11.6 The Heisenberg Picture*
To begin, lets compute the expectation value of an operatorB.
〈ψ(t)|B|ψ(t)〉 = 〈e−iHt/ ̄hψ(0)|B|e−iHt/ ̄hψ(0)〉
= 〈ψ(0)|eiHt/ ̄hBe−iHt/ ̄h|ψ(0)〉According to our rules, we can multiply operators together beforeusing them. We can then define
the operator that depends on time.
B(t) =eiHt/ ̄hBe−iHt/ ̄hIf we use this operator, we don’t need to do the time development ofthe wavefunctions!
This is called theHeisenberg Picture. In it, theoperators evolve with timeand the wave-
functions remain constant.
The usual Schr ̈odinger picture has the states evolving and the operators constant.
We can now compute the time derivative of an operator.
d
dtB(t) =iH
̄heiHt/ ̄hBe−iHt/ ̄h−eiHt/ ̄hBiH
̄he−iHt/h ̄=i
̄heiHt/ ̄h[H,B]e−iHt/h ̄=i
̄h[H,B(t)]It is governed by the commutator with the Hamiltonian.
As an example, we may look at the HO operatorsAandA†. We have already computed the
commutator.
[H,A] =− ̄hωA
dA
dt=−
i
̄h̄hωA=−iωAWe can integrate this.
A(t) =e−iωtA(0)
Take the Hermitian conjugate.
A†(t) =eiωtA†(0)
We can combine these to get the momentum and position operators inthe Heisenberg picture.
p(t) = p(0) cos(ωt)−mωx(0) sin(ωt)x(t) = x(0) cos(ωt) +p(0)
mωsin(ωt)