> To construct a potential representation of E and B, you need a vector potential A r, t scalar potential ϕ ( F,t).

Transcription

1 MIT Departent of Chestry p , Sprng 4: Introductory Quantu Mechancs II Instructor: Prof. Andre Tokakoff Interacton of Lght wth Matter We want to derve a Haltonan that we can use to descrbe the nteracton of an electroagnetc feld wth charged partcles: Electrc Dpole Haltonan. Seclasscal: atter treated quantu echancally Feld: classcal Bref outlne of electrodynacs: See nonlecture handout. Also, see Jackson, Classcal Electrodynacs, or Cohen-Tannoudj, et al., Appendx III. > Maxwell s Equatons descrbe electrc and agnetc felds (E, B ). > For Haltonan, we requre a potental. > To construct a potental representaton of E and B, you need a vector potental A r, t scalar potental ϕ ( F,t). ( ) and a > A and ϕ are atheatcal constructs that can be wrtten n varous representatons (gauges). We choose a gauge such that ϕ = (Coulob gauge) whch leads to plane-wave descrpton of E and B : ( A r,t A r,t)+ µ t ( ) = A = Ths wave equaton allows the vector potental to be wrtten as a set of plane waves: ( )= A e (k r ωt * (k r ˆ ωt) A r, t ˆ ) + A e (oscllates as cos ωt) snce A =, k ˆ where ˆ = k ˆ s the polarzaton drecton of the vector potental. ( ω t ) E = A = ω A e ˆ k r + c.c. (oscllates as sn ωt) t k ( r ω t ) B = A = ( k )A e + c.c ˆbk

2 p. 55 so we see that kˆ ˆ nˆ ˆ s the drecton of the electrc feld polarzaton and nˆ s the drecton of the agnetc feld polarzaton. ˆ ˆb We defne E = ωa B = k A ( B E = k ω = c) k E( r, t ) = E sn ˆ (k r ω t ) B( r, t ) = B b ˆ sn (k r ω t )

3 p. 56 Haltonan for radaton feld nteractng wth charged partcle We wll derve a Lagrangan for charged partcle n feld, then use t to deterne classcal Haltonan, then replace classcal operators wth quantu. Start wth Lorentz force on a charged partcle: F= q ( E + v B) () where r s the velocty. In one drecton (x), we have: F = q ( E x + yb zb x z y ) () The generalzed force for the coponents of the force n the x drecton n Lagrangan Mechancs s: U d U F x = + x dt x (3) U s the potental. Usng our relatonshps for E and B n ters of A and ϕ n eq. () and workng t nto the for of eq. (3), we can show that: U= qϕ q r A (4) See CTDL, app. III, p. 49. Confr by pluggng nto (3). Now we can wrte a Lagrangan L = T U = r + q r A q ϕ (5) Now the Haltonan s related to the Lagrangan at: H= p r L = p r r q r A q ϕ (6) p = L = r + qa r r = (p qa ) (7) Now substtutng (7) nto (6), we have:

4 p. 57 H = p (p qa ) (p qa ) q (p qa ) A + qϕ H = ( )] [p qa r,t + qϕ( r,t) Ths s the classcal Haltonan for a partcle of charge q n an electroagnetc feld. So, n the Coulob gauge (ϕ = ), we have the Haltonan for a collecton of partcles n the absence of a feld: H = p () + V r and n the presence of the feld: Expandng: A ( r )) + V ( r )) H = ( (p q H H q (p A + A ) + q = p A Generally the last ter s consdered sall energy of partcles hgh relatve to apltude of potental so we have: H = H + Vt () Vt ()= q (p A + A p ) Now we are n a poston to substtute the quantu echancal oentu for the classcal: p = Matter: Quantu; Feld (A): Classcal ()= q ( A + A ) Vt Notce A = ( A)+ A (chan rule), but we are n the Coulob gauge ( A = ), so A = A

5 p. 58 Vt ()= q A = q A p For a sngle charge partcle our nteracton Haltonan s Vt ()= q. A p Usng our plane-wave descrpton of the vector potental: Vt A ˆ (k r ωt ()= q ) p e + c.c. Electrc Dpole Approxaton If the wavelength of the feld s uch larger than the olecular denson (λ )(k ), then k r e. If r s the center of ass of a olecule: e k r = e k r e k ( r r ) = e k r [ + k ( r r ) + ] For UV, vsble, nfrared not X-ray k r r <<, set r = e k r. We do retan hgher-order ters to descrbe hgher order nteractons wth the feld. Retan second ter for quadrupole transton oent: charge dstrbuton nteractng wth gradent of electrc feld and agnetc dpole.

6 p. 59 Electrc Dpole Haltonan Vt ()= q [A ˆ p e ωt + c.c.] Usng A = E ω V t p ˆ e ω p e + ω ω ()= qe t ˆ t ()= qe V t ( p ˆ )sn ωt Electrc Dpole Haltonan ω = q (E(t) p ) ω or for a collecton of charge partcles (olecules): V t E ˆ ()= q ( p ) sn ω ωt Haronc Perturbaton: Matrx Eleents For a perturbaton V()= t V snωt the rate of transtons nduced by feld s w k = π V k [δ(e k E ω) + δ(e k E + ω)] Let s look at the atrx eleents for the E.D.H. qe V k = kv = k ˆ p ω Evaluate the bracket k p usng [r, H ]= p k p = k r H H r = ω k k r V k = qe ω k ω k ˆ r

7 p. 6 or for a collecton of partcles ω V k = E k k ˆ q r ω = E ω ω k k µ ˆ dpole oent So we can wrte the electrc dpole Haltonan as V t ()= µ E(t) So the rate of transtons between quantu states nduced by the electrc feld s π ω k w k = E k µ ˆ [δ(e k E ω)+(e k E + ω)] ω π E k µ ˆ [ δω ( k ω)+δ( ω k +ω)]