresponding pressure on the opposite side, the rotor will start rotating in a fixed direction, thereby producing useful mechanical𝑇𝑇 work from a single source that is maintained at a fixed temperature . This will violate the second law of thermodynamics. We conclude that the net pressure of thermal radiation on the flat mirror is the same as that on the hemispherical mirror. Figure 5(b) shows three different𝑇𝑇 constructs for the vanes. Instead of hemispherical reflector, the backside of the vane may be an Thermal Radiation absorber (either perfect or partial), or it may be a scatterer. Irrespective of the nature of the back surface, the second law of thermodynamics teaches us that the pressure of radiation on that surface must always be given by ( ) = ( ) 3.

Fig.5. The empty box at the top of the𝑝𝑝 figure𝑇𝑇 is𝑢𝑢 held𝑇𝑇 ⁄ at a constant temperature . Inside the box, EM radiation is at thermal equilibrium with the enclosure. Each vane of the rotary fan consists of a solid hemisphere coated with a perfect𝑇𝑇 reflector. The radiation pressure on the flat side of each vane must be Reflector Absorber Scatterer equal to that on the spherical side, otherwise the fan will rotate and perform useful mechanical work, in violation of the second law of thermodynamics. The bottom figure shows three types of vane (including the hemispherical one). A vane may be perfectly reflective on one side and perfectly (or partially) absorptive on the opposite side, or it may have a reflector on one side and a scatterer on the other side. In all cases the pressure of thermal radiation must be identical on both sides of each vane. In the case of a perfect (or partial) absorber, Kirchhoff’s law affirms that, at thermal equilibrium, the absorption and emission efficiencies are identical, which explains why the pressure on the reflective side of the vane should equal that on the absorptive side. In the case of a scatterer, the reciprocity theorem of classical electrodynamics [19] must be invoked to explain the equality of the radiation pressure on the scattering and reflective sides of the vane. (The fact that all propagation directions contain, on average, equal amounts of EM energy, that they are all equally likely to exist within the cavity, and also the fact that different cavity modes are relatively incoherent, combined with the reciprocity theorem is sufficient to prove that radiation pressure on the scatterer is the same as that on a perfect reflector. In fact, one could argue that this limited form of the reciprocity theorem is an immediate consequence of the second law.) As a second example, consider the system depicted in Fig.6(a), where the radiation emanating from a blackbody at temperature is collected by a lens and focused onto the shiny teeth of a rotating ratchet. The reflected light is Doppler shifted, as will be shown below, and the change in its energy and momentum will be transferred𝑇𝑇 to the rotating ratchet. In this way the thermal radiation performs useful work without violating the second law of thermodynamics. The reason is that the reflected radiation in this case is Doppler-shifted to a lower frequency and is, therefore, at a lower temperature than the incident radiation. So long as the rotating wheel extracts thermal energy from a hot bath and returns a fraction of this energy to a cold bath, the second law would allow the remaining energy to be converted into useful (mechanical) work. Computing the Doppler shift of the beam reflecting off the ratchet requires that we first move to the reference frame in which the illuminated tooth is stationary, as shown in Fig.6(b). A Lorentz transformation of the space-time coordinates from the lab frame to the rest-frame is needed to determine the phase factor ( ) of the incident plane-wave, as follows:′ ′ ′ 𝑥𝑥𝑥𝑥𝑥𝑥 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝜔𝜔 14

= ( ) = ( ) ( + ) inc inc inc inc = ( ) ( ) ′ ′ ′ 2 ⁄ ⁄ . ⁄ (48) 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝜔𝜔 − 𝜔𝜔 𝑐𝑐 𝑧𝑧 inc− 𝜔𝜔 𝑡𝑡 inc− 𝜔𝜔 𝑐𝑐 𝑧𝑧 −inc𝜔𝜔 𝛾𝛾 𝑡𝑡 𝑣𝑣𝑥𝑥 𝑐𝑐 This means that the incident2 plane′-wave seen by′ the mirror′ has frequency = and −𝛾𝛾 𝑣𝑣⁄𝑐𝑐 𝜔𝜔 𝑥𝑥 − 𝜔𝜔 ⁄𝑐𝑐 𝑧𝑧 − 𝛾𝛾𝜔𝜔 𝑡𝑡 propagates along = ( ) ( ) , which makes an angle = sin ( ′ ) withinc the - axis. The incidence′ angle in the2 mirror’sinc restinc -frame is thus , causing the reflected−1 𝜔𝜔 𝛾𝛾-𝜔𝜔vector to′ emerge at an angle𝒌𝒌 2 −𝛾𝛾 𝑣𝑣 relative⁄𝑐𝑐 𝜔𝜔 to𝒙𝒙� the− 𝜔𝜔-axis⁄𝑐𝑐 (again𝒛𝒛� in the mirror’s rest 𝜑𝜑frame). Now,𝑣𝑣⁄ 𝑐𝑐the magnitude𝑧𝑧 of the -vector in the rest-frame is = and, therefore,𝜃𝜃 − the𝜑𝜑 reflected -vector becomes𝑘𝑘 𝜃𝜃 − 𝜑𝜑 ′ 𝑧𝑧′ inc 𝑘𝑘 = ( 𝜔𝜔 ⁄𝑐𝑐 )[sin𝛾𝛾𝜔𝜔(2⁄𝑐𝑐 ) + cos(2 ) ]𝑘𝑘. (49) ′ inc ′ ′ A Lorentz transformation𝒌𝒌 of the𝛾𝛾 reflected𝜔𝜔 ⁄𝑐𝑐 plane-wave’s𝜃𝜃 − 𝜑𝜑 phase𝒙𝒙� -factor to𝜃𝜃 the− 𝜑𝜑 lab𝒛𝒛� frame now yields = ( )[sin(2 ) + cos(2 ) ] ′ ′ ′ ′ inc ′ ′ inc ′ 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔 𝑡𝑡 = (𝛾𝛾𝜔𝜔 ⁄𝑐𝑐)[sin(2𝜃𝜃 − 𝜑𝜑) 𝑥𝑥( ) +𝜃𝜃 cos− 𝜑𝜑(2𝑧𝑧 − 𝛾𝛾)𝜔𝜔] 𝑡𝑡 ( ) inc inc 2 = 𝛾𝛾𝜔𝜔( ⁄𝑐𝑐 )[( 𝜃𝜃)−+𝜑𝜑sin𝛾𝛾(2𝑥𝑥 − 𝑣𝑣𝑣𝑣)] + ( 𝜃𝜃 − 𝜑𝜑) cos𝑧𝑧 (−2𝛾𝛾𝜔𝜔 𝛾𝛾) 𝑡𝑡 − 𝑣𝑣𝑣𝑣⁄𝑐𝑐 2 inc inc 𝛾𝛾 𝜔𝜔 ⁄[𝑐𝑐1 +𝑣𝑣(⁄𝑐𝑐 ) sin(2 𝜃𝜃 − 𝜑𝜑)] 𝑥𝑥. 𝛾𝛾 𝜔𝜔 ⁄𝑐𝑐 𝜃𝜃 − 𝜑𝜑 𝑧𝑧 (50) inc Thus, back in the lab2 frame, the reflected beam’s frequency and wave-vector will be −𝛾𝛾 𝜔𝜔 𝑣𝑣⁄𝑐𝑐 𝜃𝜃 − 𝜑𝜑 𝑡𝑡 𝑥𝑥𝑥𝑥𝑥𝑥 = [1 + ( ) sin(2 )], (51a) ref inc = ( )[( )2+ sin(2 )] + ( ) cos(2 ) 𝜔𝜔 𝛾𝛾 𝜔𝜔 𝑣𝑣⁄𝑐𝑐 𝜃𝜃 − 𝜑𝜑 . (51b) ref 2 inc inc

