Michael Fowler, UVa

### Introduction

In analyzing the photoelectric effect in hydrogen, we derived the rate of ionization of a hydrogen atom in a monochromatic electromagnetic wave of given strength, and the result we derived is in good agreement with experiment.  Recall that the interaction Hamiltonian was

$H 1 =( e mc )cos( k → ⋅ r → −ωt ) A → 0 ⋅ p → =( e 2mc )( e i( k → ⋅ r → −ωt ) + e −i( k → ⋅ r → −ωt ) ) A → 0 ⋅ p → . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGib WaaWbaaSqabeaacaaIXaaaaOGaeyypa0ZaaeWaaeaadaWcaaqaaiaa dwgaaeaacaWGTbGaam4yaaaaaiaawIcacaGLPaaaciGGJbGaai4Bai aacohadaqadaqaaiqadUgagaWcaiabgwSixlqadkhagaWcaiabgkHi TiabeM8a3jaadshaaiaawIcacaGLPaaaceWGbbGbaSaadaWgaaWcba GaaGimaaqabaGccqGHflY1ceWGWbGbaSaaaeaacqGH9aqpdaqadaqa amaalaaabaGaamyzaaqaaiaaikdacaWGTbGaam4yaaaaaiaawIcaca GLPaaadaqadaqaaiaadwgadaahaaWcbeqaaiaadMgadaqadaqaaiqa dUgagaWcaiabgwSixlqadkhagaWcaiabgkHiTiabeM8a3jaadshaai aawIcacaGLPaaaaaGccqGHRaWkcaWGLbWaaWbaaSqabeaacqGHsisl caWGPbWaaeWaaeaaceWGRbGbaSaacqGHflY1ceWGYbGbaSaacqGHsi slcqaHjpWDcaWG0baacaGLOaGaayzkaaaaaaGccaGLOaGaayzkaaGa bmyqayaalaWaaSbaaSqaaiaaicdaaeqaaOGaeyyXICTabmiCayaala GaaiOlaaaaaa@751E@$

and we dropped the $e iωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaamyAaiabeM8a3jaadshaaaaaaa@3AC2@$ term because it would correspond to the atom giving energy to the field, and our atom was already in its ground state.  However, if we go through the same calculation for an atom not initially in the ground state, then indeed an electromagnetic wave of appropriate frequency will cause a transition rate to a lower energy state, and $e iωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaamyAaiabeM8a3jaadshaaaaaaa@3AC2@$ is the relevant term.

But this is not the whole story. An atom in an excited state will eventually emit a photon and go to a lower energy state, even if there is zero external field.  Our analysis so far does not predict this -- obviously, the interaction written above is only nonzero if $A → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala aaaa@36CF@$ is nonzero!  So what are we missing?

Essentially, the answer is that the electromagnetic field itself is quantized.  Of course, we know that, it’s made up of photons.  Recall Planck’s successful analysis of radiation in a box: he considered all possible normal modes for the radiation, and asserted that a mode of energy $ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdChaaa@37C4@$ could only gain or lose energy in amounts $ℏω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4dHGMaeq yYdChaaa@38ED@$.  This led to the correct formula for black body radiation, then Einstein proved that the same assumption, with the same $ℏ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeS4dHGgaaa@3720@$, accounted for the photoelectric effect.  We now understand that these modes of oscillation of radiation are just simple harmonic oscillators, with energy $( n+ 1 2 )ℏω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGUbGaey4kaSYaaSqaaSqaaiaaigdaaeaacaaIYaaaaaGccaGLOaGa ayzkaaGaeS4dHGMaeqyYdChaaa@3DE8@$, and, just as a mass on a spring oscillator has fluctuations in the ground state, $〈 x 〉=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaaca WG4baacaGLPmIaayPkJaGaeyypa0JaaGimaaaa@3A84@$ but $〈 x 2 〉≠0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaaca WG4bWaaWbaaSqabeaacaaIYaaaaaGccaGLPmIaayPkJaGaeyiyIKRa aGimaaaa@3C38@$, for these electromagnetic modes $〈 A → 〉=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaace WGbbGbaSaaaiaawMYicaGLQmcacqGH9aqpcaaIWaaaaa@3A5F@$ but $〈 A → 2 〉≠0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaace WGbbGbaSaadaahaaWcbeqaaiaaikdaaaaakiaawMYicaGLQmcacqGH GjsUcaaIWaaaaa@3C13@$.

These fluctuations in $A → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala aaaa@36CF@$ mean the interaction Hamiltonian is momentarily nonzero, and therefore can cause a transition.

Therefore, to find the spontaneous transition rate (as it’s called) for an atom in a zero (classically speaking) electromagnetic field, we need to express the electromagnetic field in terms of normal modes (we’ll take a big box), then quantize these modes as quantum simple harmonic oscillators, introducing raising and lowering operators for each oscillator (these will be photon creation and annihilation operators) then construct the appropriate quantum operator expression for $A → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala aaaa@36CF@$ to put in the electron-radiation interaction Hamiltonian.

The bras and kets will now be quantum states of the electron and the radiation field, in contrast to our analysis of the classical field above, where the radiation field didn’t change.  (Of course, it did, really, in that it lost one photon, but in the classical limit there are infinitely many photons in each mode, so that wouldn’t register.)

Our treatment follows Sakurai’s Advanced Quantum Mechanics, Chapter 2 (but we stay with ordinary Gaussian units, so various expressions differ from his by factors of $4π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aI0aGaeqiWdahaleqaaaaa@388D@$ ).

We use the Coulomb gauge $∇ → ⋅ A → =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafy4bIeTbaS aacqGHflY1ceWGbbGbaSaacqGH9aqpcaaIWaaaaa@3C71@$, and of course $A → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala aaaa@36CF@$ satisfies $∇ 2 A → − 1 c 2 ∂ 2 A → ∂ t 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaW baaSqabeaacaaIYaaaaOGabmyqayaalaGaeyOeI0YaaSaaaeaacaaI XaaabaGaam4yamaaCaaaleqabaGaaGOmaaaaaaGcdaWcaaqaaiabgk Gi2oaaCaaaleqabaGaaGOmaaaakiqadgeagaWcaaqaaiabgkGi2kaa dshadaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0JaaGimaaaa@452E@$.

Taking for convenience periodic boundary conditions in the big box, we can write $A → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala aaaa@36CF@$ (classically) as a Fourier series at t = 0:

$A → ( r → ,t=0 )= 1 V ∑ k → ∑ α=1,2 ( c k → ,α ( 0 ) ε → α e i k → ⋅ r → + c k → ,α * ( 0 ) ε → * α e −i k → ⋅ r → ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaiabg2da9iaaicdaaiaa wIcacaGLPaaacqGH9aqpdaWcaaqaaiaaigdaaeaadaGcaaqaaiaadA faaSqabaaaaOWaaabuaeaadaaeqbqaamaabmaabaGaam4yamaaBaaa leaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaaGimaa GaayjkaiaawMcaaiqbew7aLzaalaWaaSbaaSqaaiabeg7aHbqabaGc caWGLbWaaWbaaSqabeaacaWGPbGabm4AayaalaGaeyyXICTabmOCay aalaaaaOGaey4kaSIaam4yamaaDaaaleaaceWGRbGbaSaacaGGSaGa eqySdegabaGaaiOkaaaakmaabmaabaGaaGimaaGaayjkaiaawMcaai qbew7aLzaalaWaaWbaaSqabeaacaGGQaaaaOWaaSbaaSqaaiabeg7a HbqabaGccaWGLbWaaWbaaSqabeaacqGHsislcaWGPbGabm4Aayaala GaeyyXICTabmOCayaalaaaaaGccaGLOaGaayzkaaaaleaacqaHXoqy cqGH9aqpcaaIXaGaaiilaiaaikdaaeqaniabggHiLdaaleaaceWGRb GbaSaaaeqaniabggHiLdaaaa@6FE2@$

