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 ϕ i = ϕ j U †U ϕ i
For an observable:
(
)
ϕ j A ϕ i = ϕ j U† A(U ϕ i ) = ϕ j U † AU ϕ i
Two approaches to transformation: 1) Transform the eigenvectors: ϕ i → U ϕ i . 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 U(t , 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
∂t
in differential form, or alternatively
ψ ( t ) = U ( t, t 0 ) ψ ( t 0 )
in an integral form.
p. 22
Typically for operators:
∂A
=0
∂t
What about observables? Expectation values:
A(t) = ψ ( t ) A ψ ( t )
or...
∂
Tr ( Aρ )
∂t
 ∂ 
= i Tr  A ρ 
 ∂t 


∂ψ
∂ψ
∂A
+
Aψ + ψ
ψ 
 ψA
∂t
∂t
∂t


= ψ AH ψ − ψ HA ψ
=i
= ψ [ A, H ] ψ
= Tr ( A [ H, ρ])
∂
i
A =i
∂t
= 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 ψ(t)
= ψ A(t) ψ
S
= ψ ( t 0 ) U† A U ψ ( t 0 )
S
H
we choose to define the operator in the Heisenberg picture as:
AH (t ) = U† (t, t 0 )ASU (t, t 0 )
AH (t 0 ) = AS
Also, since the wavefunction should be time-independent
∂
ψ H = 0 , we can write
∂t
ψ S (t ) = U(t, t 0 )ψ H
So,
ψ H = U † (t,t 0 ) ψ S (t ) = ψ S (t 0 )
p. 23
In either picture the eigenvalues are preserved:
A ϕi
S
= a i ϕi
U AUU ϕi
S
= a i U ϕi
A H ϕi
H
= a i ϕi
†
†
S
†
S
H
The time-evolution of the operators in the Heisenberg picture is:
∂A H ∂
∂U †
∂U
∂AS
= ( U † AS U ) =
+ U†
AS U + U † AS
U
∂t ∂t
∂t
∂t
∂t
i
i
 ∂A 
= U † H AS U − U † AS H U + 

 ∂t  H
i
i
= HH AH − AH HH
=
i
−i
[ A, H ]H
∂
A H = [ A, H ]H
∂t
Heisenberg Eqn. of Motion
Here H H = 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:
p2
H=
+ V(x)
2m
For which we have
p=−
∂V
p
and x =
∂x
m
…using  x n , p  = i nx n−1 ;  x, p n  = i np n−1
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
H (t ) = H0 + V (t )
H0 : Treat exactly—can be (but usually isn’t) a function of time.
V (t ) : Expand perturbatively (more complicated).
The time evolution of the exact part of the Hamiltonian is described by
∂
−i
U 0 ( t, t 0 ) = H 0 ( t ) U 0 ( t, t 0 )
∂t
where
i
U 0 ( t, t 0 ) = exp + 

∫
t
t0

dτ H 0 ( t )  ⇒

e−iH ( t −t )
0
We define a wavefunction in the interaction picture ψ I as:
ψS ( t ) ≡ U 0 ( t, t 0 ) ψ I ( t )
ψ I = U †0 ψ S
or
Substitute into the T.D.S.E.
i
∂
ψS = HψS
∂t
0
for H 0 ≠ f ( t )
p. 25
∂
−i
U 0 ( t, t 0 ) ψ I = H ( t ) U 0 ( t, t 0 ) ψ I
∂t
∂ ψI
∂U 0
−i
ψI + U0
= ( H 0 + V ( t ) ) U 0 ( t, t 0` ) ψ I
∂t
∂t
−i
H0 U0 ψI + U0
∂ ψI
−i
=
H0 + V ( t ) U0 ψI
∂t
(
∴ i
)
∂ ψI
= VI ψ I
∂t
where: VI ( t ) = U †0 ( t, t 0 ) V ( t ) U 0 ( t, t 0 )
ψ I satisfies the Schrödinger equation with a new Hamiltonian: the interaction picture
Hamiltonian is the U0 unitary transformation of V (t ) .
Note: Matrix elements in VI
=
k VI l
= e− iωlk t Vkl …where k and l are eigenstates of H0.
We can now define a time-evolution operator in the interaction picture:
ψ I ( t ) = U I ( t, t 0 ) ψ I ( t 0 )
 −i t

where U I ( t, t 0 ) = exp +  ∫ dτ VI ( τ ) 
t0


ψS ( t ) = U 0 ( t, t 0 ) ψ I ( t )
= U 0 ( t, t 0 ) U I ( t, t 0 ) ψ I ( t 0 )
= U 0 ( t, t 0 ) U I ( t, t 0 ) ψ S ( t 0 )
∴ U ( t, t 0 ) = U 0 ( t, t 0 ) U I ( t, t 0 )
−i t
U(t , t0 ) = U0 (t , t0 )exp +  ∫t dτ VI (τ )
0


which is defined as
Order matters!
p. 26
U (t, t 0 ) = U 0 ( t, t 0 ) +
n
τn
τ2
 −i  t
∑
  ∫t 0 dτn ∫t 0 dτn −1 … ∫t0 dτ1 U 0 ( t, τn ) V ( τn ) U 0 ( τn , τn −1 ) …

n =1 
U 0 ( τ2 , τ1 ) V ( τ1 ) U 0 ( τ1 , t 0 )
∞
where we have used the composition property of U(t , 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 ) = U †0 ( t, t 0 ) V ( t ) U 0 ( t, t 0 )
and collected terms.
For transitions between two eigenstates of H0, l and k: The system
l
H0
evolves in eigenstates of H0 during the different time periods, with the
time-dependent interactions V driving the transitions between these
states. The time-ordered exponential accounts for all possible
intermediate pathways.
τ1
VH
H0
V
k
0
H0
H0 H0
τ1
m
V
τ
2
Also:
 +i t

 +i t

U † ( t, t 0 ) = U †I ( t, t 0 ) U †0 ( t, t 0 ) = exp −  ∫ dτ VI ( τ )  exp −  ∫ dτ H 0 ( τ ) 
t0
t0




or e
The expectation value of an operator is:
A(t ) = ψ (t ) A ψ (t )
= ψ (t 0 )U † (t,t0 )A U (t,t 0 )ψ (t0 )
= ψ (t 0 )UI†U0† AU0 UI ψ (t 0 )
= ψ I (t ) AI ψ I ( t )
AI ≡ U0† AS U0
Differentiating AI gives:
iH (t− t0 )
for H ≠ f (t )
p. 27
∂
i
AI = [ H0 , AI ]
∂t
also,
∂
−i
ψ I = VI ( t ) ψ I
∂t
Notice that the interaction representation is a partition between the Schrödinger and Heisenberg
representations. Wavefunctions evolve under VI , while operators evolve under H0.
For H 0 = 0, V ( t ) = H ⇒
For H 0 = H, V ( t ) = 0 ⇒
∂A
∂
−i
= 0;
ψS = H ψS
∂t
∂t
∂A i
∂ψ
= [ H, A ] ;
=0
∂t
∂t
Schrödinger
Heisenberg