𝒌𝒌 (a)𝛾𝛾 𝜔𝜔 ⁄𝑐𝑐 𝑣𝑣⁄𝑐𝑐 𝜃𝜃 − (b)𝜑𝜑 𝒙𝒙� 𝛾𝛾 𝜔𝜔 ⁄𝑐𝑐 𝜃𝜃 − 𝜑𝜑 𝒛𝒛�

x θ R x'

𝑣𝑣 Ω ref ref ( , )

inc inc 𝒌𝒌 𝜔𝜔 ( , )

z z' 𝒌𝒌 𝜔𝜔

Thermal Radiation

Fig.6. (a) Thermal radiation from a blackbody is captured by a lens and focused onto a ratchet with reflective teeth (or facets). Each tooth of the ratchet makes an angle with its nominally circular boundary. The ratchet, which is rotating at a constant angular velocity , has average radius and moment of inertia . The pressure of radiation exerts a torque on the rotating ratchet, thus𝜃𝜃 producing useful work. In the process, the light reflected from the tooth is Doppler shifted, acquiringΩ a lower temperature.𝑅𝑅 This system does not𝐼𝐼 violate the second law of thermodynamics because the reflected light essentially constitutes a cold bath. (b) A perfectly reflecting tooth of the ratchet moves with constant velocity = along the -axis. In the mirror’s rest frame , the surface normal of the mirror makes an angle with the -axis. In the laboratory frame ′ , ′ the′ incident plane-wave has frequency 𝑣𝑣and 𝑅𝑅propagationΩ vector𝑥𝑥′ = ( ) , while the𝑥𝑥 𝑦𝑦reflected𝑧𝑧 beam has frequency and wave-vectorinc =𝜃𝜃 + 𝑧𝑧 . inc inc 𝑥𝑥𝑥𝑥𝑥𝑥 ref 𝜔𝜔 ref ref ref 𝒌𝒌 − 𝜔𝜔 ⁄𝑐𝑐 𝒛𝒛� 𝜔𝜔 𝒌𝒌 𝑘𝑘𝑥𝑥 𝒙𝒙� 𝑘𝑘𝑧𝑧 𝒛𝒛�

15

It is easy to verify that = ( ) + ( ) = . These results are valid for positive as well as negative values of , andref also forref positive2 ref and2 negativeref values of . Upon reflection from the 𝑥𝑥 𝑧𝑧 mirror, the change in a single 𝑘𝑘photon’s� energy𝑘𝑘 is readily𝑘𝑘 computed𝜔𝜔 ⁄𝑐𝑐 as follows: 𝜃𝜃 𝑣𝑣 = ( ) = ( )[( ) + sin(2 )]. (52) ref inc 2 inc Prior to incidence,∆ℰ theℏ photon𝜔𝜔 − 𝜔𝜔carries a ℏmomentum𝛾𝛾 𝜔𝜔 𝑣𝑣 ⁄𝑐𝑐 = 𝑣𝑣⁄𝑐𝑐 along 𝜃𝜃the− negative𝜑𝜑 -axis. Upon reflection, there emerges a component of momentum along the -axis,inc whose angular momentum relative to the center of the ratchet is . This change in the photon’s𝓅𝓅 ℏ𝜔𝜔 angular⁄𝑐𝑐 momentum must 𝑧𝑧be equal and opposite to the change in the angularref momentum = of the 𝑥𝑥ratchet — being the ratchet’s moment of 𝑥𝑥 inertia. Given that the ratchet’s𝑅𝑅 ℏrotational𝑘𝑘 kinetic energy is = ½ = /(2 ), its acquired energy as ℒ 𝐼𝐼Ω 𝐼𝐼 a result of the photon impact will be 𝐾𝐾 2 2 ℰ 𝐼𝐼Ω ℒ 𝐼𝐼 (2 ) = = ( )[( ) + sin(2 )]. (53) ref ref inc 𝐾𝐾 2 Clearly,∆ℰ the≅ change2ℒ𝑅𝑅ℏ𝑘𝑘 in𝑥𝑥 the⁄ kinetic𝐼𝐼 𝑅𝑅 energyΩℏ𝑘𝑘𝑥𝑥 of theℏ𝛾𝛾 ratchet𝜔𝜔 𝑣𝑣is⁄ 𝑐𝑐equal𝑣𝑣 ⁄and𝑐𝑐 opposite𝜃𝜃 to− the𝜑𝜑 change in the photon energy upon reflection as revealed by the Doppler shift of its frequency. The blackbody radiation at temperature is thus seen to be capable of performing mechanical work on the ratchet, albeit at the expense of reducing the temperature of the radiation. The temperature drop is in the ratio of . 𝑇𝑇 ref inc 12. A proof of quantization of thermal radiation. An important feature of thermal radiation𝜔𝜔 interacting⁄𝜔𝜔 with material objects is intimately tied to the fluctuation-dissipation theorem [20,21]. With reference to Fig.7, imagine a highly reflective aluminum foil dropping under the pull of gravitational attraction inside an evacuated chamber containing only thermal radiation. According to the fluctuation-dissipation theorem, a friction force must be acting on the foil to oppose its acceleration under the pull of gravitation. The origin of this friction force is the Doppler shift of the various modes of the EM field inside the chamber, as seen in the foil’s rest frame.

