4.5. QUANTIZED RADIATION FIELD

Transcription

1 QUANTIZED RADIATION FIELD Baground Our treatent of the vetor potental has drawn on the onohroat plane-wave soluton to the wave-euaton for A. The uantu treatent of lght as a partle desrbes the energy of the lght soure as proportonal to the freueny, and the photon of ths freueny s assoated wth a avty ode wth wavevetor /that desrbes the nuber of osllatons that the wave an ae n a ube wth length L. For a very large avty you have a ontnuous range of allowed. The avty s portant for onsderng the energy densty of a lght feld, sne the eletroagnet feld energy per unt volue wll learly depend on the wavelength λ π/ of the lght. Boltzann used a desrpton of the lght radated fro a blabody soure of fnte volue at onstant teperature n ters of a superposton of avty odes to oe up wth the statsts for photons. The lassal treatent of ths proble says that the energy densty (odes per unt volue) nreases rapdly wth nreasng wavelength. For an eulbru body, the energy absorbed has to eual the energy radated, but learly as freueny nreases, the energy of the radated lght should dverge. Boltzann used the detaled balane ondton to show that the partles that ade up lght ust obey Bose-Ensten statsts. That s the eulbru probablty of fndng a photon n a partular avty ode s gven by f 1 e 1 ( ) /T Fro our perspetve (n retrospet), ths should be expeted, beause the uantu treatent of any partle has to follow ether Bose-Ensten statsts or Fer-Dra statsts, and learly lght energy s soethng that we want to be able to nrease arbtrarly. That s, we want to be able to add ode and ore photons nto a gven avty ode. By sung over the nuber of avty odes n a ubal box (usng perod boundary ondtons) we an deterne that the densty of avty odes (a photon densty of states), g ( ) π Usng the energy of a photon, the energy densty per ode s g ( ) π and so the probablty dstrbuton that desrbes the uantu freueny dependent energy densty s π /T u g f 1 e 1 Andre Toaoff, MIT Departent of Chestry, 4/00

2 4- The Quantu Vetor Potental So, for a uantzed feld, the feld wll be desrbed by a photon nuber N, whh represents the nuber of photons n a partular ode (, ) wth freueny n a avty of volue v. For lght of a partular freueny, the energy of the lght wll be N. So, the state of the eletroagnet feld an be wrtten: ϕ N, N, N, EM 1 1, If y atter absorbs a photon fro ode, then the state of y syste would be ϕ N, N 1, N, EM 1, What I want to do s to wrte a uantu ehanal Haltonan that nludes both the atter and the feld, and then use frst order perturbaton theory to tell e about the rates of absorpton and stulated esson. So, I a gong to partton y Haltonan as a su of a ontrbuton fro the atter and the feld: H0 HEM + HM If the atter s desrbed by expressed as produt states: ϕ M, then the total state of the E.M. feld and atter an be ϕ ϕ ϕ EM M And we have egenenerges E E + E EM M

3 4- Now, f I a wathng transtons fro an ntal state to a fnal state, then I an express the ntal and fnal states as: ϕ I ;N 1,N,N,N, atter feld ϕ ; N, N, N,, N ± 1, F 1 N + : esson : absorpton N 1 Where I have abbrevated N N,, the energes of these two states are: E E N + I E E + N ± F So loong at absorpton states as:, we an wrte the Golden Rule Rate for transtons between π w E E V t δ ϕf ϕi Now, let s opare ths to the absorpton rate n ters of the lassal vetor potental: π w A p ( ) ˆ δ v If these are to be the sae, then learly V ( t) ust have part that loos le ( ˆ p) that ats on the atter, but t wll also need another part that ats to lower and rase the photons n the feld. Based on analogy wth our eletr dpole Haltonan, we wrte: 1 * () ( ˆ ˆ ˆ + ˆ ) V t p A p A v

4 4-4 where Aˆ and A ˆ are lowerng/rasng operators for photons n ode. These are operators n the feld states, whereas p reans only an operator n the atter states. So, we an wrte out the atrx eleents of V as 1 ϕf V() t ϕ ˆ ˆ I p,n 1, A,N, v 1 ˆ μ v A ( ) Coparng wth our Golden Rule expresson for absorpton, We see that the atrx eleent w π δ μ ( ) E 0 ( ) E0 E0 A but N 4v 8π So we an wrte π v N Â π a v, Â π a v, where a,a are lowerng, rasng operators. So ˆ π r ( t) r ( t) A ˆ ( a e + a e ) v

5 4-5 So what we have here s a syste where the lght feld loos le an nfnte nuber of haron osllators, one per ode, and the feld rases and lowers the nuber of uanta n the feld whle the oentu operator lowers and rases the atter: H () 1 ( a a ) () H H + H + V t H + V t EM M 0 EM p HM + V r,t V t + A p π r ( t ) r ( t) ( ˆ p) a e a e + v ( ) ( + ) V + V Let s loo at the atrx eleents for absorpton ( > 0) ( ) π,n 1 V,N, N 1 ( ˆ p) a, N v π N ˆ p v π v N μ ˆ and for stulated esson ( < 0)