The time-dependence is given by putting in the whole plane wave: $e i k → ⋅ r → → e i( k → ⋅ r → −ωt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaamyAaiqadUgagaWcaiabgwSixlqadkhagaWcaaaakiab gkziUkaadwgadaahaaWcbeqaaiaadMgadaqadaqaaiqadUgagaWcai abgwSixlqadkhagaWcaiabgkHiTiabeM8a3jaadshaaiaawIcacaGL Paaaaaaaaa@49DD@$, which time dependence can be taken into the coefficient, $c k → ,α ( t )= c k → ,α ( 0 ) e −iωt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaiabg2da9iaadogadaWgaaWcbaGabm4Aayaala Gaaiilaiabeg7aHbqabaGcdaqadaqaaiaaicdaaiaawIcacaGLPaaa caWGLbWaaWbaaSqabeaacqGHsislcaWGPbGaeqyYdCNaamiDaaaaaa a@4A57@$, so

$A → ( r → ,t )= 1 V ∑ k → ∑ α=1,2 ( c k → ,α ( t ) ε → α e i k → ⋅ r → + c k → ,α * ( t ) ε → * α e −i k → ⋅ r → ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaaGaayjkaiaawMcaaiab g2da9maalaaabaGaaGymaaqaamaakaaabaGaamOvaaWcbeaaaaGcda aeqbqaamaaqafabaWaaeWaaeaacaWGJbWaaSbaaSqaaiqadUgagaWc aiaacYcacqaHXoqyaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaa GafqyTduMbaSaadaWgaaWcbaGaeqySdegabeaakiaadwgadaahaaWc beqaaiaadMgaceWGRbGbaSaacqGHflY1ceWGYbGbaSaaaaGccqGHRa WkcaWGJbWaa0baaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeaacaGG QaaaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGafqyTduMbaSaada ahaaWcbeqaaiaacQcaaaGcdaWgaaWcbaGaeqySdegabeaakiaadwga daahaaWcbeqaaiabgkHiTiaadMgaceWGRbGbaSaacqGHflY1ceWGYb GbaSaaaaaakiaawIcacaGLPaaaaSqaaiabeg7aHjabg2da9iaaigda caGGSaGaaGOmaaqab0GaeyyeIuoaaSqaaiqadUgagaWcaaqab0Gaey yeIuoaaaa@6EA0@$

The vector $ε → α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaS aadaWgaaWcbaGaeqySdegabeaaaaa@397A@$ is the polarization of the plane wave.  It’s in the same direction as the electric field.  Actually it varies with $k → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala aaaa@36F9@$, because from $∇ → ⋅ A → =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafy4bIeTbaS aacqGHflY1ceWGbbGbaSaacqGH9aqpcaaIWaaaaa@3C71@$, it’s perpendicular to $k → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala aaaa@36F9@$. That is, for a given $k → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala aaaa@36F9@$ there are two independent polarizations.  For $k → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala aaaa@36F9@$ along the $z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36F6@$ -axis, they could be along the $x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@$ -and $y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@$ -axes, these would be called linear polarization, and is the standard approach.  But we could also take the vectors $( 1/ 2 )( 1, ±i, 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaai4lamaakaaabaGaaGOmaaWcbeaaaOGaayjkaiaawMcaamaa bmaabaGaaGymaiaacYcacaaMe8UaeyySaeRaamyAaiaacYcacaaMe8 UaaGimaaGaayjkaiaawMcaaaaa@4423@$.  These correspond to circular polarization: equal $x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F4@$ - and $y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaaaa@36F5@$ -components but with the y-component 90 degrees ahead in phase.  You may recognize the vectors $( 1/ 2 )( 1, ±i, 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIXaGaai4lamaakaaabaGaaGOmaaWcbeaaaOGaayjkaiaawMcaamaa bmaabaGaaGymaiaacYcacaaMe8UaeyySaeRaamyAaiaacYcacaaMe8 UaaGimaaGaayjkaiaawMcaaaaa@4423@$ as the eigenvectors for the rotation operator around the $z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaaaa@36F6@$ -axis -- the circularly polarized beam carries angular momentum, $±ℏ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyySaeRaeS 4dHGgaaa@390E@$ per photon, pointed along the direction of motion.

The energy density $1 8π ( | E → | 2 + | B → | 2 ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGioaiabec8aWbaadaqadaqaamaanaaabaWaaqWaaeaa ceWGfbGbaSaaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccq GHRaWkdaabdaqaaiqadkeagaWcaaGaay5bSlaawIa7amaaCaaaleqa baGaaGOmaaaaaaaakiaawIcacaGLPaaaaaa@459B@$ can be expressed as a sum over the individual $( k → , ε → ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaace WGRbGbaSaacaGGSaGafqyTduMbaSaaaiaawIcacaGLPaaaaaa@3AEB@$ modes.

Writing the electric and magnetic fields in terms of the vector potential,

$E → =−( 1/c )∂ A → /∂t, B → = ∇ → × A → . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaala Gaeyypa0JaeyOeI0YaaeWaaeaacaaIXaGaai4laiaadogaaiaawIca caGLPaaacqGHciITceWGbbGbaSaacaGGVaGaeyOaIyRaamiDaiaacY cacaaMe8UaaGjbVlqadkeagaWcaiabg2da9iqbgEGirBaalaGaey41 aqRabmyqayaalaGaaiOlaaaa@4CD7@$

where

$A → ( r → ,t )= 1 V ∑ k → ∑ α=1,2 ( c k → ,α ( t ) ε → α e i k → ⋅ r → + c k → ,α * ( t ) ε → * α e −i k → ⋅ r → ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaaGaayjkaiaawMcaaiab g2da9maalaaabaGaaGymaaqaamaakaaabaGaamOvaaWcbeaaaaGcda aeqbqaamaaqafabaWaaeWaaeaacaWGJbWaaSbaaSqaaiqadUgagaWc aiaacYcacqaHXoqyaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaa GafqyTduMbaSaadaWgaaWcbaGaeqySdegabeaakiaadwgadaahaaWc beqaaiaadMgaceWGRbGbaSaacqGHflY1ceWGYbGbaSaaaaGccqGHRa WkcaWGJbWaa0baaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeaacaGG QaaaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGafqyTduMbaSaada ahaaWcbeqaaiaacQcaaaGcdaWgaaWcbaGaeqySdegabeaakiaadwga daahaaWcbeqaaiabgkHiTiaadMgaceWGRbGbaSaacqGHflY1ceWGYb GbaSaaaaaakiaawIcacaGLPaaaaSqaaiabeg7aHjabg2da9iaaigda caGGSaGaaGOmaaqab0GaeyyeIuoaaSqaaiqadUgagaWcaaqab0Gaey yeIuoaaaa@6EA0@$

and thereby expressing the total energy

$V 8π ( | E → | 2 + | B → | 2 ¯ )= V 4π ( ω c ) 2 | A → | 2 ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGwbaabaGaaGioaiabec8aWbaadaqadaqaamaanaaabaWaaqWaaeaa ceWGfbGbaSaaaiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccq GHRaWkdaabdaqaaiqadkeagaWcaaGaay5bSlaawIa7amaaCaaaleqa baGaaGOmaaaaaaaakiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaadA faaeaacaaI0aGaeqiWdahaamaabmaabaWaaSaaaeaacqaHjpWDaeaa caWGJbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakmaana aabaWaaqWaaeaaceWGbbGbaSaaaiaawEa7caGLiWoadaahaaWcbeqa aiaaikdaaaaaaaaa@545C@$

in terms of the $( k → , ε → ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaace WGRbGbaSaacaGGSaGafqyTduMbaSaaaiaawIcacaGLPaaaaaa@3AEB@$ amplitudes $c k → ,α * ( t ), c k → ,α ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaDa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabaGaaiOkaaaakmaabmaa baGaamiDaaGaayjkaiaawMcaaiaacYcacaaMe8UaaGPaVlaadogada qhaaWcbaGabm4AayaalaGaaiilaiabeg7aHbqaaaaakmaabmaabaGa amiDaaGaayjkaiaawMcaaaaa@4850@$, then integrating the energy density over the whole large box the cross terms disappear from the orthogonality of the different modes and the total energy in the box -- the  Hamiltonian -- is:

$H= 1 2π ∑ k → ∑ α ( ω c ) 2 c k → ,α * c k → ,α . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2 da9maalaaabaGaaGymaaqaaiaaikdacqaHapaCaaWaaabuaeaadaae qbqaamaabmaabaWaaSaaaeaacqaHjpWDaeaacaWGJbaaaaGaayjkai aawMcaamaaCaaaleqabaGaaGOmaaaaaeaacqaHXoqyaeqaniabggHi LdaaleaaceWGRbGbaSaaaeqaniabggHiLdGccaWGJbWaa0baaSqaai qadUgagaWcaiaacYcacqaHXoqyaeaacaGGQaaaaOGaam4yamaaBaaa leaaceWGRbGbaSaacaGGSaGaeqySdegabeaakiaac6caaaa@5169@$

Note that although the Hamiltonian is (of course) time independent, the coefficients $c k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@3A5B@$ here are time dependent,  $c k → ,α ( t )= c k → ,α ( 0 ) e −iωt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaiabg2da9iaadogadaWgaaWcbaGabm4Aayaala Gaaiilaiabeg7aHbqabaGcdaqadaqaaiaaicdaaiaawIcacaGLPaaa caWGLbWaaWbaaSqabeaacqGHsislcaWGPbGaeqyYdCNaamiDaaaaaa a@4A57@$.

But this is formally identical to a set of simple harmonic oscillators!   Recall that for the classical oscillator, $p 2 + ( mωx ) 2 =2mE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCa aaleqabaGaaGOmaaaakiabgUcaRmaabmaabaGaamyBaiabeM8a3jaa dIhaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGH9aqpca aIYaGaamyBaiaadweaaaa@4276@$, the vector $z=mωx+ip MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEaiabg2 da9iaad2gacqaHjpWDcaWG4bGaey4kaSIaamyAaiaadchaaaa@3E7D@$ has time dependence $z( t )= z 0 e −iωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaadQhadaWgaaWcbaGa aGimaaqabaGccaWGLbWaaWbaaSqabeaacqGHsislcaWGPbGaeqyYdC NaamiDaaaaaaa@4225@$, and the oscillator energy is proportional to $z * z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaCa aaleqabaGaaiOkaaaakiaadQhaaaa@38DA@$  ( $x,p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiaacY cacaWGWbaaaa@3899@$ are the usual conjugate variables). Clearly, $c k → ,α ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaaa@3CE7@$ here corresponds to $z( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaabm aabaGaamiDaaGaayjkaiaawMcaaaaa@3978@$: same time dependence, same Hamiltonian.  Therefore the real and imaginary parts of $c k → ,α ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaaa@3CE7@$ must also be conjugate variables, which can therefore be quantized exactly as for the simple harmonic oscillator.

From

$A → ( r → ,t )= 1 V ∑ k → ∑ α=1,2 ( c k → ,α ( t ) ε → α e i k → ⋅ r → + c k → ,α * ( t ) ε → * α e −i k → ⋅ r → ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaaGaayjkaiaawMcaaiab g2da9maalaaabaGaaGymaaqaamaakaaabaGaamOvaaWcbeaaaaGcda aeqbqaamaaqafabaWaaeWaaeaacaWGJbWaaSbaaSqaaiqadUgagaWc aiaacYcacqaHXoqyaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaa GafqyTduMbaSaadaWgaaWcbaGaeqySdegabeaakiaadwgadaahaaWc beqaaiaadMgaceWGRbGbaSaacqGHflY1ceWGYbGbaSaaaaGccqGHRa WkcaWGJbWaa0baaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeaacaGG QaaaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGafqyTduMbaSaada ahaaWcbeqaaiaacQcaaaGcdaWgaaWcbaGaeqySdegabeaakiaadwga daahaaWcbeqaaiabgkHiTiaadMgaceWGRbGbaSaacqGHflY1ceWGYb GbaSaaaaaakiaawIcacaGLPaaaaSqaaiabeg7aHjabg2da9iaaigda caGGSaGaaGOmaaqab0GaeyyeIuoaaSqaaiqadUgagaWcaaqab0Gaey yeIuoaaaa@6EA0@$

we see that the real part of $c k → ,α ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaaa@3CE7@$ basically gives the contribution of the $k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala Gaaiilaiabeg7aHbaa@3948@$ oscillator to $A → ( r → ,t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaaGaayjkaiaawMcaaaaa @3B09@$, and, recalling the time dependence $c k → ,α ( t )= c k → ,α ( 0 ) e −iωt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaiabg2da9iaadogadaWgaaWcbaGabm4Aayaala Gaaiilaiabeg7aHbqabaGcdaqadaqaaiaaicdaaiaawIcacaGLPaaa caWGLbWaaWbaaSqabeaacqGHsislcaWGPbGaeqyYdCNaamiDaaaaaa a@4A57@$, the imaginary part is proportional to the contribution to $∂ A → ( r → ,t )/∂t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRabm yqayaalaWaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaaGaayjkaiaa wMcaaiaac+cacqGHciITcaWG0baaaa@3F81@$, that is, to $E → ( r → ,t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaala WaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaaGaayjkaiaawMcaaaaa @3B0D@$.  Essentially, then, the real part of $c k → ,α ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaaa@3CE7@$, proportional to the $k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala Gaaiilaiabeg7aHbaa@3948@$ Fourier component of the vector potential $A → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala aaaa@36CF@$, is what corresponds to displacement x in a 1-D simple harmonic oscillator, and the imaginary part of $c k → ,α ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaaa@3CE7@$, the $k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala Gaaiilaiabeg7aHbaa@3948@$ Fourier component of $E → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyrayaala aaaa@36D3@$, corresponds to the momentum  in the simple harmonic oscillator.