Fig.7. Inside an evacuated chamber filled with thermal radiation at the Friction force equilibrium temperature , a highly reflective aluminum foil of mass and surface area drops under the gravitational acceleration . The fluctuation- dissipation theorem requires𝑇𝑇 that the foil eventually reach a steady𝑚𝑚-state velocity where𝐴𝐴 the gravitational force is cancelled by𝑔𝑔 a friction force. mg The origin of the friction force is the Doppler shift of the various radiation x Thermal Radiation modes as seen in the foil’s rest frame. 𝑚𝑚𝑚𝑚 In what follows, we shall derive an expression for the friction force ( , ), and confirm that indeed it is proportional to the velocity of the foil (when ) and dependent on the temperature . We will then argue that the discrete nature of the EM radiation (in the form of light𝑭𝑭 𝑣𝑣particle𝑇𝑇 s – photons – that carry energy and linear momentum𝑣𝑣 along their propagation𝑣𝑣 ≪ 𝑐𝑐 direction) is sufficient to guaran𝑇𝑇tee that a free-floating foil would reach thermal equilibrium with its radiation-filled environment. Our analysis followsℏ in𝜔𝜔 the footsteps of Einsteinℏ 𝜔𝜔in⁄ 𝑐𝑐his trail-blazing papers on the quantum nature of light [15-17,22- 24]. While some of Einstein’s arguments in these papers depend on individual atoms or molecules having discrete energy levels and , which result in absorption and emission of EM energy in discrete packets at the frequency = ( ) , his other reasonings based on thermodynamics and statistical physics leave no doubt as to the𝜀𝜀1 quantized𝜀𝜀2 nature of thermal radiation. To put it differently, Einstein’s description of physical phenomena𝜔𝜔 𝜀𝜀2 such− 𝜀𝜀1 as⁄ℏ the photoelectric effect, photoluminescence, optical absorption, and spontaneous as well as stimulated emission of radiation require only that the exchange of energy and linear momentum with matter take place via packets that are characteristic of the discrete energy levels of the atoms and molecules involved [15-17,22]; however, his examination of a highly reflective thin foil immersed in thermal radiation reveals that the radiation itself must indeed be quantized [23,24].