6 4-6 ( + ) π,n + 1 V, N, N + 1 ( ˆ p) a, N v π N ˆ ˆ + 1 p v π N 1 ˆ v + μ We have spontaneous esson! Even f there are no photons n the ode ( N 0) have transtons downward n the atter whh reates a photon. Let s play ths ba nto the suaton-over-odes expresson for the rates of absorpton/esson by sotrop feld. π ( + ) w d δ dω,n + 1V,N ( π ) π π 8π ( π )( N + 1) μ ( π) nuberdenstyper ode 4 N + 1 μ B N + 1 energydensty per ode averageover polarzaton, you an stll So we have the result we dedued before.

7 4-7 Appendx: Rates of Absorpton and Stulated Esson Here are a ouple of ore detaled dervatons: Verson 1: Let s loo a lttle ore arefully at the rate of absorpton lght soure w w ρe d where, for a onohroat lght soure π w E 0 ˆ μ δ ( ) w ndued by an sotrop, broadband For a broadband sotrop lght soure ρ d represents a nuber densty of eletroagnet odes n a freueny range d ths s the nuber of standng eletroagnet waves n a unt volue. For one freueny we wrote: but ore generally: ˆ r ( t) A A.. 0 e + r ( t) A A ˆ e where the su s over the odes and s the polarzaton oponent. By sung over wave vetors for a box of fxed volue, the nuber densty of odes n a freueny range d radated nto a sold angle dω s and we get ρ E by ntegratng over all Ω 1 dn ddω ( π) 1 ρ d d dω d π ( ) E π 4π nuber densty at

8 4-8 We an now wrte the total transton rate between two dsrete levels sued over all freuenes, dreton, polarzatons π 1 w 0 ( ) ˆ d E δ d Ω μ ( π ) ( ) E0 μ 6π 8π μ We an wrte an energy densty whh s the nuber densty n a range d # of polarzaton oponents energy densty per ode. E0 U( ) π 8π w B U rate of energy flow/ B 4π μ s the Ensten B oeffent for the rate of absorpton U s the energy densty and an also be wrtten n a uantu for, by wrtng t n ters of the nuber of photons N E N U N 8π π 0 ( ) The golden rule rate for absorpton also gves the sae rate for stulated esson. We fnd for two levels and n : w n w n ( ) ( ) sne ( ) ( ) B U B U U U n n n n n n B n B n The absorpton probablty per unt te euals the stulated esson probablty per unt te.

9 4-9 Verson : Let s alulate the rate of transtons ndued by an sotrop broadband soure we ll do t a bt dfferently ths te. The unts are gs. The power transported through a surfae s gven by the Poyntng vetor and depends on. A ˆ E S E B 4π 8π π 0 0 and the energy densty for ths sngle ode wave s the te average of S/. The vetor potental for a sngle ode s r ( t ˆ ) A A e wth. More generally any wave an be expressed as a su over Fourer oponents of the wave vetor: r ( t) e A A ˆ +.. V The fator of V noralzes for the energy densty of the wave whh depends on. The nteraton Haltonan for a sngle partle s: () V t A ρ or for a olleton of partles V t A p () Now, the oentu depends on the poston of partles, and we an express p n ters of an ntegral over the dstrbuton of partles: p d r p r p r pδ r r So f we assue that all partles have the sae ass and harge say eletrons:

10 4-0 V t d r A r t p r () (, ) The rate of transtons ndued by a sngle ode s: π V ( w ) δ( ) A ˆ p( r) And the total transton rate for an sotrop broadband soure s: w ( w ) We an replae the su over odes for a fxed volue wth an ntegral over : 1 V Ω d d d d d ( π) ( π) ( πc ) Ω So for the rate we have: dω snθ dθ dø π w d d p r A δ ( ) Ω ˆ ( π) an be wrtten as The atrx eleent an be evaluated n a anner slar to before: ˆ p r pδ r r ˆ ˆ r,h 1 0 δ r r [ ] ˆ r 1 μ ˆ where μ r For the feld E E0 A 4

11 4-1 π W ˆ d δ E 0 dω μ 4 ( π ) 6π E 0 μ For a broadband soure, the energy densty of the lght 8π/ μ for sotrop I E U 8π 0 4π μ W B U B We an also wrte the ndent energy densty n ters of the uantu energy per photon. For N photons n a sngle ode: N B N π where B has oleular uanttes and no dependene or feld. Note B B rato of S.E. absorpton. The rato of absorpton an be related to the absorpton ross-seton, δ A P total energy absorbed/unt te σ A I total ntensty (energy/unt te/area) P W B U I U( ) σ a B or ore generally, when you have a freueny-dependent absorpton oeffent desrbed by a lneshape funton g ( ) σa B g unts of