To carry out the quantization, we must express the classical Hamiltonian

$H= 1 2π ∑ k → ∑ α ( ω c ) 2 c k → ,α * c k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2 da9maalaaabaGaaGymaaqaaiaaikdacqaHapaCaaWaaabuaeaadaae qbqaamaabmaabaWaaSaaaeaacqaHjpWDaeaacaWGJbaaaaGaayjkai aawMcaamaaCaaaleqabaGaaGOmaaaaaeaacqaHXoqyaeqaniabggHi LdaaleaaceWGRbGbaSaaaeqaniabggHiLdGccaWGJbWaa0baaSqaai qadUgagaWcaiaacYcacqaHXoqyaeaacaGGQaaaaOGaam4yamaaBaaa leaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@50AD@$

in the form

$H= ∑ k → ∑ α 1 2 ( P k → ,α 2 + ω 2 Q k → ,α 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2 da9maaqafabaWaaabuaeaadaWcbaWcbaGaaGymaaqaaiaaikdaaaGc daqadaqaaiaadcfadaqhaaWcbaGabm4AayaalaGaaiilaiabeg7aHb qaaiaaikdaaaGccqGHRaWkcqaHjpWDdaahaaWcbeqaaiaaikdaaaGc caWGrbWaa0baaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeaacaaIYa aaaaGccaGLOaGaayzkaaaaleaacqaHXoqyaeqaniabggHiLdaaleaa ceWGRbGbaSaaaeqaniabggHiLdaaaa@4FAB@$

with $P k → ,α , Q k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakiaacYcacaaMe8Ua amyuamaaBaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@40E3@$ being the imaginary and real parts of the oscillator amplitude $c k → ,α ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaaa@3CE7@$ (scaled appropriately) exactly parallel to the standard treatment of the simple harmonic oscillator:

$Q k → ,α = 1 c 4π ( c k → ,α + c k → ,α * ), P k → ,α =− iω c 4π ( c k → ,α − c k → ,α * ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakiabg2da9maalaaa baGaaGymaaqaaiaadogadaGcaaqaaiaaisdacqaHapaCaSqabaaaaO WaaeWaaeaacaWGJbWaaSbaaSqaaiqadUgagaWcaiaacYcacqaHXoqy aeqaaOGaey4kaSIaam4yamaaDaaaleaaceWGRbGbaSaacaGGSaGaeq ySdegabaGaaiOkaaaaaOGaayjkaiaawMcaaiaacYcacaaMf8Uaamiu amaaBaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakiabg2da9i abgkHiTmaalaaabaGaamyAaiabeM8a3bqaaiaadogadaGcaaqaaiaa isdacqaHapaCaSqabaaaaOWaaeWaaeaacaWGJbWaaSbaaSqaaiqadU gagaWcaiaacYcacqaHXoqyaeqaaOGaeyOeI0Iaam4yamaaDaaaleaa ceWGRbGbaSaacaGGSaGaeqySdegabaGaaiOkaaaaaOGaayjkaiaawM caaiaac6caaaa@6739@$

From the time-dependence $c k → ,α ( t )= c k → ,α ( 0 ) e −iωt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaiabg2da9iaadogadaWgaaWcbaGabm4Aayaala Gaaiilaiabeg7aHbqabaGcdaqadaqaaiaaicdaaiaawIcacaGLPaaa caWGLbWaaWbaaSqabeaacqGHsislcaWGPbGaeqyYdCNaamiDaaaaaa a@4A57@$, these (classical) variables $P,Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacY cacaWGrbaaaa@3852@$ are canonical:

$∂H ∂ Q k → ,α =− P ˙ k → ,α , ∂H ∂ P k → ,α = Q ˙ k → ,α . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWGibaabaGaeyOaIyRaamyuamaaBaaaleaaceWGRbGbaSaa caGGSaGaeqySdegabeaaaaGccqGH9aqpcqGHsislceWGqbGbaiaada WgaaWcbaGabm4AayaalaGaaiilaiabeg7aHbqabaGccaGGSaGaaGzb VpaalaaabaGaeyOaIyRaamisaaqaaiabgkGi2kaadcfadaWgaaWcba Gabm4AayaalaGaaiilaiabeg7aHbqabaaaaOGaeyypa0Jabmyuayaa caWaaSbaaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeqaaOGaaiOlaa aa@54B5@$

The Hamiltonian can now be quantized by the standard procedure.  The pairs of canonical variables $P,Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaacY cacaWGrbaaaa@3852@$ (one pair to each mode $k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala Gaaiilaiabeg7aHbaa@3948@$ ) become operators, the Poisson brackets become commutators, the scale determined by Planck’s constant:

$[ Q k → ,α , P k → ′ , α ′ ]=iℏ δ k → , k → ′ δ α, α ′ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGrbWaaSbaaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeqaaOGaaiil aiaadcfadaWgaaWcbaGabm4AayaalyaafaGaaiilaiqbeg7aHzaafa aabeaaaOGaay5waiaaw2faaiabg2da9iaadMgacqWIpecAcqaH0oaz daWgaaWcbaGabm4AayaalaGaaiilaiqadUgagaWcgaqbaaqabaGccq aH0oazdaWgaaWcbaGaeqySdeMaaiilaiqbeg7aHzaafaaabeaakiaa c6caaaa@4FA6@$

The next step is to express the electron radiation interaction $( e/mc ) A → ⋅ p → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGLbGaai4laiaad2gacaWGJbaacaGLOaGaayzkaaGabmyqayaalaGa eyyXICTabmiCayaalaaaaa@3F20@$ in terms of these field operators. Since the electromagnetic field is quantized, the interaction with the electron must be that the electron emits or absorbs quanta (photons).  This is most directly represented by writing the interaction in terms of creation and annihilation (raising and lowering) operators:

$a k → ,α = 1 2ℏω ( ω Q k → ,α +i P k → ,α ) a k → ,α † = 1 2ℏω ( ω Q k → ,α −i P k → ,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb WaaSbaaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeqaaOGaeyypa0Za aSaaaeaacaaIXaaabaWaaOaaaeaacaaIYaGaeS4dHGMaeqyYdChale qaaaaakmaabmaabaGaeqyYdCNaamyuamaaBaaaleaaceWGRbGbaSaa caGGSaGaeqySdegabeaakiabgUcaRiaadMgacaWGqbWaaSbaaSqaai qadUgagaWcaiaacYcacqaHXoqyaeqaaaGccaGLOaGaayzkaaaabaGa amyyamaaDaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabaGaaiiiGa aakiabg2da9maalaaabaGaaGymaaqaamaakaaabaGaaGOmaiabl+qi OjabeM8a3bWcbeaaaaGcdaqadaqaaiabeM8a3jaadgfadaWgaaWcba Gabm4AayaalaGaaiilaiabeg7aHbqabaGccqGHsislcaWGPbGaamiu amaaBaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaOGaayjkai aawMcaaaaaaa@66B5@$

These satisfy $[ a, a † ]=1. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaaca WGHbGaaiilaiaadggadaahaaWcbeqaaiaaccciaaaakiaawUfacaGL DbaacqGH9aqpcaaIXaGaaiOlaaaa@3DD3@$

(Notice that the annihilation operator $a k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@3A59@$ is nothing but the operator representation of the classical complex amplitude $c k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@3A5B@$, with an extra factor to make it dimensionless, $c k → ,α →c 2πℏ ω a k → ,α . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakiabgkziUkaadoga daGcaaqaamaalaaabaGaaGOmaiabec8aWjabl+qiObqaaiabeM8a3b aacaaMc8oaleqaaOGaaGjcVlaadggadaWgaaWcbaGabm4AayaalaGa aiilaiabeg7aHbqabaGccaGGUaaaaa@4B1A@$   We discussed this same equivalence in the lecture on coherent states, which were eigenstates of the annihilation operator.)