16

To evaluate the friction force acting on a foil that moves with constant velocity along the -axis (see Fig.7), we Lorentz transform the phase-factor of a plane wave having frequency and -vector = ( )[cos + sin cos 𝑭𝑭 + sin sin ] to the rest-frame of the mirror,𝑣𝑣 as follows:𝑥𝑥 inc 𝜔𝜔 𝑘𝑘 = ( )[cos + sin cos + sin sin ] ′ ′ ′ 𝒌𝒌 𝜔𝜔⁄𝑐𝑐 𝜃𝜃 𝒙𝒙� 𝜃𝜃 𝜑𝜑 𝒚𝒚� 𝜃𝜃 𝜑𝜑 𝒛𝒛� 𝑥𝑥 𝑦𝑦 𝑧𝑧 = ( ){cos ( + ) + sin cos + sin sin } ( + ) 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔𝜔𝜔 𝜔𝜔⁄𝑐𝑐 𝜃𝜃 𝑥𝑥 𝜃𝜃 𝜑𝜑 𝑦𝑦 𝜃𝜃 𝜑𝜑 𝑧𝑧 − 𝜔𝜔𝜔𝜔 = ( ){ (cos ′ )′ + sin cos ′ + sin sin ′ } [1′ ( ′ ) cos2 ] 𝜔𝜔⁄𝑐𝑐 𝜃𝜃 𝛾𝛾 𝑥𝑥 𝑣𝑣𝑡𝑡 𝜃𝜃 𝜑𝜑 𝑦𝑦 𝜃𝜃 𝜑𝜑 𝑧𝑧 − 𝜔𝜔𝜔𝜔 𝑡𝑡 𝑣𝑣𝑥𝑥 ⁄𝑐𝑐 . (54) ′ ′ ′ ′ Upon reflection𝜔𝜔⁄𝑐𝑐 𝛾𝛾 from 𝜃𝜃the− 𝑣𝑣mirror,⁄𝑐𝑐 𝑥𝑥 the 𝜃𝜃sign of𝜑𝜑 𝑦𝑦 is reversed.𝜃𝜃 𝜑𝜑 𝑧𝑧 Subsequently,− 𝛾𝛾𝛾𝛾 − 𝑣𝑣 ⁄a𝑐𝑐 second𝜃𝜃 𝑡𝑡Lorentz transformation (this time back to the or lab frame) yields′ 𝑘𝑘𝑥𝑥 = ( )[ ( cos ) + sin cos + sin sin ] [1 ( ) cos ] 𝑥𝑥𝑥𝑥𝑥𝑥 ′ ′ ′ ′ ′ ′ ′ ′ 𝒌𝒌 ∙ 𝒓𝒓 − 𝜔𝜔 𝑡𝑡 = (𝜔𝜔⁄𝑐𝑐)[𝛾𝛾 𝑣𝑣(⁄𝑐𝑐 − cos𝜃𝜃 )𝑥𝑥( )𝜃𝜃+ sin𝜑𝜑 𝑦𝑦cos 𝜃𝜃+ sin𝜑𝜑 𝑧𝑧sin− 𝛾𝛾𝛾𝛾] − 𝑣𝑣⁄𝑐𝑐 𝜃𝜃 𝑡𝑡 2 𝜔𝜔⁄𝑐𝑐 [𝛾𝛾1 𝑣𝑣(⁄𝑐𝑐 −) cos𝜃𝜃 ](𝑥𝑥 − 𝑣𝑣𝑣𝑣 ) 𝜃𝜃 𝜑𝜑 𝑦𝑦 𝜃𝜃 𝜑𝜑 𝑧𝑧 2 2 = (−𝛾𝛾 𝜔𝜔){ −[2(𝑣𝑣⁄𝑐𝑐 ) (1𝜃𝜃 +𝑡𝑡 − 𝑣𝑣𝑣𝑣⁄)𝑐𝑐cos ] + sin cos + sin sin } [(2 ) ( 2 ) 2 ] 𝜔𝜔⁄𝑐𝑐 𝛾𝛾1 + 𝑣𝑣⁄𝑐𝑐 − 2 𝑣𝑣 ⁄𝑐𝑐cos 𝜃𝜃. 𝑥𝑥 𝜃𝜃 𝜑𝜑 𝑦𝑦 𝜃𝜃 𝜑𝜑 𝑧𝑧 (55) 2 2 2 The changes in and−𝛾𝛾 𝜔𝜔 in consequence𝑣𝑣 ⁄𝑐𝑐 − of reflection𝑣𝑣⁄𝑐𝑐 𝜃𝜃from𝑡𝑡 the mirror are thus seen to be ( )[( ) ] 𝜔𝜔 𝑘𝑘𝑥𝑥 = = 2 cos , ref inc 2 ∆𝜔𝜔 =𝜔𝜔 − 𝜔𝜔 = 2𝛾𝛾 [(𝑣𝑣⁄𝑐𝑐 ) 𝑣𝑣⁄cos𝑐𝑐 −]( 𝜃𝜃 )𝜔𝜔. (56) ref inc 2 Now, for every photon∆𝑘𝑘𝑥𝑥 colliding𝑘𝑘𝑥𝑥 − with𝑘𝑘𝑥𝑥 the top𝛾𝛾 of𝑣𝑣 ⁄the𝑐𝑐 mirror− 𝜃𝜃at an𝜔𝜔 ⁄angle𝑐𝑐 (relative to the -axis), there is one bouncing off at an angle ( ) from the bottom, each imparting a momentum to the foil. The distance that a photon must travel to reach the foil within a short time𝜃𝜃 is [ (𝑥𝑥 cos )] . 𝑥𝑥 Consequently, the energy-density𝜋𝜋 must− 𝜃𝜃 be multiplied by the volume ±( cos ) ℏ ∆of𝑘𝑘 the relevant parallelepiped, then divided by the photon energy , to yield the number of incident∆𝑡𝑡 𝑐𝑐 photons− 𝑣𝑣⁄ at 𝜃𝜃 and∆𝑡𝑡 ( ) on the surface area of the foil. The net force resulting from the𝑐𝑐 impact𝜃𝜃 − of𝑣𝑣 all𝐴𝐴 ∆such𝑡𝑡 photon pairs is thus the product of 4 ( ) cos and theℏ relevant𝜔𝜔 energy-density (at frequency ) associated𝜃𝜃 𝜋𝜋 − 𝜃𝜃 𝐴𝐴 with the propagation angle . 2Considering that cos = sin cos d = ½ and that the integrated − 𝐴𝐴𝛾𝛾 𝑣𝑣⁄𝑐𝑐 𝜃𝜃 𝜔𝜔 energy-density ( , ) over all frequencies is 4 —𝜋𝜋the⁄2 Stefan-Boltzmann constant is given by 〈 〉 0 Eq.(11) — the average friction𝜃𝜃 force acting on the foil𝜃𝜃4 is readily∫ found𝜃𝜃 to 𝜃𝜃be 𝜃𝜃 ℰ 𝜔𝜔 𝑇𝑇 𝜎𝜎𝑇𝑇 ⁄𝑐𝑐 𝜎𝜎 = 8 ( ) . (57) 1 2 2 4 When , we have , in which𝐹𝐹𝑥𝑥 −case𝐴𝐴 𝛾𝛾 is𝑣𝑣 ⁄proportional𝑐𝑐 𝜎𝜎𝑇𝑇 to the mirror velocity , as anticipated by the fluctuation-dissipation theorem.1 Next, let𝑣𝑣 ≪us 𝑐𝑐examine our𝛾𝛾 ≅highly reflective foil𝐹𝐹𝑥𝑥 which is assumed to be in thermal equilibrium𝑣𝑣 with its surrounding radiation while residing in an otherwise empty chamber at temperature . The situation is similar to that depicted in Fig.7, except that the gravitational force is now absent. In accordance with the equipartition theorem of statistical physics [12,13], the average kinetic energy of the foil𝑇𝑇 along the -axis must be ½ = ½ ; therefore, = . At the same time, the friction force dissipates 𝑥𝑥 the foil’s kinetic2 energy 𝐵𝐵at the constant rate2 of 𝐵𝐵 per second, which, for the one-dimensional motion along , is given𝑚𝑚〈𝑣𝑣 〉by 𝑘𝑘 𝑇𝑇 〈𝑣𝑣 〉 𝑘𝑘 𝑇𝑇⁄𝑚𝑚 𝑭𝑭 𝑭𝑭 ∙ 𝒗𝒗 1 As an 𝑥𝑥aside, note the consequences of the existence of this friction force in the context of cosmic microwave background radiation. In particular, an observer traveling with constant velocity in intergalactic space must be able to determine his/her own velocity relative to absolute space by measuring the friction force against the background microwave radiation, in apparent contradiction to Einstein’s principle of relativity!

