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 Ut,( t) = Ht()Ut, ( t) ∂t 0 0 0 0 0 where

t  i  − iH0 (t −t0 ) Ut,0 ( t0 ) = exp+ dτ Ht 0 () ⇒ e forH 0 ≠ f ()t  ∫t   0 

We define a wavefunction in the interaction picture ψ I as:

ψS (t ) ≡ U0 (t,t0 ) ψ I (t )

† 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 = UI (t,t0 ) ψ I (t0 )

− i t  where UI (t, t 0 ) = exp+ dτ VI ()τ  ∫t   0 

ψS ()t = U0 (t,t0 ) ψ I (t )

= Ut,0 ( t0) U I (t,t0 ) ψI (t0 )

= Ut,0 ( t0) U I (t,t0 ) ψS (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