Septem b er 2020

Lecture Notes for 8.225 / STS.042, “Ph ysics in the 20th Cen tury”:

Blac kb o dy Radiation & Compton Scattering

Da vid Kaiser

Center for The or etic al Physics, MIT

In tro duction

These notes discuss t w o of the main topics f or the lecture on “Rethinking Ligh t” in a bit more detail: Max Planc k’s analysis of blac kb o dy radiation, and Arth ur Compton’s stud y of the scattering of high-energy ligh t o ff of elec trons. Reading these notes is optional ; the notes are m ean t to fill in som e of the gaps in v arious deriv ations t hat w e will not co v er durin g our class session.

Blac kb o dy Radiation

In Decem b er 1900, Max Planc k in tro duced a form ula to d escrib e the sp ecific pattern of ligh t emitted in the form of blac kb o dy radiation. Recall that in the late y ears of the 19th cen tury , ph ysicists (esp ecial ly in Berlin’s stil l-new Ph ysik alisc h T ec hnisc he Reic hsanstalt, or PTR) w ere studying the pattern of ligh t emitted from a t yp e of ob ject kno wn as a “blac kb o dy”: an ob j ect that absorbs (nearly) all the incoming ligh t that migh t fall up on it (so that it app ears blac k). When a blac kb o dy is heated to a sufficien tly high temp erature it will gl o w, emitting a sp ecific pattern of radiation. Ab o v e a temp erature of ab out 500 ◦ C the glo w will include substan tial emission in the visible rang e of the sp ectrum, whic h i s detectable b y the unaided ey e. Though the term “blac kb o dy radiation” migh t sound ab s t ract or esoteric, man y p eople are familiar with the phenomen on: think of the reddish-orange glo w of c harcoal on a grill or em b ers in a fireplace.

Thanks to his colleagues at the PTR, Planc k had access to some of the most up-to-date and precise measure men ts of the sp e ctrum of blac kb o dy radiation: the amoun t of energy emitted p er frequency ν , p er unit v olume. Planc k’s form ula to ok the form

8 π hν 3

u ( ν , T ) =

c 3 [ e hν /k T − 1] , (1)

Figure 1: The energy densit y p er unit frequency , u ( ν , T ), for blac kb o dy radiation in tro du c ed b y Max Planc k in 1900, as a function of frequency ν , for three temp eratures T 1 < T 2 < T 3 . As the temp erature rises, the fr e qu e n c y of p eak emission shifts to higher frequencies (to w ard the b lue end of the sp ectrum), and the total energy output increases.

where k is Boltzmann’s constan t (familiar from statistical mec hanics), an d h w as a new uni - v ersal constan t, so on dubb ed “Planc k’s constan t.” Both t he p eak frequency of emission and the o v erall amoun t of energy emitted dep end on the temp erature T to whic h the blac k b o dy has b een heated. The c haracteristic shap e of the energy densit y p e r unit frequency , u ( ν , T ), is sho wn in Fig. 1 f or three temp eratures T 1 < T 2 < T 3 .

Ph ysicists a nd historians con tin ue to debate ho w Planc k ar riv ed at the expression in

Eq. (1), and what Planc k thoug h t his o wn equa tion really mean t. 1 In these notes, w e will briefly consider a mo dern approac h to deriving Planc k’s expression in Eq. (1), to highligh t wh y ph ysicists to day consider it to b e suc h a no v el break with previous approac hes to the in teraction b et w een ligh t and matter.

The quan tit y u ( ν , T ) represen ts the amoun t of energy emitted b y a blac kb o dy p er unit frequency p er unit v olume. T o arriv e at the expression in Eq. (1), w e ma y pr o ceed in t w o steps: deriv e an expression for the n um b er of indep enden t radiation mo des within a v olume

V p er frequency ν , and separately der iv e an expression for the a v erage energy of ea c h mo de. In other w ords, our goal is to find expre s sion s for eac h quan tit y on the righ t side of the

