DOC PREVIEW
MIT 5 74 - SCHRÖDINGER AND HEISENBERG REPRESENTATIONS

This preview shows page 1-2 out of 7 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 7 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 7 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 7 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

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 = inx ;  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 ()τ0  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 ()τ0 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 


View Full Document
Download SCHRÖDINGER AND HEISENBERG REPRESENTATIONS
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view SCHRÖDINGER AND HEISENBERG REPRESENTATIONS and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view SCHRÖDINGER AND HEISENBERG REPRESENTATIONS 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?