Lecture 5 8.31 Quatum Theory I, Fall 017 106 Lecture 5 (Dec. 6, 017) 5.1 Degeerate Perturbatio Theory Previously, whe discussig perturbatio theory, we restricted ourselves to the case where the uperturbed eergy levels were ot degeerate. We will ow cosider the case where the uperturbed Hamiltoia has degeeracies. Suppose the eigestate 0 is degeerate with k 0, i.e., E,0 = E k,0. We clearly caot use the formula from last class for the first-order correctio to the eigestate, m0 m 0 V 0 = 0 + λ, (5.1) E,0 E m,0 m because the deomiator vaishes at m = k. The solutio is to restrict our attetio to the degeerate subspace ad the diagoalize the perturbatio o this subspace. Withi the subspace of degeerate states of the uperturbed Hamiltoia of eergy E,0 (referred to as the degeerate maifold of E,0 ), we diagoalize V. Withi this subspace, H 0 1 by defiitio (sice every state i this space has the same eergy eigevalue). Thus, P H 0 P 1, where P is the projector oto the degeerate subspace, ad so P H 0 P commutes with P V P. We ca the diagoalize the projected V simultaeously with the projected H 0. I the orthoormal basis that diagoalizes P V P, the off-diagoal matrix elemets of P V P are zero, so we ca retur to usig the formulas of ordiary perturbatio theory. The terms where E,0 E k,0 vaish do ot occur, as the umerator also vaishes. 5.1.1 Liear Stark Effect As a example, cosider a atom i a electric field. I the case of a hydroge atom, we foud that the eergy of the electro i the atom chaged quadratically i the electric field (for small electric field). This was the pheomeo kow as the quadratic Stark effect. Now we will discuss a situatio i which the eergy chages liearly i the applied electric field. Let us choose E = Eẑ. Now, we cosider the = levels of a hydrogeic atom. There are four degeerate states (igorig spi) of the form, l, m, which are, 1, 1,, 1, 0,, 1, 1,, 0, 0 }{{}}{{}. (5.) p Our perturbatio is V = ee z. Cosider V i the subspace of the four = levels. We eed to compute, l, m z, l, m. (5.3) Because [z, J z ] = 0, we have Thus, 0 =, l, m (zj z J z z), l, m = (m m ), l, m z, l, m. (5.4) Note also that uder parity, z z, so, l, m z, l, m = 0, if m m. (5.5) i.e., we will oly fid a o-vaishig matrix elemet for l l. s, l, m z, l, m = 0, (5.6)
Lecture 5 8.31 Quatum Theory I, Fall 017 107 Thus, the oly o-vaishig matrix elemets are, 1, 0 z, 0, 0 =, 0, 0 z, 1, 0. (5.7) We ca compute these etries exactly, because we kow the hydroge atom wavefuctios. Explicitly, we fid 0 3ea 0E 0 0 V = 3ea 0E 0 0 0 0 0 0 0, (5.8) 0 0 0 0 where a 0 is the Bohr radius. This matrix has eigevalues 0, 0, ±3ea 0 E. Sp ecifically, the states, 1, 1 ad, 1, 1 have o eergy shift, while the state (, 0, 0 +, 1, 0 )/ has shift +3ea 0 E ad the state (, 0, 0, 1, 0 )/ has shift 3ea 0 E. This eergy shift is liear i the stregth of the electric field, hece the effect is called the liear Stark effect. I geeral, it may be the case that diagoalizig V i the space of degeerate states does ot split the degeeracy. I such a case, we must go to higher order i perturbatio theory, util we are able to successfully split the degeeracy, after which poit we ca proceed by o-degeerate perturbatio theory. If P V P does ot split the degeeracy, the at the ext order we will be cosiderig operators of the form 5. Time-Depedet