1 F or an indication of the range of debate o v er Planc k’s original deriv ation and in terpretation of Eq. ( 1 ) , compare Thomas S. Kuhn, Black-Bo dy The ory and the Quantum Disc ontinuity, 1894 - 1912 (New Y ork: Oxford Univ ersit y Press, 1978) and Kuhn, “Revisiting Pl anc k,” Historic al Studies in the Natur al Scienc es 14 (1984): 231-252, with the rec en t discussion in Mic hael Nauen b erg, “Max Planc k and the birth of the quan tum h yp othesis,” A meric an Journal of P hy sics 84 (Sept 2016): 709-720.

follo wing expression:

u ( ν , T ) = energy densit y

p er frequency

n um b er o f mo des

= p er frequency p er unit v olume

× a v erag e energy

p er mo de

.

(2)

Planc k calculated the first of these terms in a manner that w as totally consisten t w ith the ph ysics of the 19th cen tury . The big c hange comes with ho w to ev aluate the second term: the a v erage energy p er mo de.

In the App endix I d escrib e ho w to calculate the n um b er of radiation mo des p er frequency p er unit v olum e within a ca vit y of fixed size. Planc k found

n um b er o f mo des

p er frequency p e r unit v olume

8 π ν 2

= c 3

. (3)

As w e will see in the App endix, Planc k’s approac h t o de riving this expression made use of Maxw ell’s treatmen t of e lectromagnetic w a v es. In fact, sev eral of Planc k’s predecessors had deriv ed this expression during the 1880s a nd 1890s.

The next step, according to Eq. (2), is to find an expression for the aver age ener gy p er mo de, whic h w e ma y denote ϵ ¯ . If w e w ere to con tin ue to use (b y-then) standard argumen ts from 19th cen tury ph ysics, w e w ould conclude that in equilibrium at a temp erat ure T , eac h radiation mo de should ha v e an a v erage energ y ϵ ¯ = k T . (This is kno wn as the “equipartition theorem.”) This result follo w ed from the pioneering studies of statistical mec hanics b y figures lik e Maxw ell and Ludwig Boltzmann. The argumen t is that in equilibrium, the probabilit y that a ph ysical system will b e found in a state of energy E is w eigh ted b y the “Boltzmann

factor” exp[ − E /k T ]: the greater the energy of the asso ciated state, the less lik ely it w ould

b e to find the system in suc h a state. If w e w an t to find the aver age v alue of the energy p er mo de in equilibrium, w e calculate the exp ectation v alu e

Using

a v erag e energy p er mo de

Z ∞

∞ − є/k F

0

classical ∞ − є/k F

0

. (4)

w e find

dx e − x/a = a for Re[ a ] > 0 ,

0

∞

dx x e − x/a = a 2 for Re[ a ] > 0 ,

0

(5)

a v erage energy p er mo de

= ϵ ¯ classical

= k T , (6)

consisten t with the (classical) equipartition th eorem. Com bining Eqs. (3) an d (6 ), w e then find, for Eq. (2),

u ( ν , T ) =

8 π ν 2

c 3

k T . (7)

This form for the energy densit y p er frequency is kno wn a s the “Ra yleigh-Jeans” sp ectrum. Note that according to Eq. (7), the energy of radiation em itted b y a blac kb o dy w ould gr ow without limit ; the blac k b o dy w ould emit more and more energy at higher an d higher fre - quencies. This b ecame kno wn as the “ultra violet catastrophe.” If one to ok this e xpression at face v alue, Maxw ell’s treatmen t of radiation, com b ined with then-standard tec hniques from statistical ph ysics, suggested that a blac kb o dy should emit an unlimite d amoun t o f energy p er v olume — tha t is, t he energy de nsit y , ρ ( T ) = dν u ( ν , T ), diver ges once one in tegrates u ( ν , T ) o v er all p ossible f requencies ν . Suc h an exp ression cle arly w as not consisten t with ordinary exp erience, let alone the increasing ly precise data measured b y Planc k’s colle agues at the PTR.