17

= 8 = Joule/sec 2 5 5 . (58) dℰ 2 2 4 2𝜋𝜋 𝐴𝐴𝑘𝑘𝐵𝐵𝑇𝑇 This loss of energy, of course, must be overcome by the4 kinetic3 energy gained by the foil in its d𝑡𝑡 𝑭𝑭 ∙ 𝒗𝒗 ≅ − 𝐴𝐴〈𝑣𝑣 ⁄𝑐𝑐 〉𝜎𝜎𝑇𝑇 − 15𝑚𝑚𝑐𝑐 ℏ random collisions with the surrounding (thermal) photons. Each photon momentum must be multiplied by 2 cos to yield the velocity (along ) that is imparted to the foil upon impact. Recall that, when adding a number of independent random variables, say, = , we will have = provided that 𝜃𝜃⁄𝑚𝑚= 0. Now, the variance of the photon𝑥𝑥 momentum ( = ) is the same2 as that of2 the photon 𝑛𝑛 𝑛𝑛 energy ( = ) divided by — see Eq.(33) for the𝜗𝜗 variance∑ 𝜗𝜗 of the energy〈-𝜗𝜗density〉 ∑ 〈𝜗𝜗( 〉 , ). The 𝑛𝑛 variance〈𝜗𝜗 〉 of the foil velocity along2 is thus 4 cos ( )𝓅𝓅 timesℏ 𝜔𝜔the⁄𝑐𝑐 variance of the corresponding EM energy. Consideringℰ ℏ𝜔𝜔 that, for impacts𝑐𝑐 along the direction2 2 2 on a surface area during a timeℰ interval𝜔𝜔 𝑇𝑇 , the relevant volume of radiation𝑥𝑥 is cos𝜃𝜃⁄ ,𝑚𝑚 the𝑐𝑐 variance of the foil velocity becomes 4 cos ( ) times the variance of the corresponding𝜃𝜃 EM energy-density.𝐴𝐴 This result must ∆be𝑡𝑡 multiplied3 by ½ 2 to yield the average kinetic𝐴𝐴𝐴𝐴∆ energy𝑡𝑡 𝜃𝜃 of the foil, then divided by to arrive at the time- 𝐴𝐴∆𝑡𝑡 𝜃𝜃⁄ 𝑚𝑚 𝑐𝑐 rate of increase of the foil’s kinetic energy. Noting that cos = sin cos d = ¼, the time- 𝑚𝑚 ∆𝑡𝑡 rate of increase of the kinetic energy of the foil associated with3 photons𝜋𝜋 ⁄in2 the frequency3 interval ( , + d ) will be (2 ) times Var[ ( , )] given by Eq.(33〈). Integration𝜃𝜃〉 ∫0 over all𝜃𝜃 frequencies𝜃𝜃 𝜃𝜃 now yields 𝜔𝜔 𝜔𝜔 ( ) ( ) 𝜔𝜔 = 𝐴𝐴⁄ 𝑚𝑚𝑚𝑚 ℰ 𝜔𝜔 𝑇𝑇 d = d = . (59) ∞ 2[ 4 ( ) ] 5 5 ∞ [ 4 ( ) ] 2 5 5 dℰ 𝐴𝐴 ℏ 𝜔𝜔 exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 𝐴𝐴𝑘𝑘𝐵𝐵𝑇𝑇 𝑥𝑥 exp 𝑥𝑥 2𝜋𝜋 𝐴𝐴𝑘𝑘𝐵𝐵𝑇𝑇 2 3 2 2 4 3 integration2 by parts4 3 d𝑡𝑡 �2𝑚𝑚𝑚𝑚� � 𝜋𝜋 𝑐𝑐 exp ℏ𝜔𝜔⁄𝑘𝑘𝐵𝐵𝑇𝑇 − 1 𝜔𝜔 �2𝜋𝜋 𝑚𝑚𝑐𝑐 ℏ � � exp 𝑥𝑥 − 1 𝑥𝑥 15𝑚𝑚𝑐𝑐 ℏ The time-rates0 of growth and decay of the kinetic energy of0 the foil given by Eqs.(58) and (59) are seen to be identical, thus leaving the foil in thermal equilibrium with its environment. In this way we have confirmed that the EM radiation, in consequence of its inherently discrete nature, imparts ½ of

