- #1
max_jammer
- 6
- 0
Hello.
I have this problem at hand:
A quantum mechanical system has a hamilton operator [itex]\hat{H}[/itex] and another, time independent operator [itex]\hat{A}_{0}[/itex].
Construct a time dependent operator [itex]\hat{A}(t)[/itex] so that:
<ψ(t)|[itex]\hat{A}_{0}[/itex]|ψ(t)> = <ψ(0)|[itex]\hat{A}(t)[/itex]|ψ(0)>
for all states ψ(t) that develop in time according to the SE.
In the derivation of the Schrödinger equation, we use the unitary operator [itex]\hat{U}(t)[/itex] to calculate the effect of time on the state ψ(0)...
So
ψ(t) = [itex]\hat{U}(t)[/itex] ψ(0) = exp(-i/[itex]\hbar \hat{H}[/itex] t) ψ(0).
In other words:
<ψ(t)|[itex]\hat{A}_{0}[/itex]|ψ(t)> = <[itex]\hat{U}(t)[/itex] ψ(0)|[itex]\hat{A}_{0}[/itex]|[itex]\hat{U}(t)[/itex] ψ(0)>
=<ψ(0) |[itex]\hat{U}(t)^{+} \hat{A}_{0} \hat{U}(t)[/itex] | ψ(0)>.
so my "solution" is that
[itex]\hat{A}(t)[/itex] = [itex]\hat{U}(t)^{+} \hat{A}_{0} \hat{U}(t)[/itex]...
But this is way too simple to be correct...
So what am I missing?
Thanks
/D
I have this problem at hand:
Homework Statement
A quantum mechanical system has a hamilton operator [itex]\hat{H}[/itex] and another, time independent operator [itex]\hat{A}_{0}[/itex].
Construct a time dependent operator [itex]\hat{A}(t)[/itex] so that:
<ψ(t)|[itex]\hat{A}_{0}[/itex]|ψ(t)> = <ψ(0)|[itex]\hat{A}(t)[/itex]|ψ(0)>
for all states ψ(t) that develop in time according to the SE.
The Attempt at a Solution
In the derivation of the Schrödinger equation, we use the unitary operator [itex]\hat{U}(t)[/itex] to calculate the effect of time on the state ψ(0)...
So
ψ(t) = [itex]\hat{U}(t)[/itex] ψ(0) = exp(-i/[itex]\hbar \hat{H}[/itex] t) ψ(0).
In other words:
<ψ(t)|[itex]\hat{A}_{0}[/itex]|ψ(t)> = <[itex]\hat{U}(t)[/itex] ψ(0)|[itex]\hat{A}_{0}[/itex]|[itex]\hat{U}(t)[/itex] ψ(0)>
=<ψ(0) |[itex]\hat{U}(t)^{+} \hat{A}_{0} \hat{U}(t)[/itex] | ψ(0)>.
so my "solution" is that
[itex]\hat{A}(t)[/itex] = [itex]\hat{U}(t)^{+} \hat{A}_{0} \hat{U}(t)[/itex]...
But this is way too simple to be correct...
So what am I missing?
Thanks
/D