What Planc k did next — and what he thou gh t hi s next steps actually implied ab out the nature of radiation — remains a p oin t of con tro v ersy among ph ysicists and h istorians. 2 The standard view among ph ysicists to day is that quan tum theory a v oids the ultra violet catastrophe, a nd yields an expression for u ( ν , T ) that matc hes high-precision exp erimen tal data, b y replacing the (classical) equipartition theorem with an en tirely new metho d for accoun ting for the allo w able energies of matter and radiation at the atomic scale. That is, in mo dern deriv ations, w e arriv e at the form of the expression in Eq. (1) b y r estricting the allo w abl e energie s p er mo de to discr ete , quan tized v alues: in place of the con tin uum of p ossible v alues of ϵ o v er whic h w e in tegrated in Eq. (4 ) to find ϵ ¯ classical , mo dern quan tum- theoretic treat men ts sum o v er a set of discr ete v alue s , ϵ n = nhν , with n a non-negativ e in teger. On this mo dern view, a radiation mo de in the ca vit y can carry one unit of ene rgy ( hν ), or t w o unit s (2 hν ), or 57 units, but not 1.3 units.

In place of Eq. (4), w e write

a v er age energy

P ∞

nhν e − nhν /k F

T o ev aluat e Eq. (8), w e can use some clev er tric ks, including the usual e xpression for summing a geometric seri es:

∞

x n = 1

1 − x

for | x | < 1 . (9)

n =0

Then the deno minator in Eq. (8) ma y b e ev aluated as

∞ ∞

e − nhν /k F = e − hν /k F n = 1 . (10)

(1 − e − hν /k F )

n =0 n =0

2 See, e.g., the references b y Kuhn and Nauen b erg in fo otnote 1.

F or the n umerator, let us define β ≡ hν /k T , so that w e ma y write

X nhν e − nhν /k F

= hν X n e − nβ

n =0 n =0

d

= hν −

dβ

X n =0

e − nβ #

(11)

= hν − d

dβ

e − β

1

1 − e − β

= hν (1 − e − β ) 2 .

Com bining Eqs. (10) and (11 ), Eq. (8) then ma y b e written

a v erag e energy p er mo de

hν e − hν /k F

= ϵ ¯ quan tum = [1 − e − hν /k F ]

hν

= [ e hν /k F − 1] .

(12)

W e no w ha v e expressions for e ac h term on the righ t side of Eq. (2 ), whic h yields Planc k’s expression for the energy densit y p er frequency:

8 π hν 3

exactly as in Eq. (1).

u ( ν , T ) =

c 3 [ e hν /k F − 1] , (13)

Thomas Kuhn con tends that Planc k himself did not reason this w a y when he first deriv ed Eq. (13) in 1900. In particular, Kuhn argues that Planc k set the total energy of the system to b e an in teger m ultiple of hν ,

E total = ϵ n = N hν , (14)

n

but that Planc k did not restrict eac h radiation mo de to ha v e a quan tized energy ϵ n = nhν . Moreo v er, Kuhn con tin ues, in Planc k’s original deriv ation, he considered energi es for sub - systems “in the r ange ( ϵ i , ϵ i + Δ ϵ i )”; that Planc k used bins of size ϵ = hν for his accoun ting, but then coun ted ho w man y sub-systems with c ontinuous energies fell within bins [0 , ϵ ], [ ϵ, 2 ϵ ], and so on. In his 1906 lectures, Kuhn notes, Planc k still sp ok e of c ontinuous (rather than quan tized) en ergy exc hange b et w een th e matter of the blac kb o dy and the emitted radi ation. 3 Other historians and ph ysicists con tin ue to scrutinize Planc k’s original deriv atio n as w ell as Planc k’s re-d eriv ations during the y ears after 1900. 4 What all comme n tators agree up on

3 See Kuhn, Black-Bo dy The ory , and Kuhn , “Revisiting Planc k.”

4 F or a review of more recen t analyses and a comprehensiv e list of additional references on the topic, see Nauen b erg, “Max P lanc k and the bir th of the quan tu m h yp othesis.”

is that Planc k w as highly r eluctant to break with the statistical argumen ts (b y Maxw ell, Boltzmann, and others) that had b ecome stan dard in his da y . Planc k wrote to a colleague as late as 1931: “What I did can b e describ ed as simply an act of desp eration. [... In tro du cing bins of size ϵ = hν ] w as purely a formal assump tion and I really did not giv e it m uc h though t.” 5