Following the standard simple harmonic oscillator development, the operator $n ^ k → ,α = a k → ,α † a k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOBayaaja WaaSbaaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeqaaOGaeyypa0Ja amyyamaaDaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabaGaaiiiGa aakiaadggadaWgaaWcbaGabm4AayaalaGaaiilaiabeg7aHbqabaaa aa@451C@$ has eigenstates with integer eigenvalues,  $n ^ |n〉=n|n〉 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOBayaaja Waa4HaaeqabaGaamOBaaGaay5bSlaawQYiaiabg2da9iaad6gadaGh caqabeaacaWGUbaacaGLhWUaayPkJaaaaa@4005@$, the contribution to the Hamiltonian from the mode $k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala Gaaiilaiabeg7aHbaa@3948@$ is just $H k → ,α =( n ^ k → ,α + 1 2 )ℏω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakiabg2da9maabmaa baGabmOBayaajaWaaSbaaSqaaiqadUgagaWcaiaacYcacqaHXoqyae qaaOGaey4kaSYaaSqaaSqaaiaaigdaaeaacaaIYaaaaaGccaGLOaGa ayzkaaGaeS4dHGMaeqyYdChaaa@46D9@$, and $a † |n〉= n+1 | n+1〉 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaiiiGaaakmaaEiaabeqaaiaad6gaaiaawEa7caGLQmca cqGH9aqpdaGcaaqaaiaad6gacqGHRaWkcaaIXaaaleqaaOWaa4Haae qabaGaamOBaiabgUcaRiaaigdaaiaawEa7caGLQmcaaaa@4442@$,   $a|n〉= n | n−1〉 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaEi aabeqaaiaad6gaaiaawEa7caGLQmcacqGH9aqpdaGcaaqaaiaad6ga aSqabaGcdaGhcaqabeaacaWGUbGaeyOeI0IaaGymaaGaay5bSlaawQ Yiaaaa@41B5@$

The bottom line is: the classical plane wave expansion of $A → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala aaaa@36CF@$, with wave amplitudes $c k → ,α ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaaa@3CE7@$

$A → ( r → ,t )= 1 V ∑ k → ∑ α=1,2 ( c k → ,α ( t ) ε → α e i k → ⋅ r → + c k → ,α * ( t ) ε → * α e −i k → ⋅ r → ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaaGaayjkaiaawMcaaiab g2da9maalaaabaGaaGymaaqaamaakaaabaGaamOvaaWcbeaaaaGcda aeqbqaamaaqafabaWaaeWaaeaacaWGJbWaaSbaaSqaaiqadUgagaWc aiaacYcacqaHXoqyaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaa GafqyTduMbaSaadaWgaaWcbaGaeqySdegabeaakiaadwgadaahaaWc beqaaiaadMgaceWGRbGbaSaacqGHflY1ceWGYbGbaSaaaaGccqGHRa WkcaWGJbWaa0baaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeaacaGG QaaaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGafqyTduMbaSaada ahaaWcbeqaaiaacQcaaaGcdaWgaaWcbaGaeqySdegabeaakiaadwga daahaaWcbeqaaiabgkHiTiaadMgaceWGRbGbaSaacqGHflY1ceWGYb GbaSaaaaaakiaawIcacaGLPaaaaSqaaiabeg7aHjabg2da9iaaigda caGGSaGaaGOmaaqab0GaeyyeIuoaaSqaaiqadUgagaWcaaqab0Gaey yeIuoaaaa@6EA0@$

is replaced on quantization by a parallel operator expansion, the wave amplitude $c k → ,α ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaaaa@3CE7@$ becoming the (scaled) annihilation operator:

$A → ( r → ,t )= 1 V ∑ k → ∑ α=1,2 c 2πℏ ω ( a k → ,α ( t ) ε → α e i k → ⋅ r → + a k → ,α † ( t ) ε → * α e −i k → ⋅ r → ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaaGaayjkaiaawMcaaiab g2da9maalaaabaGaaGymaaqaamaakaaabaGaamOvaaWcbeaaaaGcda aeqbqaamaaqafabaGaam4yamaakaaabaWaaSaaaeaacaaIYaGaeqiW daNaeS4dHGgabaGaeqyYdChaaaWcbeaakmaabmaabaGaamyyamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaamiD aaGaayjkaiaawMcaaiqbew7aLzaalaWaaSbaaSqaaiabeg7aHbqaba GccaWGLbWaaWbaaSqabeaacaWGPbGabm4AayaalaGaeyyXICTabmOC ayaalaaaaOGaey4kaSIaamyyamaaDaaaleaaceWGRbGbaSaacaGGSa GaeqySdegabaGaaiiiGaaakmaabmaabaGaamiDaaGaayjkaiaawMca aiqbew7aLzaalaWaaWbaaSqabeaacaGGQaaaaOWaaSbaaSqaaiabeg 7aHbqabaGccaWGLbWaaWbaaSqabeaacqGHsislcaWGPbGabm4Aayaa laGaeyyXICTabmOCayaalaaaaaGccaGLOaGaayzkaaaaleaacqaHXo qycqGH9aqpcaaIXaGaaiilaiaaikdaaeqaniabggHiLdaaleaaceWG RbGbaSaaaeqaniabggHiLdaaaa@753E@$.

### Revisiting the Photoelectric Effect, now with a Quantized Field

Recall now that for the photoelectric effect in hydrogen, following Shankar we wrote the ingoing electromagnetic field $A → ( r → ,t )= A → 0 cos( k → ⋅ r → −ωt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaeWaaeaaceWGYbGbaSaacaGGSaGaamiDaaGaayjkaiaawMcaaiab g2da9iqadgeagaWcamaaBaaaleaacaaMc8UaaGimaaqabaGcciGGJb Gaai4BaiaacohadaqadaqaaiqadUgagaWcaiabgwSixlqadkhagaWc aiabgkHiTiabeM8a3jaadshaaiaawIcacaGLPaaaaaa@4BC6@$.  The only relevant component was that going as $e i( k → ⋅ r → −ωt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaamyAamaabmaabaGabm4AayaalaGaeyyXICTabmOCayaa laGaeyOeI0IaeqyYdCNaamiDaaGaayjkaiaawMcaaaaaaaa@418C@$.  In this section, following standard usage (including Shankar) we take an ingoing field $A → 0 e i( k → ⋅ r → −ωt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaSbaaSqaaiaaykW7caaIWaaabeaakiaadwgadaahaaWcbeqaaiaa dMgadaqadaqaaiqadUgagaWcaiabgwSixlqadkhagaWcaiabgkHiTi abeM8a3jaadshaaiaawIcacaGLPaaaaaaaaa@44DF@$ -- an irritating change by a factor of 2, but apparently unavoidable if we want to follow Shankar’s nonquantized photoelectric effect, then go on to the quantized case.  Anyway, recall the matrix element to calculate the rate was (with ingoing wave now $A → = A → 0 e i( k → ⋅ r → −ωt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala Gaeyypa0JabmyqayaalaWaaSbaaSqaaiaaykW7caaIWaaabeaakiaa dwgadaahaaWcbeqaaiaadMgadaqadaqaaiqadUgagaWcaiabgwSixl qadkhagaWcaiabgkHiTiabeM8a3jaadshaaiaawIcacaGLPaaaaaaa aa@46BD@$ )

$〈 k → f |( e mc ) A → 0 e i( k → ⋅ r → −ωt ) ⋅ p → | 100〉 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4raaeaace WGRbGbaSaadaWgaaWcbaGaamOzaaqabaaakeqacaGLPmIaay5bSdWa aeWaaeaadaWcaaqaaiaadwgaaeaacaWGTbGaam4yaaaaaiaawIcaca GLPaaaceWGbbGbaSaadaWgaaWcbaGaaGPaVlaaicdaaeqaaOGaamyz amaaCaaaleqabaGaamyAamaabmaabaGabm4AayaalaGaeyyXICTabm OCayaalaGaeyOeI0IaeqyYdCNaamiDaaGaayjkaiaawMcaaaaakiab gwSixlqadchagaWcamaaEiaabeqaaiaaigdacaaIWaGaaGimaaGaay 5bSlaawQYiaaaa@5613@$