kinetic energy (along ) to the free-floating foil, in compliance with the requirements of the equipartition𝐵𝐵 theorem of statistical physics. 𝑘𝑘 𝑇𝑇 𝑥𝑥 13. Summary and concluding remarks. In this paper we have derived Planck’s blackbody radiation formula using two different methods (Secs.2 and 9), and related the results to Stefan-Boltzmann’s law of thermal radiation (Sec.3), to the radiation pressure inside an evacuated chamber at constant temperature (Sec.4), to energy-density fluctuations (Sec.7), to the entropy of photon gas at fixed temperature (Sec.8), and to Einstein’s and coefficients, which pertain to spontaneous and stimulated emission of radiation (Sec.10). We have shown that a chamber filled with a homogeneous dielectric medium of refractive index ( ) continues to𝐴𝐴 emanate𝐵𝐵 radiation in accordance with the Stefan-Boltzmann law (Sec.5), even though the energy-density and radiation pressure inside the chamber are modified by the presence of ( ). 𝑛𝑛 𝜔𝜔 In Sec.6 we argued against the Minkowski momentum of a photon inside a dielectric-filled chamber and showed the EM momentum-density to be that given by Abraham’s formula, despite the 𝑛𝑛fact𝜔𝜔 that the radiation pressure acting on the walls is known to be proportional to the refractive index ( ) of the dielectric. In Sec.11 we applied the second law of thermodynamics to surfaces of varying shapes and/or differing optical properties, and demonstrated that the pressure of thermal radiation on such𝑛𝑛 surfaces𝜔𝜔 is independent of the shape and/or optical properties of the surface. We also showed that thermal radiation emanating from a blackbody at a given temperature is capable of performing mechanical work without violating the second law. Finally, following in the footsteps of Einstein, we showed in Sec.12 that the principles of statistical physics lead inescapably to the𝑇𝑇 conclusion that thermal radiation into vacuum is quantized — in discrete packets that have energy = and linear momentum = . It is thus clear that thermal physics has the ability to shed light on certain problems associated with radiation pressure and photon momentum. A problemℰ ℏthat,𝜔𝜔 to our knowledge, has𝓅𝓅 heretoforeℏ𝜔𝜔⁄𝑐𝑐 been ignored is the mechanism by which a transparent dielectric slab (or a spherical glass bead, or any other transparent object) acquires is average kinetic energy of ½ per degree of freedom in accordance with the

equipartition theorem. Perhaps one could invoke the𝐵𝐵 Balazs thought experiment [25] in conjunction with Einstein’s methods as applied to individual atoms or𝑘𝑘 free𝑇𝑇 -floating mirrors [17,24] to address this problem.

18

Appendix A To derive the Stefan-Boltzmann law from thermodynamic principles, let the energy residing in a chamber of volume be in thermal equilibrium with the walls. Denote the absolute temperature of the box by , and let the (isotropic) pressure on the interior walls be . A well-known identity, derived directly𝑈𝑈 from the second law of thermodynamics𝑉𝑉 (see [11], Chapter 45, Sec.1, Eq.(45.7)), is the following: 𝑇𝑇 𝑝𝑝 ( ) = ( ) . (A1)

Now, inside an otherwise empty box,𝜕𝜕𝜕𝜕 the⁄𝜕𝜕 𝜕𝜕radiation𝑇𝑇 𝑇𝑇 will𝜕𝜕𝜕𝜕⁄ 𝜕𝜕be𝜕𝜕 in𝑉𝑉 −thermal𝑝𝑝 equilibrium with the walls, the energy of the EM field being proportional to the volume . Consequently, ( ) = ( ), where ( ) is the EM energy-density of blackbody radiation at temperature . Also, according to Eq.(12), ( ) = ( ). Therefore,𝑈𝑈 𝑇𝑇 Eq.(A1) may be written 𝑉𝑉 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑢𝑢 𝑇𝑇 𝑢𝑢 𝑇𝑇 𝑇𝑇 𝑝𝑝 𝑇𝑇 ⅓𝑢𝑢 𝑇𝑇 = ( 3)( ) ( 3) 4 ( ) = ( )

𝑉𝑉 ( ) ( ) ( 𝑉𝑉) d 𝑢𝑢 = 4𝑇𝑇⁄d 𝜕𝜕𝜕𝜕 ⁄ 𝜕𝜕 𝜕𝜕 − ln𝑢𝑢⁄ = ln→ 𝑢𝑢 𝑇𝑇 𝑇𝑇 𝜕𝜕=𝜕𝜕⁄4𝜕𝜕𝜕𝜕 . (A2) 4 4 In the final expression→ 𝑢𝑢⁄𝑢𝑢 in Eq.(𝑇𝑇⁄A𝑇𝑇2), the→ integration𝑢𝑢 𝑇𝑇 constant,𝑇𝑇 written→ 𝑢𝑢as𝑇𝑇 4 ,𝜎𝜎 incorporates⁄𝑐𝑐 𝑇𝑇 the Stefan- Boltzmann constant defined in Eq.(11). The total energy density of blackbody radiation obtained from the laws of thermodynamics thus exhibits the same dependence as the Planck law of Eq.(9),𝜎𝜎⁄𝑐𝑐 albeit without an explicit derivation of the coefficient𝜎𝜎 4 . 4 The identity in Eq.(A1) may also 𝑇𝑇be derived by starting from the fundamental thermodynamic relation = (1 ) + ( ) , 𝜎𝜎where⁄𝑐𝑐 ( , , ) is the entropy of a closed system in thermal equilibrium with a reservoir at absolute temperature , the closed system having , volume , and a fixed number of ∆particles,𝑆𝑆 ⁄ 𝑇𝑇 [1∆𝑈𝑈2]. Since𝑝𝑝⁄𝑇𝑇 the∆𝑉𝑉 number 𝑆𝑆 of𝑈𝑈 particles𝑉𝑉 𝑁𝑁 in the system is constant, we ignore the variable and attempt to express , and as functions of𝑇𝑇 the remaining variables and . From the fundamental𝑈𝑈 𝑉𝑉 relation we have 1 = ( ) 𝑁𝑁 and = ( ) . Expanding𝑁𝑁 ( , ) to first order in the variables and , we find𝑁𝑁 ⁄ 𝑆𝑆 𝑝𝑝 𝑇𝑇 𝑈𝑈 𝑉𝑉 ( ) 𝑇𝑇 𝑉𝑉 = 𝑈𝑈 + = 0 = 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑝𝑝⁄𝑇𝑇 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑇𝑇 𝑈𝑈 𝑉𝑉 ( 𝑈𝑈 ) · 𝑉𝑉 (A3) 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑈𝑈 Substituting the above identity∆𝑇𝑇 �into𝜕𝜕𝜕𝜕� Eq𝑉𝑉 ∆.𝑈𝑈(A1)� now𝜕𝜕𝜕𝜕�𝑈𝑈 yields∆𝑉𝑉 → �𝜕𝜕𝜕𝜕�𝑇𝑇 − 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑉𝑉 ( ) = ( ) ( ) = ( ) ( ) ( )

𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑇𝑇 𝑇𝑇 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑉𝑉 − 𝑝𝑝 → −(𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕)𝑈𝑈 = 𝑇𝑇(𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 )𝑉𝑉 𝜕𝜕𝜕𝜕⁄(𝜕𝜕𝜕𝜕 𝑉𝑉 −)𝑝𝑝 𝜕𝜕=𝜕𝜕⁄𝜕𝜕𝜕𝜕( 𝑉𝑉 ) | 2 (1 ) 𝑈𝑈 | = ( ) 𝑉𝑉 | 𝑉𝑉 = · 𝑉𝑉 (A4) → − 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑇𝑇 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 − 𝑝𝑝 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 2 𝑇𝑇 𝜕𝜕2𝑝𝑝⁄𝑇𝑇 ⁄𝜕𝜕𝜕𝜕 𝜕𝜕 𝑆𝑆 𝜕𝜕 𝑆𝑆 The last equality confirms the validity→ of𝜕𝜕 the⁄ thermodynamic𝑇𝑇 ⁄𝜕𝜕𝜕𝜕 𝑈𝑈 𝜕𝜕 𝑝𝑝 identity⁄𝑇𝑇 ⁄𝜕𝜕𝜕𝜕 in𝑉𝑉 Eq.→(A1).𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 Digression: Using the thermodynamic relations 1 = ( ) and = ( ) , one can prove the identity = ( ) ( ) that relates the pressure to the rates of change of the internal energy and the 𝑉𝑉 𝑈𝑈 entropy with volume at the constant temperature⁄𝑇𝑇 𝜕𝜕𝜕𝜕. ⁄This𝜕𝜕𝜕𝜕 is done𝑝𝑝⁄ 𝑇𝑇by expanding𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 to first order the functions 𝑇𝑇 𝑇𝑇 𝑝𝑝 ( ,𝑇𝑇 )𝜕𝜕 and𝜕𝜕⁄𝜕𝜕 𝜕𝜕( ,−) 𝜕𝜕in𝜕𝜕 terms⁄𝜕𝜕𝜕𝜕 of the small variations in the𝑝𝑝 independent variables and , we find 𝑈𝑈 𝑆𝑆 𝑉𝑉 𝑇𝑇 𝑈𝑈 𝑉𝑉 𝑇𝑇 𝑆𝑆= 𝑉𝑉( 𝑇𝑇 ) + ( ) = 0 ( ) = 𝑉𝑉( 𝑇𝑇 ) ( ) . (A5)

∆𝑈𝑈 = ( 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕) 𝑇𝑇∆𝑉𝑉+ ( 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕) 𝑉𝑉∆𝑇𝑇 → ( 𝜕𝜕𝜕𝜕⁄)𝜕𝜕𝜕𝜕= 𝑈𝑈( − 𝜕𝜕)𝜕𝜕⁄+𝜕𝜕(𝜕𝜕 𝑇𝑇⁄ 𝜕𝜕𝜕𝜕) ⁄(𝜕𝜕𝜕𝜕 𝑉𝑉 ) ∆𝑆𝑆 = 𝜕𝜕(𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑇𝑇)∆𝑉𝑉 ( 𝜕𝜕𝜕𝜕⁄𝜕𝜕)𝜕𝜕 (𝑉𝑉∆𝑇𝑇 ) ( → )𝜕𝜕𝜕𝜕=⁄𝜕𝜕(𝜕𝜕 𝑈𝑈 )𝜕𝜕𝜕𝜕⁄(𝜕𝜕𝜕𝜕 𝑇𝑇 )𝜕𝜕(𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝑉𝑉 )𝜕𝜕𝜕𝜕 ⁄𝜕𝜕𝜕𝜕 𝑈𝑈 ( )𝑇𝑇 ( )( 𝑉𝑉 ) 𝑇𝑇 𝑉𝑉 𝑇𝑇( ) 𝑉𝑉( ) 𝑇𝑇 → 𝑝𝑝⁄𝑇𝑇 = 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 − 𝜕𝜕1𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 ⁄ 𝜕𝜕 𝜕𝜕 ⁄𝜕𝜕 𝑇𝑇 𝜕𝜕 𝜕𝜕 ⁄𝜕𝜕=𝜕𝜕 − 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 . (A6) →In terms𝑝𝑝⁄𝑇𝑇 of 𝜕𝜕the𝜕𝜕⁄ 𝜕𝜕Helmholtz𝜕𝜕 𝑇𝑇 − ⁄ free𝑇𝑇 𝜕𝜕energy𝜕𝜕⁄𝜕𝜕𝜕𝜕 of𝑇𝑇 the system,→ = 𝑝𝑝 , 𝑇𝑇the𝜕𝜕 𝜕𝜕preceding⁄𝜕𝜕𝜕𝜕 𝑇𝑇 − identity𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 is𝑇𝑇 written as = ( ) . The pressure observed under an isothermal expansion of the volume is thus associated with the change in the free energy , as opposed to the change in the internal𝐹𝐹 𝑈𝑈 −energy𝑇𝑇𝑇𝑇 . This is a consequence of the fact 𝑝𝑝that 𝑇𝑇 −a certain𝜕𝜕𝜕𝜕⁄𝜕𝜕𝜕𝜕 amount of heat, 𝑝𝑝 , is needed to maintain the temperature during a reversible𝑉𝑉 expansion by . The change of the entropy, 𝐹𝐹= , allows one to express the total change 𝑈𝑈in the internal energy of the system as = , which ∆leads𝑄𝑄 straightforwardly to the thermodynamic𝑇𝑇 identity under consideration. ∆𝑉𝑉 ∆𝑆𝑆 ∆𝑄𝑄⁄𝑇𝑇 ∆𝑈𝑈 𝑇𝑇∆𝑆𝑆 − 𝑝𝑝∆𝑉𝑉