No mat ter what Plan c k though t ab out the derivation of his expression in Eq. (13), w e do

kno w wh y he had some confidenc e ab out its form. F or small frequencies (long w a v elengths), Eq. (13) coincides with the Ra yleig h-Jeans expression. I n pa rticular, for hν k T , w e ma y expand

hν " hν 2 #

e hν /k F

In that limit, Eq. (13) b ecomes

= 1 + + O . (15)

k T k T

8 π " hν 2 #

whic h exactly matc hes th e Ra yleigh-Jeans expression in Eq. (7). Unlike the Ra y leigh-Jeans form in Eq. (7), ho w ev er, Planc k’s expression in Eq. (13) do es not lead to an “ultra violet catastrophe.” In the limit of large frequencies (short w a v elengths), hν k T , w e ma y ex pand

1 = 1 = e − hν /k F + O k T , (17)

so that Planc k’s expression in Eq. (13) tak es the form

u ( ν , T ) = 8 π hν 3 e − hν /k F + O k T . (18)

The exp onen tial deca y of this function with large ν more than comp ensates for the factor of ν 3 , and the o v erall energy densit y p er frequency falls gen tly with increasing freq uency , as sho wn in Fig. 1. Planc k had more than only these theoretical features to b olster his confidence. He w as also in close con tact with exp erimen talists at the PTR who w ere refining their o wn measuremen ts of blac kb o dy sp ectra in their lab orato ry . As he lated recalled: “The v ery next morning [aft er first presen ting his expression at a meeting of the Berlin Ph ysical So ciet y] I re ceiv ed a visit from m y colleague [Heinric h] Rub ens [an exp erimen tal ph ysicist at the PTR]. He came to tell me that after the conclusion o f the meeting, he had that v ery nigh t c hec k ed m y form ula against the results of his measuremen ts and found a satisfac tory concordance at ev ery p oin t.” 6

5 Max Planc k to Rob ert W. W o o d, 7 Octob er 1931, as quoted in Nauen b erg, “Max Planc k and the birth of the quan tum h yp othesis,” p. 715.

6 Max Planc k, Scientific A u t obi o gr aphy , trans. F. Ga ynor (New Y ork: P hilosophical Library , 1949), pp. 39 - 41, as quoted in Nauen b erg, “Max Planc k and th e birth of the quan tum h yp othesis,” on p. 713.

Compton Scattering

Alb ert Einstein in tro duced his “heuristic” notion of a ligh t quan tum in 1905, in an article en - titled, “On a heuristic p oin t of view concerning the pro duction and transformation of ligh t.” 7 Ab out t w en t y y ears later, ph ysicists adopted the te rm “photon” t o refer to an individual ligh t quan tum, or particle of li gh t. Einstein considered his pap er on ligh t q uan ta to b e the most “rev olutionary” of the pap ers he wrote that y ear. 8 In the pap er, he suggested that “in the propagation of a ligh t ra y emitted from a p oin t sourc e, the en ergy is not distributed con tin uously o v er ev er-increasing v olumes of space, but consists of a finite n um b er of energy quan ta lo calized at p oin ts of space that mo v e without dividing, and can b e absorb ed or generated only as complete units.” 9

Although Einstein’s suggestion of ligh t quan ta offered a particularly economical expla - nation of puzzling exp erimen tal results, suc h as the photo electric effect, most ph ysicists remained sk eptical of the idea for man y y ears to come. Ev en after Max Planc k h ad b ecome con vinced of the imp ortance of Einstein’s w ork on topics lik e relativit y , for example, he still harb ored d oubts ab out Ein s t ein’s idea of ligh t quan ta. In urging his colleagues to in vite Einstein to join the prestigious Prussian Academ y of Sciences in 1913, Planc k explained, “In sum , one c an sa y that there is hardly one among the grea t problems in whic h mo dern ph ysics is so ric h to whic h Einstein has not made a remark able con tribution. That he ma y sometimes ha v e missed the target in his sp ecula tions, as, for example, in his h yp othesis of ligh t quan ta, cannot really b e held to o m uc h against him.” 10