Perturbatio Theory V m m V P V P + P P. (5.9) m E 0 E m E 0 E m We ow cosider time-depedet perturbatios to the Hamiltoia. We begi with the Schrödiger equatio, d i ψ d ψ(t) = t H (t), (5.10) with H = H 0 + H 1 (t), (5.11) where H 0 has a kow spectrum ad H 1 is some weak, time-depedet perturbatio. We may wat to compute the probability that the iitial H 0 eigestate i trasitios to the fial H 0 eigestate f i time t. If H 1 = 0, the the solutio to the Schrödiger equatio is of the form ψ(t) = c 0, (5.1) e ie,0 t/ where the c are time-idepedet. If H 1 (t) 0, the there is additioal time depedece comig from the perturbatio; we ca still expad ay state i the basis of uperturbed eergy eigestates, so we ca write the solutios to the Schrödiger equatio i the form ψ(t) = c (t)e ie,0t/ 0, (5.13) where the coefficiets c (t) are ow fuctios of time. We could absorb the time depedece i the expoetial ito the coefficiets c (t), but it is otatioally coveiet ot to do so. Applyig the Schrödiger equatio to Eq. (5.13), (i t H 0 H 1 ) ψ(t) = 0, (5.14)
Lecture 5 8.31 Quatum Theory I, Fall 017 108 we are left with Takig the ier product with we arrive at where ie [i t c (t) H1(t)c (t)]e,0t/ 0 = 0. (5.15) f 0 e ie f t/ 0, (5.16) dc f i = f iω f t 0 H 1 (t) 0 e c (t), (5.17) dt E f,0 E, 0 ωf =. (5.18) We ca recogize this as a cocrete example of workig i the iteractio picture; we have removed the time-depedece due to the uperturbed Hamiltoia, so the oly time-depedece we see remaiig comes from H 1. What we have doe so far is exact. We ow assume that ψ(t = t 0 ) = i ad solve perturbatively i H 1. I the iitial state, c (t = t 0 ) = δ i. (5.19) At first order, we the have which ca be itegrated to yield dc f dt = i f H (t) i e iω fit, (5.0) 0 1 0 i c fi = δ fi ˆ t iω f it 0 e dt f0 H1( t ) i. (5.1) t 0 This is the trasitio amplitude betwee the states i 0 ad f 0 to first order i the perturbatio H 1. 5..1 SHO i a Time-Depedet Electric Field Cosider a simple harmoic oscillator i a time-depedet electric field ɛ(t) = ɛ 0 e t /τ. (5.) The Hamiltoia is p H = + 1 mω x exɛ(t). (5.3) m If our iitial state is the groud state, i = 0, the we wat to compute the trasitio amplitude to the state after a time t. To first order, we fid this probability to be P 0 (t) = Here, we have used E,0 E = 0,0 ω 1 P 0 (t ) = ˆ 1 t dt H 1 (t ) 0 e iωt, 0. (5.4) for the SHO. I the limit t, this becomes ˆ dt H 1 (t ) 0 e iωt, 0. (5.5)
Lecture 5 8.31 Quatum Theory I, Fall 017 109 We ca write the perturbatio i the form ( H 1 (t) = eɛ(t) a + a ), (5.6) mω ad so we fid P 0 (t ) = e mω ˆ d t ɛ ( t ) e iωt a + a ) 0 (, 0. (5.7) Because a 0 = 0 ad a 0 = 1, we see that to this order, P 0 = 0 for > 1. For = 1, we fid e ˆ P 0 1 (t ) = mω = πe ɛ 0 τ e ω τ /4. mω dt e iωt ɛ 0 e t /τ (5.8) If we work to higher orders, we will fid ozero trasitios to higher excitatio states. Note that P 0 1 is small both for ωτ 1 (slow pulse) ad ωτ 1 (fast pulse), ad peaks whe ωτ 1. 5.. Secod-Order Trasitio Amplitude Iteratig the solutio for c (t) to ext order, we fid where ( ) i ˆ t ˆ t (t) = dt c () m t0 t0 Icorporatig the higher-order terms, the trasitio probability is dt e iωmt H (t )e iω mit 1,m H 1,mi (t ), (5.9) H 1,jk (t) = j H 1 (t) k. (5.30) P i = c (1) () ( t) + c (t) +. (5.31)
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