On quantizing the field, from the end of the previous section

$A → 0 e i( k → ⋅ r → −ωt ) = c k → ,α ( 0 ) ε → V e i( k → ⋅ r → −ωt ) →c 2πℏ ω a k → ,α ε → V e i( k → ⋅ r → −ωt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaSbaaSqaaiaaicdaaeqaaOGaamyzamaaCaaaleqabaGaamyAamaa bmaabaGabm4AayaalaGaeyyXICTabmOCayaalaGaeyOeI0IaeqyYdC NaamiDaaGaayjkaiaawMcaaaaakiabg2da9maalaaabaGaam4yamaa BaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaabmaabaGaaG imaaGaayjkaiaawMcaaiqbew7aLzaalaaabaWaaOaaaeaacaWGwbaa leqaaaaakiaadwgadaahaaWcbeqaaiaadMgadaqadaqaaiqadUgaga WcaiabgwSixlqadkhagaWcaiabgkHiTiabeM8a3jaadshaaiaawIca caGLPaaaaaGccqGHsgIRcaWGJbWaaOaaaeaadaWcaaqaaiaaikdacq aHapaCcqWIpecAaeaacqaHjpWDaaaaleqaaOGaaGjcVlaadggadaWg aaWcbaGabm4AayaalaGaaiilaiabeg7aHbqabaGcdaWcaaqaaiqbew 7aLzaalaaabaWaaOaaaeaacaWGwbaaleqaaaaakiaadwgadaahaaWc beqaaiaadMgadaqadaqaaiqadUgagaWcaiabgwSixlqadkhagaWcai abgkHiTiabeM8a3jaadshaaiaawIcacaGLPaaaaaaaaa@7655@$

(the c at the beginning here being the speed of light).

Now that the electromagnetic field amplitude $A → 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaSbaaSqaaiaaicdaaeqaaaaa@37B4@$ is expressed as an annihilation operator, appropriate (photon number) bras and kets must be supplied for it to operate on.  The relevant photon mode is $k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala Gaaiilaiabeg7aHbaa@3948@$,  so labeling the corresponding photon number states $| n k → ,α 〉= |n〉 k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4Haaeqaba GaamOBamaaBaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaOGa ay5bSlaawQYiaiabg2da9maaEiaabeqaaiaad6gaaiaawEa7caGLQm cadaWgaaWcbaGabm4AayaalaGaaiilaiabeg7aHbqabaaaaa@4513@$ the matrix element that must appear in the Golden Rule is

$( 〈 k → f |⊗ 〈 n−1| k → ,α )( e mc ) e i k → ⋅ r → A → 0 ⋅ p → ( | 100〉⊗ |n〉 k → ,α ) =〈 k → f ;n−1|( e mc ) e i k → ⋅ r → c 2πℏ ω a k → ,α ε → ⋅ p → V | 100;n〉. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada qaamaaEeaabaGabm4AayaalaWaaSbaaSqaaiaadAgaaeqaaaGcbeGa ayzkJiaawEa7aiabgEPiepaaEeaabaGaamOBaiabgkHiTiaaigdaae qacaGLPmIaay5bSdWaaSbaaSqaaiqadUgagaWcaiaacYcacqaHXoqy aeqaaaGccaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiaadwgaaeaaca WGTbGaam4yaaaaaiaawIcacaGLPaaacaWGLbWaaWbaaSqabeaacaWG PbGabm4AayaalaGaeyyXICTabmOCayaalaaaaOGabmyqayaalaWaaS baaSqaaiaaicdaaeqaaOGaeyyXICTabmiCayaalaWaaeWaaeaadaGh caqabeaacaaIXaGaaGimaiaaicdaaiaawEa7caGLQmcacqGHxkcXda GhcaqabeaacaWGUbaacaGLhWUaayPkJaWaaSbaaSqaaiqadUgagaWc aiaacYcacqaHXoqyaeqaaaGccaGLOaGaayzkaaaabaGaeyypa0Zaa4 raaeaaceWGRbGbaSaadaWgaaWcbaGaamOzaaqabaGccaGG7aGaamOB aiabgkHiTiaaigdaaeqacaGLPmIaay5bSdWaaeWaaeaadaWcaaqaai aadwgaaeaacaWGTbGaam4yaaaaaiaawIcacaGLPaaacaWGLbWaaWba aSqabeaacaWGPbGabm4AayaalaGaeyyXICTabmOCayaalaaaaOGaam 4yamaakaaabaWaaSaaaeaacaaIYaGaeqiWdaNaeS4dHGgabaGaeqyY dChaaaWcbeaakiaayIW7caWGHbWaaSbaaSqaaiqadUgagaWcaiaacY cacqaHXoqyaeqaaOWaaSaaaeaacuaH1oqzgaWcaiabgwSixlqadcha gaWcaaqaamaakaaabaGaamOvaaWcbeaaaaGcdaGhcaqabeaacaaIXa GaaGimaiaaicdacaGG7aGaamOBaaGaay5bSlaawQYiaiaac6caaaaa @93E2@$

(We’ve removed the $e iωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyzamaaCa aaleqabaGaamyAaiabeM8a3jaadshaaaaaaa@3AC2@$, that just contributes to the $δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqgaaa@379C@$ -function in the Golden Rule.)

Since $a k → ,α |n〉 k → ,α = n k → ,α | n−1〉 k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaakmaaEiaabeqaaiaa d6gaaiaawEa7caGLQmcadaWgaaWcbaGabm4AayaalaGaaiilaiabeg 7aHbqabaGccqGH9aqpdaGcaaqaaiaad6gadaWgaaWcbaGabm4Aayaa laGaaiilaiabeg7aHbqabaaabeaakmaaEiaabeqaaiaad6gacqGHsi slcaaIXaaacaGLhWUaayPkJaWaaSbaaSqaaiqadUgagaWcaiaacYca cqaHXoqyaeqaaaaa@4FB1@$, it is clear that quantizing the incoming electromagnetic wave amounts to replacing the classical vector potential for this wave

$A → 0 →c ε → 2πℏ n k → ,α ωV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaSbaaSqaaiaaykW7caaIWaaabeaakiabgkziUkaadogacuaH1oqz gaWcamaakaaabaWaaSaaaeaacaaIYaGaeqiWdaNaeS4dHGMaamOBam aaBaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaOqaaiabeM8a 3jaadAfaaaGaaGPaVdWcbeaaaaa@4A52@$

At the photon occupation level $n k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@3A66@$ the (macroscopic) energy in this single mode $1 2π ( ω c ) 2 c k → ,α * c k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaiabec8aWbaadaqadaqaamaalaaabaGaeqyYdCha baGaam4yaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGcca WGJbWaa0baaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeaacaGGQaaa aOGaam4yamaaBaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaa a@47FE@$ becomes

$1 2π ( ω c ) 2 c 2 2πℏ ω a k → ,α † a k → ,α = n k → ,α ℏω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaiabec8aWbaadaqadaqaamaalaaabaGaeqyYdCha baGaam4yaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGcca WGJbWaaWbaaSqabeaacaaIYaaaaOWaaSaaaeaacaaIYaGaeqiWdaNa eS4dHGgabaGaeqyYdChaaiaadggadaqhaaWcbaGabm4AayaalaGaai ilaiabeg7aHbqaaiaaccciaaGccaWGHbWaaSbaaSqaaiqadUgagaWc aiaacYcacqaHXoqyaeqaaOGaeyypa0JaamOBamaaBaaaleaaceWGRb GbaSaacaGGSaGaeqySdegabeaakiabl+qiOjabeM8a3baa@57EA@$.