An im p ortan t step in con vincing the broader comm unit y to tak e the idea of lig h t quan ta

seriously came in the ea rly 1920s, with the exp erimen ts b y Arth ur H. Compton on the scattering of high-energy ligh t off of el ectrons. In fact, whereas nearly t w o d ecades elapsed b efore Einstein’s “ heuristic” suggestion of ligh t quan ta b ecam e broadly accepted within the comm unit y , Compton ’s exp erimen tal results w ere greeted with fanfare almost immediately . He sub mitted his article on the new results to the Physic al R eview in Decem b er 1922; the pap er w as published in Ma y 1923; and Compton w as a w arded the Nob el Prize for his efforts in 1927! Ph ysicist Rob ert Millik an lauded Compton’s exp erimen t for “k eep[ing] the ph ysicist

7 Alb ert Einstein, “ U ¨ b er einen die Erzeugung und V erw andlun g des Lic h tes b etreffenden heuristisc hen Gesic h t s p unkt,” A n nal en der Physi k 17 (1905): 132-148. An English translation is a v ailable in John Stac hel, Einstei n ’s Mir aculous Y e ar: Five Pap ers that Change d the F ac e of Physics (Princeton: Princeton Univ ersit y Press, 2005 [1998], 177-198.

8 Alb ert Einstein to Conrad Habic h t, M a y 1905, as translated in The Col le cte d Pap ers of A l b ert E i n - stein , V olume 5, The Swiss Y e ars: Corr esp ondenc e, 1902-1914 (English T r anslation Supplement) (Princeton: Princeton Univ ersit y Press, 1995), pp. 19-20.

9 Einstein, “Heuristic p oin t of view,” as translated in Stac hel, Einstei n ’s M ir aculous Y e ar , p. 178.

10 Max Planc k, as quoted in Abraham P ais, Subtle is the L or d: The Scienc e and the Life of A lb er t E i n stei n

(New Y ork: Oxford Univ ersit y Press, 1982), on p. 382.

Figure 2: Arth ur Compton in terpreted his exp erimen tal r e sults of the scattering of high-energy X-ra ys off of electrons in terms of t w o-b o dy scattering among discrete particles. In this case, the incoming ligh t quan tum scattered off an electron at rest; follo wing the collision, the ligh t quan tum w as scattered at some angle θ from the direction of its original path, wh ile the electron recoiled along some angle ϕ .

mo dest and undogmatic.” 11

Compton h ad not set o ut to test Einste in’s ideas ab out ligh t quan ta; on the con trary , Compton, lik e most ph y s i cists, had b een sk eptical of Einstein’s idea. Instead, Compton w as in terested in the b eha vior of hig h-energy ligh t w a v es, and b egan cond ucting a series of exp erimen ts on the reflection of X-ra ys. He measured w a v e-lik e prop erties of the ligh t b efore and after reflection . In particular, he found a shift in the w a v elength of the ligh t follo wing scattering. The amoun t of the shift in w a v elength dep ended on the angle of scatter:

Δ λ ≡ λ 0 − λ ∝ (1 − cos θ ) , (19)

where λ w as the w a v elength of the incoming ligh t w a v e and λ 0 the w a v elength of the scattered ligh t w a v e. Compton found he could o nly mak e sense of these exp erimen tal results if he in - v ok e d Einstein’s h yp othesis ab out ligh t quan ta. Rather th an in terpreting his l igh t-reflection exp erimen ts in terms of con tin uou s w a v es, Compton analyzed his results as if the incoming ligh t consisted of individual par ticles, ea c h wi th a definite energy and mom en tum, and that individual ligh t quan ta scattered off of electrons m uc h as t w o billiard balls w ould collide on a p o o l table. In other w ords, Compton w as only abl e to mak e sense of his ligh t-reflection results if he treated the in teraction of ligh t with matter in terms of t w o-b o dy scattering among discrete particles, as in Fig. 2.

