Equation of Motion in Heisenberg Picture

[QuarksAbove]

Homework Statement

A particle of mass m is in a harmonic oscillator potential with spring constant k. An observable quantity is given in the Schrödinger picture by the operator:

$Z = a^{\dagger}a a^{\dagger} a$

a) Determine the equation of motion of the operator in the Heisenberg picture

b) Solve the equation of motion to calculate the form of Z as a function of time.

Homework Equations

$i\hbar \frac{dZ(t)}{dt} = [Z(t),H]$

$Z(t) = U^{\dagger}ZU$

$U = e^{-iHt/\hbar}$

$<Z>_{t} = <\psi(x,0) | Z(t) |\psi(x,0)>$

The Attempt at a Solution

a)

From a textbook, it said the equation of motion for a time-independent operator (Schrödinger picture) in the Heisenberg picture is:
$i\hbar \frac{dZ(t)}{dt} = [Z(t),H]$

Where I am assuming $H = \frac{p^2}{2m} + 1/2kx^2$ because it's a harmonic oscillator.

Is that it for this part of the problem? Just write it down? Seems a bit silly to me.

b)

This part is confusing to me. It asks for me to solve the equation of motion to get Z as a function of time. However, can't I just use

$Z(t) = U^{\dagger}ZU$

$U = e^{-iHt/\hbar}$

$<Z>_{t} = <\psi(x,0) | Z(t) |\psi(x,0)>$

to obtain Z as a function of time? I assume this because I know with the creation and annihilation operators, I can act on $<\psi(x,0)|$ to (if I assume the ground state) eliminate some operators in Z(t).

Okay so this is the method I think I should go with??

writing out the last equation above, I get:

$<Z>_{t} = <\psi(x,0) | U^{\dagger}ZU|\psi(x,0)>$

$<Z>_{t} = <\psi(x,0) | U^{\dagger}a^{\dagger}a a^{\dagger} aU|\psi(x,0)>$

now I am stuck and not convinced i am going about this problem correctly.

 

[George Jones]

Can you write the Hamiltonian in terms of raising and lowering operators?

 

[QuarksAbove]

Ah! yes! I completely forgot about that.

$H = (a^{\dagger}a + 1/2)\hbar \omega$
I know what to do now. I compute the commutator using Z, and the U's will go away because they commute with H.

Thank you!

in the end I got that Z(t) = Z because

dz(t)/dt = 0 which means that z(t) = some constant.