p. 21 MIT Department of Chemistry� 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Andrei Tokmakoff SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mathematical formulation of the dynamics of a quantum system is not unique. Ultimately we are interested in observables (probability amplitudes)—we can’t measure a wavefunction. An alternative to propagating the wavefunction in time starts by recognizing that a unitary transformation doesn’t change an inner product. ϕjϕ=ϕjU†Uϕi i For an observable: Aϕi =(ϕj U† )AU ϕ)= ϕU† AUϕ(ϕj i ji Two approaches to transformation: 1) Transform the eigenvectors: ϕi → Uϕi . Leave operators unchanged. 2) Transform the operators: A → U† AU . Leave eigenvectors unchanged. (1) Schrödinger Picture: Everything we have done so far. Operators are stationary. Eigenvectors evolve under Ut,t0 ( ). (2) Heisenberg Picture: Use unitary property of U to transform operators so they evolve in time. The wavefunction is stationary. This is a physically appealing picture, because particles move – there is a time-dependence to position and momentum. Schrödinger Picture We have talked about the time-development of ψ, which is governed by ∂ i ψ=H ψ in differential form, or alternatively ∂t ψ() t = U t, t0 )ψ(t0 ) in an integral form. (p. 22 ∂ATypically for operators: =0 ∂t What about observables? Expectation values: A(t) =ψ() ψ() or... t A t =i ∂Tr (Aρ)∂ ∂ψ ∂ψ ∂Ai A = i ψ A + A ψ + ψ ψ ∂t∂t ∂t ∂t ∂t ∂ =ψ =ψ =iTr A ρ AH HA ψ ∂t ψ−ψψ =Tr A H, ρ])([ =[A, H ]=Tr ([A, H ]ρ) [A, H ] If A is independent of time (as it should be in the Schrödinger picture) and commutes with H, it is referred to as a constant of motion. Heisenberg Picture Through the expression for the expectation value, †A =ψ()t A t t0 U A U ψ()S = ψ() ψ()S t0 =ψAt ψ() H we choose to define the operator in the Heisenberg picture as: † ( 0 )AH(t)=U(t,t0 )A Ut,tSAHt0()= AS ∂Also, since the wavefunction should be time-independent = 0 , we can writeψH∂t ψS(t)=Ut,t( 0 )ψH So, †= U(t,t0)ψS()t =ψS()tψH 0p. 23 In either picture the eigenvalues are preserved: A = aiϕ i ϕ iS S †UAUU† = a U†ϕ i Si ϕ i S = aiAH ϕ i ϕ iH H The time-evolution of the operators in the Heisenberg picture is: † † ∂ U + U† A ∂∂ SU∂ AH = ∂( UA U )=∂ U†A U + UAS∂ t ∂ tS ∂ tS ∂ tt i† i† A ∂∂ = UH A U − UA H U +S S t H i i = HAH − AHHH H − i =[ A, H ] H ∂i AH =[ A, H ] Heisenberg Eqn. of Motion H∂ t †Here HH = U H U . For a time-dependent Hamiltonian, U and H need not commute. Often we want to describe the equations of motion for particles with an arbitrary potential: 2 H = p + V(x)2m For which we have n n− 1 n n− 1p=−∂ V and x= p …using x, p = inx ; x, p = i np ∂ x m p. 24 THE INTERACTION PICTURE When solving problems with time-dependent Hamiltonians, it is often best to partition the Hamiltonian and treat each part in a different representation. Let’s partition ()=H0 +VtHHt() 0 : Treat exactly—can be (but usually isn’t) a function of time. Vt(): Expand perturbatively (more complicated). The time evolution of the exact part of the Hamiltonian is described by ∂ Ut, t )=−iHt Ut, t0 )0 () 0 (0∂t0 ( where iH0 (t −t0 ) Ut, t )=exp + i ∫tt0d Ht 0 ( τ 0 () ⇒ e− for H0 ≠f ()t0 We define a wavefunction in the interaction picture ψI as: ≡ U0 (t, t0 )ψS (t ) ψI (t ) or = U†0ψI ψS ∂Substitute into the T.D.S.E. i ψS = HψS∂tp. 25 ∂ −iU0 (t,t0 ) ()U0 (t,t0 )H t ψ=I ψI∂t ∂ψI∂U0 U0 ())U0 (t, t0` )=−i (H +V t ψ+I ψI∂t ∂t 0 ∂ψI 0 −i −i (HU0 ψI +U0 = +Vt())U0 ψI ∂t H0 ∂ψI∴ i = VI ψI∂t †t 0 0 ()U0 (t, t0 )where: VI ()= U (t, t )V t ψI satisfies the Schrödinger equation with a new Hamiltonian: the interaction picture Hamiltonian is the U0 unitary transformation of Vt(). iNote: Matrix elements in VI = k l = e−ωlktVkl …where k and l are eigenstates of H0.VI We can now define a time-evolution operator in the interaction picture: t = UI (t, t0 )ψI () ψI (t0 ) where UI (t, t )= exp+ − i ∫tt0 dτ VI ()τ0 t =U0 (t, t0 )ψS () ψI (t ) 0 (=Ut, t )U (t, t0 )0 I ψI (t0 ) 0 (=Ut, t )U (t, t0 )0 I ψS (t0 ) ∴ U t,t )=U (t,t )UI (t,t 0 ) Order matters! ( 0 0 0 t()= U()exp+−i∫t0 dτV ()Ut, t0 0 t, t0 Iτ which is defined asp. 26 Ut, t )= U (t, t )+( 0 0 0 ∞ −i nt τ2 n1 ∫t0 dτn ∫τndτn −1 … ∫ dτ1 U0 (t, τ)V (τ )U (τ τ n −1 )…∑ n n 0 n,t0 t0= U0 (ττ1 )V (τ1 )U0 (τ1, t0 )2, ( 0 ). The same positive time-ordering applies. Note that the interactions V(τi) are not in the interaction representation here. Rather we have expanded where we have used the composition property of Ut,t†VI ()= Ut, t )V t 0 (t0 ( 0 ()Ut, t0 ) τ1and collected terms. For transitions between two eigenstates of H0, l and k: The system l H0V H0 k evolves in eigenstates of H0 during the different time periods, with the H0 H0 time-dependent interactions V driving the transitions between these V H0 H0Vstates. The time-ordered exponential accounts for all possible τ 2τ1 intermediate pathways. m Also: † † +it U† (t,t )=U (t,t )U0 (t,t )=exp −+i ∫tt0 dτV () exp − ∫dτH0 ()τ0 I 0 0 t0 I τ or eiH t− t0 )( for H ≠ ft() The expectation value of an operator is: At() =ψ(t)Aψ(t) =ψt0()AU t,t()U† t,t()ψt() =ψt00 0 0()U†U† AU0UIψt()I 0 0()AIψI()t=ψItAI ≡ U† ASU00 Differentiating AIgives:p. 27 ∂ iAI =[H , AI ]∂ t 0 ∂ −ialso, = VtI ()ψI ψI ∂t Notice that the interaction representation is a partition between the Schrödinger and Heisenberg representations. Wavefunctions evolve under VI, while operators evolve under H0. ∂ −iFor H0 =0, V t ()=H ⇒∂A =0; = H Schrödinger ψS ψS∂t ∂t ∂ψ For H0 =H, V t ()=0 ⇒∂A = i [H, A ]; =0 Heisenberg ∂t
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