Compton analyzed the in teraction b et w een high-energy X-ra ys and electrons b y balancing the total ener gy and momen tum of the system, b efore and after the collision. 12 Prior to the

11 Arth ur H. Compton, “A quan tu m theo ry of the scattering of X-ra ys b y ligh t elemen ts,” Physic al R eview 21 (1923): 483-502. Rob ert Millik an’s quotation from 1926 ma y b e found in Roger H. Steu w er, The Compton Effe ct (New Y ork: Science History Publications, 1975), p. 288.

12 In his 1923 article, Compton used the term “ligh t quan tum” to desc r ib e individual particles of ligh t. F or ease of notati on in these notes, I will use the term “photon,” whic h en tered in to common usage a few y ears after Compton’s original publication.

collision, the incom ing photon mo v ed along the x ˆ axis with energy E photon and momen tum p photon while the ele ctron sat at rest, with energy E electron = mc 2 and momen tum p electron = 0. F ollo wing the collision, the photo n scattered at some an gle θ from its original direction

of motion, wit h p ost-collision energy E 0 and momen tum p 0

, while the electron

recoiled along some angle ϕ from the incoming photon’s path with (relativistic) energy

E 0 = γ mc 2 and momen tum p 0

= γ m v . (By the time C ompton w as conductin g his

electron electron

exp erimen ts, Einstein’s w ork on sp ecial relativit y had b ecome fairly w ell establishe d; g iv en the hi gh energy of the incoming X-ra ys, Com pton readily adopted relativistic kinematics for treating t he electron’s motion follo wing collision.) T o treat the photo n’s energy and momen tum, Compton r eluctantly adopted Einstein’s expressions for individual ligh t quan ta:

hc

E photon = hν =

E

,

λ

hν h

(20)

| p | =

photon

= = .

photon

c c λ

Keeping in mind that momen tum is a v ector quan tit y — so that eac h comp onen t of the momen tum ( p x and p y ) m ust balance b efore and after collision — Compton could the n write three equations to describ e the t w o-b o dy scattering:

(conserv ation of energy :)

hc + mc 2 = hc + γ mc 2 λ λ 0

(conserv ation of p

h h

:) + 0 = cos θ + γ mv cos ϕ (21)

x λ λ 0 h

(conserv ation of p y :) 0 + 0 = λ 0 sin θ − γ mv sin ϕ ,

where v = | v | is the magnitude of th e electron’s recoil v elo cit y . He no w had three equations for three unkno wns ( λ 0 , θ , and ϕ ), so he could solv e for the q uan tit y of in terest, namely , Δ λ = λ 0 − λ .

First w e ma y square the expressions that co me from p x and p y and add them. F rom the

p x equation, w e find

γ 2 m 2 v 2 cos 2 ϕ = h 2 1 − 2 cos θ + 1 cos 2 θ , (22)

and from the p y equation w e ma y write

λ 2 λλ 0

λ 0 2

γ 2 m 2 v 2 sin 2 ϕ = h 2 1 sin 2 θ . (23)

λ 0 2

Adding Eqs. (22) and (23) yie lds

γ 2 m 2 v 2 = h 2 1

+ 1 − 2 cos θ . (24)

Squaring the expression that comes from the conserv ation of energy (after canceling the factor of c that is common to eac h term), w e ha v e

h h 2

γ 2 m 2 c 2 = − + mc

λ λ 0

(25)

= h 2 1 + 1 − 2 + 2 mc 1 − 1 + m 2 c 2 .

Subtracting Eq. (24) from Eq. (25) yields

2 2 2

v 2

2 2 2 mc 1 1 2 2

Using the definitio n of γ , w e m a y rewrite the quan tit y of the left side of Eq. (26):

2 2 2 v 2

1 2 2 v 2

γ m c 1 − c 2

= v 2 m c 1 − c 2

c 2

= m 2 c 2 .

