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Schrödinger and Heisenberg Representations

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Schrödinger and Heisenberg Representations 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. † ϕ j ϕi = ϕ j U Uϕi For an observable: † † ϕ j Aϕi = (ϕ j U )AU( ϕi )= ϕ j U AUϕi Two approaches to transformation: 1) Transform the eigenvectors: ϕi → U ϕi . 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 ∂A Typically for operators: = 0 ∂t What about observables? Expectation values: A(t) =ψ()t A ψ()t or... ∂ ∂ ∂ψ ∂ψ ∂A = i Tr(Aρ ) i A = i ψ A + A ψ + ψ ψ ∂t ∂t ∂t ∂t ∂t ∂ = iTr A ρ =ψAH ψ−ψHA ψ ∂t =ψ[ A, H ] ψ = Tr (A [ H, ρ]) = [A, H ] = Tr ([A, H ]ρ) 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 ψ()t = ψ ()t0 U AUψ ()t0 S S =ψAt() ψ H we choose to define the operator in the Heisenberg picture as: † AH (t) = U (t,t0 )ASUt( ,t0 ) AH ()t0 = AS ∂ Also, since the wavefunction should be time-independent ψ = 0 , we can write ∂t H ψ S(t) = Ut( ,t0 )ψ H So, † ψ H = U (t,t0)ψ S (t) = ψ S (t0 ) p. 23 In either picture the eigenvalues are preserved: A ϕ = a ϕ i S i i S UAUU† † ϕ = aU † ϕ i S i i S A ϕ = a ϕ H i H i i H The time-evolution of the operators in the Heisenberg picture is: † ∂A H ∂ † ∂U † ∂U † ∂A S = ( UAS U) = AS U+ UAS + U U ∂t ∂t ∂t ∂t ∂t i † i † ∂A = UHA S U− UAS HU + ∂t H i i = HA− AH H H H H −i = [ A, H ] H ∂ i AH = [ A, H ] Heisenberg Eqn. of Motion ∂t H † Here HH = UHU . 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: p 2 H = + V(x) 2m For which we have ∂V p p =− and x = …using x,n p = inxn− 1 ; x,p n = i npn− 1 ∂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 Ht( ) = H0 + Vt( ) H0 : 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 ∂ −i Utt( ,) = Ht()Utt, ( ) ∂t 0 0 0 0 0 where t i − iH0 (t −t0 ) Utt0 ( 0, ) = exp+ dτ Ht 0 () ⇒ e forH 0 ≠ f ()t ∫t 0 We define a wavefunction in the interaction picture ψ I as: ψS (t ) t≡ U0 ( ,0 ) ψ I (tt ) † or ψI = U0 ψS ∂ Substitute into the T.D.S.E. i ψ = H ψ ∂t S S p. 25 ∂ −i U (t,t ) ψ= H() t U (t,t ) ψ ∂t 0 0 I 0 0 I ∂U0 ∂ψI −i ψ+I U0 = (H 0 + V()t) U0 (t,t0`) ψ I ∂t ∂t −i ∂ψ −i HU ψ + U I = H + Vt() U ψ 0 0 I 0 ∂t ( 0 ) 0 I ∂ψ ∴ i I = V ψ ∂t I I † where: VI ()t = U 0(t, t 0 ) V ()t U0 (t, t0 ) ψ I satisfies the Schrödinger equation with a new Hamiltonian: the interaction picture Hamiltonian is the U0 unitary transformation of Vt( ). −ωi lk t Note: Matrix elements in VI = k VI l = e Vkl …where k and l are eigenstates of H0. We can now define a time-evolution operator in the interaction picture: ψI ()t t = UI ( ,0 ) ψ I (tt0 ) − i t where UI (t, t 0 ) = exp+ dτ VI ()τ ∫t 0 ψS ()t t = U0 ( ,0 ) ψ I (tt ) = Utt0 ( 0,) Ut I ( ,0 ) tψI (t0 ) = Utt0 ( 0,) Ut I ( ,0 ) tψS (t0 ) ∴ U( t,t0 ) = U 0(t,t 0 ) UI (t,t0 ) Order matters! −i t Ut(), t = U ()t, t exp dτ V ()τ 0 0 0 + t I ∫ 0 which is defined as p. 26 Ut,( t0 ) = U 0 (t,t0 ) + ∞ n −i t τn τ2 dτn dτn −1 … dτ1 U0 (t,τn )V (τ n ) U 0(τ n ,τn −1 )… ∑ ∫t ∫t ∫t n1= 0 0 0 U0 (ττ2 , 1 ) V (τ1 ) U0 (τ1,t0 ) where we have used the composition property of Ut( ,t0 ). The same positive time-ordering applies. Note that the interactions V(τi) are not in the interaction representation here. Rather we have expanded † VI ()t = Ut,0 ( t0 ) V()t Ut, 0 ( t0 ) τ1 and collected terms. l V k For transitions between two eigenstates of H0, l and k: The system H0 H0 evolves in eigenstates of H0 during the different time periods, with the H0 H0 time-dependent interactions V driving the transitions between these H0 H0 states. The time-ordered exponential accounts for all possible V V τ1 τ 2 intermediate pathways. m Also: t t † † † +i +i U (t,t0 ) = U I(t,t 0 ) U0 (t,t0 ) = exp dτ VI ()τ exp dτ H0 ()τ − ∫t − ∫t 0 0 iH t− t or e ( 0 ) for H ≠ ft( ) The expectation value of an operator is: At() = ψ (t)Aψ (t) † = ψ ()t0 U ()t,t0 AU() t,t0 ψ ()t0 † † = ψ ()t0 UI U0 AU0UI ψ ()t0 = ψ I ()t AI ψ I()t † AI ≡ U0 AS U0 Differentiating AI gives: p. 27 ∂ i AI = [H0 ,AI ] ∂t ∂ −i also, ψ = Vt()ψ ∂t I I I Notice that the interaction representation is a partition between the Schrödinger and Heisenberg representations. Wavefunctions evolve under VI , while operators evolve under H0. ∂A ∂ −i For H = 0, V ()t = H ⇒ = 0; ψ = H ψ Schrödinger 0 ∂t ∂t S S ∂A i ∂ψ For H0 = H, V ()t = 0 ⇒ = [H, A ]; = 0 Heisenberg ∂t ∂t .
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