(Recall the Hamiltonian for the classical electromagnetic field is $H= 1 2π ∑ k → ∑ α ( ω c ) 2 c k → ,α * c k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2 da9maalaaabaGaaGymaaqaaiaaikdacqaHapaCaaWaaabuaeaadaae qbqaamaabmaabaWaaSaaaeaacqaHjpWDaeaacaWGJbaaaaGaayjkai aawMcaamaaCaaaleqabaGaaGOmaaaaaeaacqaHXoqyaeqaniabggHi LdaaleaaceWGRbGbaSaaaeqaniabggHiLdGccaWGJbWaa0baaSqaai qadUgagaWcaiaacYcacqaHXoqyaeaacaGGQaaaaOGaam4yamaaBaaa leaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@50AD@$ in terms of the $c k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@3A5B@$ ’s.)

From $a|n〉= n | n−1〉 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaEi aabeqaaiaad6gaaiaawEa7caGLQmcacqGH9aqpdaGcaaqaaiaad6ga aSqabaGcdaGhcaqabeaacaWGUbGaeyOeI0IaaGymaaGaay5bSlaawQ Yiaaaa@41B5@$, the Golden Rule matrix element

$〈 k → f ;n−1|( e mc ) e i k → ⋅ r → c 2πℏ ω a k → ,α ε → ⋅ p → V | 100;n〉 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4raaeaace WGRbGbaSaadaWgaaWcbaGaamOzaaqabaGccaGG7aGaamOBaiabgkHi TiaaigdaaeqacaGLPmIaay5bSdWaaeWaaeaadaWcaaqaaiaadwgaae aacaWGTbGaam4yaaaaaiaawIcacaGLPaaacaWGLbWaaWbaaSqabeaa caWGPbGabm4AayaalaGaeyyXICTabmOCayaalaaaaOGaam4yamaaka aabaWaaSaaaeaacaaIYaGaeqiWdaNaeS4dHGgabaGaeqyYdChaaaWc beaakiaayIW7caWGHbWaaSbaaSqaaiqadUgagaWcaiaacYcacqaHXo qyaeqaaOWaaSaaaeaacuaH1oqzgaWcaiabgwSixlqadchagaWcaaqa amaakaaabaGaamOvaaWcbeaaaaGcdaGhcaqabeaacaaIXaGaaGimai aaicdacaGG7aGaamOBaaGaay5bSlaawQYiaaaa@61E3@$

is proportional to $n k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca WGUbWaaSbaaSqaaiqadUgagaWcaiaacYcacqaHXoqyaeqaaaqabaaa aa@3A77@$, so the Golden Rule rate, which includes the square of the matrix element, will be exactly proportional to $n k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@3A67@$.  But from $A → 0 →c ε → 2πℏ n k → ,α ωV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaSbaaSqaaiaaykW7caaIWaaabeaakiabgkziUkaadogacuaH1oqz gaWcamaakaaabaWaaSaaaeaacaaIYaGaeqiWdaNaeS4dHGMaamOBam aaBaaaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaOqaaiabeM8a 3jaadAfaaaGaaGPaVdWcbeaaaaa@4A52@$, this is proportional to $| A → 0 | 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaace WGbbGbaSaadaWgaaWcbaGaaGimaaqabaaakiaawEa7caGLiWoadaah aaWcbeqaaiaaikdaaaaaaa@3BCA@$, and in fact the quantum rate of absorption of radiation  is exactly equal to the classical rate over the whole range of field strengths.

### Spontaneous Emission

However, this exact correspondence with the classical result does not hold for photon emission!  In that case, the atom adds a photon to a mode which already contains n photons, say, and the relevant matrix element is $a † |n〉= n+1 | n+1〉 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaiiiGaaakmaaEiaabeqaaiaad6gaaiaawEa7caGLQmca cqGH9aqpdaGcaaqaaiaad6gacqGHRaWkcaaIXaaaleqaaOWaa4Haae qabaGaamOBaiabgUcaRiaaigdaaiaawEa7caGLQmcaaaa@4442@$, so the equivalent classical vector $A → 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaala WaaSbaaSqaaiaaicdaaeqaaaaa@37B5@$ is $c ( n k → ,α +1 )2πℏ ωV ε → α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaka aabaWaaSaaaeaadaqadaqaaiaad6gadaWgaaWcbaGabm4AayaalaGa aiilaiabeg7aHbqabaGccqGHRaWkcaaIXaaacaGLOaGaayzkaaGaaG Omaiabec8aWjabl+qiObqaaiabeM8a3jaadAfaaaaaleqaaOGafqyT duMbaSaadaWgaaWcbaGaeqySdegabeaaaaa@4882@$.  This is nonzero even if $n k → ,α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaaceWGRbGbaSaacaGGSaGaeqySdegabeaaaaa@3A67@$ is zero$— MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=rbiaaa@37C4@$hence spontaneous emission.

For spontaneous emission, then, the relevant matrix element is

$〈 100;1|( e mc ) e −i k → ⋅ r → c 2πℏ ω a † k → ,α ε → ⋅ p → V | 21m;0〉. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4raaeaaca aIXaGaaGimaiaaicdacaGG7aGaaGymaaqabiaawMYicaGLhWoadaqa daqaamaalaaabaGaamyzaaqaaiaad2gacaWGJbaaaaGaayjkaiaawM caaiaadwgadaahaaWcbeqaaiabgkHiTiaadMgaceWGRbGbaSaacqGH flY1ceWGYbGbaSaaaaGccaWGJbWaaOaaaeaadaWcaaqaaiaaikdacq aHapaCcqWIpecAaeaacqaHjpWDaaaaleqaaOGaaGjcVlaadggadaah aaWcbeqaaiaaccciaaGcdaWgaaWcbaGabm4AayaalaGaaiilaiabeg 7aHbqabaGcdaWcaaqaaiqbew7aLzaalaGaeyyXICTabmiCayaalaaa baWaaOaaaeaacaWGwbaaleqaaaaakmaaEiaabeqaaiaaikdacaaIXa GaamyBaiaacUdacaaIWaaacaGLhWUaayPkJaGaaiOlaaaa@62AA@$

The density of outgoing states for the emitted photon, taking box normalization with periodic boundary conditions as usual, is

$V ( 2π ) 3 k 2 dkdΩ= V ( 2π ) 3 ω 2 dωdΩ c 3 = V ( 2π ) 3 ω 2 dEdΩ ℏ c 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGwbaabaWaaeWaaeaacaaIYaGaeqiWdahacaGLOaGaayzkaaWaaWba aSqabeaacaaIZaaaaaaakiaadUgadaahaaWcbeqaaiaaikdaaaGcca WGKbGaam4AaiaadsgacqqHPoWvcqGH9aqpdaWcaaqaaiaadAfaaeaa daqadaqaaiaaikdacqaHapaCaiaawIcacaGLPaaadaahaaWcbeqaai aaiodaaaaaaOWaaSaaaeaacqaHjpWDdaahaaWcbeqaaiaaikdaaaGc caWGKbGaeqyYdCNaamizaiabfM6axbqaaiaadogadaahaaWcbeqaai aaiodaaaaaaOGaeyypa0ZaaSaaaeaacaWGwbaabaWaaeWaaeaacaaI YaGaeqiWdahacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaakm aalaaabaGaeqyYdC3aaWbaaSqabeaacaaIYaaaaOGaamizaiaadwea caWGKbGaeuyQdCfabaGaeS4dHGMaam4yamaaCaaaleqabaGaaG4maa aaaaaaaa@63A6@$

so the density of states in energy contribution to the Golden Rule delta function is $V ( 2π ) 3 ω 2 dΩ ℏ c 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGwbaabaWaaeWaaeaacaaIYaGaeqiWdahacaGLOaGaayzkaaWaaWba aSqabeaacaaIZaaaaaaakmaalaaabaGaeqyYdC3aaWbaaSqabeaaca aIYaaaaOGaamizaiabfM6axbqaaiabl+qiOjaadogadaahaaWcbeqa aiaaiodaaaaaaaaa@4419@$, and the photon emission rate with polarization $ε → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaS aaaaa@37B0@$ into a solid angle $dΩ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizaiabfM 6axbaa@386E@$ will be:

$2π ℏ | 〈 100;1|( e mc ) e −i k → ⋅ r → c 2πℏ ω a † k → ,α ε → ⋅ p → V | 21m;0〉 | 2 V ( 2π ) 3 ω 2 dΩ ℏ c 3 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaeqiWdahabaGaeS4dHGgaamaaemaabaWaa4raaeaacaaIXaGa aGimaiaaicdacaGG7aGaaGymaaqabiaawMYicaGLhWoadaqadaqaam aalaaabaGaamyzaaqaaiaad2gacaWGJbaaaaGaayjkaiaawMcaaiaa dwgadaahaaWcbeqaaiabgkHiTiaadMgaceWGRbGbaSaacqGHflY1ce WGYbGbaSaaaaGccaWGJbWaaOaaaeaadaWcaaqaaiaaikdacqaHapaC cqWIpecAaeaacqaHjpWDaaaaleqaaOGaaGjcVlaadggadaahaaWcbe qaaiaaccciaaGcdaWgaaWcbaGabm4AayaalaGaaiilaiabeg7aHbqa baGcdaWcaaqaaiqbew7aLzaalaGaeyyXICTabmiCayaalaaabaWaaO aaaeaacaWGwbaaleqaaaaakmaaEiaabeqaaiaaikdacaaIXaGaamyB aiaacUdacaaIWaaacaGLhWUaayPkJaaacaGLhWUaayjcSdWaaWbaaS qabeaacaaIYaaaaOWaaSaaaeaacaWGwbaabaWaaeWaaeaacaaIYaGa eqiWdahacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaakmaala aabaGaeqyYdC3aaWbaaSqabeaacaaIYaaaaOGaamizaiabfM6axbqa aiabl+qiOjaadogadaahaaWcbeqaaiaaiodaaaaaaOGaaiOlaaaa@789E@$

One slight difference in evaluating the matrix element from our treatment of the photoelectric effect is in the representation of the dipole interaction.  Recall that there we gave the equivalent forms

$〈f| H 1 |i〉=( e mc ) A → 0 ⋅〈f| p → |i〉 e −iωt =( e mc )imω A → 0 ⋅〈f| r → |i〉 e −iωt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4raaeaaca WGMbaabeGaayzkJiaawEa7aiaadIeadaahaaWcbeqaaiaaigdaaaGc daGhcaqabeaacaWGPbaacaGLhWUaayPkJaGaeyypa0ZaaeWaaeaada WcaaqaaiaadwgaaeaacaWGTbGaam4yaaaaaiaawIcacaGLPaaaceWG bbGbaSaadaWgaaWcbaGaaGimaaqabaGccqGHflY1daGhbaqaaiaadA gaaeqacaGLPmIaay5bSdGabmiCayaalaWaa4HaaeqabaGaamyAaaGa ay5bSlaawQYiaiaadwgadaahaaWcbeqaaiabgkHiTiaadMgacqaHjp WDcaWG0baaaOGaeyypa0ZaaeWaaeaadaWcaaqaaiaadwgaaeaacaWG TbGaam4yaaaaaiaawIcacaGLPaaacaWGPbGaamyBaiabeM8a3jqadg eagaWcamaaBaaaleaacaaIWaaabeaakiabgwSixpaaEeaabaGaamOz aaqabiaawMYicaGLhWoaceWGYbGbaSaadaGhcaqabeaacaWGPbaaca GLhWUaayPkJaGaamyzamaaCaaaleqabaGaeyOeI0IaamyAaiabeM8a 3jaadshaaaaaaa@70DF@$

and used the $p → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaala aaaa@36FE@$ representation because the outgoing photoelectron was taken to be in a plane wave state, an eigenstates of $p → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiCayaala aaaa@36FE@$.  But for spontaneous emission, the electron goes from one bound state to another, so the $r → MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaala aaaa@3700@$ form gives a more immediate picture of the interacting dipole with the external field, and in fact the integration between the states is generally a little more direct.

So in the matrix element we make the substitution $ε → ⋅ p → →imω ε → ⋅ r → MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaS aacqGHflY1ceWGWbGbaSaacqGHsgIRcaWGPbGaamyBaiabeM8a3jqb ew7aLzaalaGaeyyXICTabmOCayaalaaaaa@45A6@$, and must then evaluate the atomic matrix element $〈 100| ε → ⋅ r → | 21m〉 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa4raaeaaca aIXaGaaGimaiaaicdaaeqacaGLPmIaay5bSdGafqyTduMbaSaacqGH flY1ceWGYbGbaSaadaGhcaqabeaacaaIYaGaaGymaiaad2gaaiaawE a7caGLQmcaaaa@44C5@$.  The natural way to do this is to express the vectors in terms of spherical harmonics, that is, to write them as spherical vectors,

$r 1 ±1 =∓( x±iy )/ 2 =r 4π/3 Y 1 ±1 , r 1 0 =z=r 4π/3 Y 1 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaDa aaleaacaaIXaaabaGaeyySaeRaaGymaaaakiabg2da9iabloHiTnaa bmaabaGaamiEaiabgglaXkaadMgacaWG5baacaGLOaGaayzkaaGaai 4lamaakaaabaGaaGOmaaWcbeaakiabg2da9iaadkhadaGcaaqaaiaa isdacqaHapaCcaGGVaGaaG4maaWcbeaakiaadMfadaqhaaWcbaGaaG ymaaqaaiabgglaXkaaigdaaaGccaGGSaGaaGjbVlaadkhadaqhaaWc baGaaGymaaqaaiaaicdaaaGccqGH9aqpcaWG6bGaeyypa0JaamOCam aakaaabaGaaGinaiabec8aWjaac+cacaaIZaaaleqaaOGaamywamaa DaaaleaacaaIXaaabaGaaGimaaaaaaa@5EAE@$

and similarly for $ε → . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyTduMbaS aacaGGUaaaaa@3862@$ The integrals are then straightforward but tedious (see Shankar page 519, Sakurai (Advanced) page 43 $– MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@$ but even then, Merzbacher is quoted on the general result.)

An amusing point made by Sakurai is that the total transition probability for spontaneous emission is $1 137 4 3 ω 3 c 2 | 〈 100| x → | 21m〉 | 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymaiaaiodacaaI3aaaamaalaaabaGaaGinaaqaaiaa iodaaaWaaSaaaeaacqaHjpWDdaahaaWcbeqaaiaaiodaaaaakeaaca WGJbWaaWbaaSqabeaacaaIYaaaaaaakmaaemaabaWaa4raaeaacaaI XaGaaGimaiaaicdaaeqacaGLPmIaay5bSdGabmiEayaalaWaa4Haae qabaGaaGOmaiaaigdacaWGTbaacaGLhWUaayPkJaaacaGLhWUaayjc SdWaaWbaaSqabeaacaaIYaaaaaaa@4E0E@$ and this same expression was obtained using the Correspondence Principle by Heisenberg, before quantum field theory was invented.

The calculated lifetime of the n = 2 state is