(27)

Then the term s m 2 c 2 cancel from eac h side of Eq. (26), lea ving

2 (1 − cos θ ) = 2 mc 1 − 1 , (28)

λλ 0 h λ λ 0

or

λ 0 − λ = h (1 − cos θ ) . (29)

mc

Not only did this result yield the pa rticular angular dep endence that Compton had mea - sured, as in Eq. (19); the co efficien t on the righ t side of Eq. (29), h/mc , yielded a v ery close fit to Compton’s exp erimen tal data, on ce he plugged in the v alues for Planc k’s constan t ( h ), the electron’s mass ( m ), and the sp eed of ligh t ( c ). Whereas Compton had b een sk eptical of Einstein’s ligh t-quan tu m h yp othesis b efore em barking on his exp e rimen ts, he found rema rk - able agreemen t with his o wn data once he applied Einstein’s ideas to treat the scattering of ligh t from ma tter. The co efficien t h/mc b ecame kn o wn as the “Co mpton w a v elength”: it set the sc ale for the observ ed shift in w a v elength for the scattere d ligh t.

Note the strange mixing of w a v e and part icle notions throughout: Compton pro duced ligh t waves of kno wn w a v elength λ , and measured the shift in their w a v elength follo wing scattering; he could only m ak e sense of his empirical results b y treating the ligh t as a collection of individual p articles ; and then his result, deriv ed in terms of t w o-b o dy scattering, yielded a new measure of “w a v elength,” in t erms of the quan tit y h/mc . This s o rt of “ wave - p article duality ” b ecame em blematic of more systematic attempts to create a first-principles quan tum mec hanics during the mid- 1920s.

App endix: Calculating the Num b er of Radiation Mo des in a Ca vit y

In this App endix w e ca lculate the n um b er o f radiation mo des p er frequency p e r unit v olu me, whic h is one of the main steps in derivin g Planc k’s expression for the energ y densit y p er unit frequency of blac kb o dy radiation. During th e 1880s and 1890s, blac kb o dy radiation w as often referred to as “ca vit y radiation,” sinc e one of the b est w a ys to pro duce a blac kb o dy and measure prop erties of the resulting radiation w as to heat a closed ca vit y — so that inside the ca vit y there w as virtually no incom ing ligh t — and then measure the radia tion that escap ed from the ca vit y through a tin y windo w. The prop erties of the radiation from the ca vit y w ould then b e indep en den t of an y external radiation that migh t ha v e otherwise reflected off o f the ca vit y itself. 13

Recall from Maxw ell’s equation for the propagation of ligh t that the electric field asso ci -

ated with a ligh t w a v e satisfies

∂ 2 ∂ 2 ∂ 2 1 ∂ 2

W e are in terested in solutions of this w a v e equation for the case in whi c h the radiation is con tained within a ca vit y , in equilibrium. W e ma y consider a cubic c a vit y of length L on eac h side. W e w an t to find standing-wave solutions — that is, solutions for E that v anish at the b oundaries of the ca vit y . (If the s o lutions didn’t v an is h at the b ound aries, then radiation, and hence energy , w ould leak outside the ca vit y; but in that case, the system within the ca vit y w ould not remain in equilibrium .) If w e adopt a separable form for E ( t, x, y , z ) = T ( t ) X ( x ) Y ( y ) Z ( z ), then Eq. (30) b ecomes four separate equations for the four fu nctions:

where

d 2 X dx 2

+ k 2 X = 0 ,

d 2 Y dy 2

+ k 2 Y = 0 ,

d 2 Z dz 2

+ k 2 Z = 0 ,

d 2 T dt 2

+ ω 2

T = 0 , (31)

ω 2 = c 2 � k 2 + k 2 + k 2 . (32)

T o satisfy the b oundary condition that E ( t, x, y , z ) v anish es at ea c h edge of the cubic ca vit y , w e ma y write a solution of the form

E ( t, x, y , z ) = E 0 sin sin sin sin ( ω t ) , (33)

n x π x n y π y n z π z

13 See, e.g., Nau e n b erg, “Max Planc k and the b irth of the quan tum h yp othesis,” p. 710, and Emilio Segr ` e, “Planc k, un willing rev olutionary: The idea of quan tization,” in Segr ` e, F r om X-R ays to Quarks: Mo dern Physicists and Their Disc overies (San F rancisco: W. H. F reeman, 1980), 61-77, on pp. 67-68.