19

Appendix B Here we show, for a homogeneous plane-wave in free space, that = . Assuming a plane- wave of frequency and propagation vector , Maxwell’s 3rd equation, × = , yields × = 0 0 0 0 0 0 . Squaring both side of this equation, we find ( × ) ( 𝜀𝜀×𝑬𝑬 ∙)𝑬𝑬= ( 𝜇𝜇 𝑯𝑯) ∙ 𝑯𝑯 , which may be 0 0 simplified as 𝜔𝜔 𝒌𝒌 𝜵𝜵 𝑬𝑬 −2𝜕𝜕𝑩𝑩⁄𝜕𝜕𝜕𝜕 𝒌𝒌 𝑬𝑬 0 0 0 0 0 0 0 0 0 𝜇𝜇 𝜔𝜔 𝑯𝑯 ( ) = ( 𝒌𝒌 ) 𝑬𝑬 ∙ 𝒌𝒌. 𝑬𝑬 𝜇𝜇 𝜔𝜔 𝑯𝑯 ∙ 𝑯𝑯 (B1) st 2 2 2 From Maxwell’s 1 equation, = 0, we have 0 0 = 0. Using the dispersion relation, = ( ) , 𝑘𝑘 𝑬𝑬0 ∙ 𝑬𝑬0 − 𝒌𝒌 ∙ 𝑬𝑬0 𝜇𝜇 𝜔𝜔 𝑯𝑯0 ∙ 𝑯𝑯0 Eq.(B1) yields ( ) = ( ) . Substituting for 1 , we arrive at =2 .2 0 0 2 𝜵𝜵 ∙ 𝑬𝑬2 𝒌𝒌 ∙ 𝑬𝑬 2 𝑘𝑘 𝜔𝜔 ⁄𝑐𝑐 0 0 0 0 0 0 0 References 𝜔𝜔 ⁄𝑐𝑐 𝑬𝑬0 ∙ 𝑬𝑬0 𝜇𝜇 𝜔𝜔 𝑯𝑯0 ∙ 𝑯𝑯0 𝜇𝜇 𝜀𝜀 ⁄𝑐𝑐 𝜀𝜀 𝑬𝑬0 ∙ 𝑬𝑬0 𝜇𝜇 𝑯𝑯0 ∙ 𝑯𝑯0

1. P. Penfield and H. A. Haus, Electrodynamics of Moving Media, MIT Press, Cambridge, MA (1967). 2. J. D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York (1999). 3. F. N. H. Robinson, “Electromagnetic stress and momentum in matter,” Physics Reports 16, 313-354 (1975). 4. R. Loudon, “Theory of the radiation pressure on dielectric surfaces,” J. Modern Optics 49, 821-838 (2002). 5. C. Baxter and R. Loudon, “Radiation pressure and the photon momentum in dielectrics,” J. Modern Optics 57, 830-842 (2010). 6. S. M. Barnett and R. Loudon, “The enigma of optical momentum in a medium,” Phil. Trans. Roy. Soc. A 368, 927-939 (2010). 7. M. Mansuripur, “On the Foundational Equations of the Classical Theory of Electrodynamics,” Resonance 18(2), 130-155 (2013). 8. M. Mansuripur, “The Force Law of Classical Electrodynamics: Lorentz versus Einstein and Laub,” Optical Trapping and Optical Micro-manipulation X, edited by K. Dholakia and G. C. Spalding, Proceedings of SPIE 8810, 88100K~1:18 (2013). 9. S. M. Barnett and R. Loudon, “Theory of radiation pressure on magneto-dielectric materials,” New Journal of Physics 17, 063027, pp 1-16 (2015). 10. M. Klein, “Thermodynamics in Einstein’s Thought,” Science 157, 509-516 (1967). 11. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading, Massachusetts (1964). 12. C. Kittel and H. Kroemer, Thermal Physics, 2nd edition, W. H. Freeman and Company, New York (1980). 13. C. Kittel, Elementary Statistical Physics, Dover, New York (2016). 14. A. B. Pippard, The Elements of Classical Thermodynamics, Cambridge, United Kingdom (1964). 15. 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). 16. A. Einstein, “Quantentheorie der Strahlung” (Quantum Theory of Radiation), Mitteilungen der Physikalischen Gesellschaft, Zürich, 16, 47–62 (1916). 17. A. Einstein, “Zur Quantentheorie der Strahlung” (On the Quantum Theory of Radiation), Physikalische Zeitschrift 18, 121–128 (1917). 18. D. Kleppner, “Rereading Einstein on Radiation,” Physics Today S-0031-9228-0502-010-7, pp 30-33 (2005). 19. M. Mansuripur and D.P. Tsai, “New perspective on the reciprocity theorem of classical electrodynamics,” Optics Communications 284, 707-714 (2011). 20. D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, United Kingdom (1987). 21. H.B. Callen and T.A. Welton, “Irreversibility and Generalized Noise,” Physical Review 83, 34-40 (1951). 22. 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). 23. A. Einstein, “Zum gegenwärtigen Stande des Strahlungsproblems” (On the Present Status of the Radiation Problem), Physikalische Zeitschrift 10, 185–193 (1909). 24. 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). 25. N. L. Balazs, “The energy-momentum tensor of the electromagnetic field inside matter,” Physical Review 91, 408-411 (1953). 26. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Academic, New York (2007).

20