Time evolution of spherical harmonics

[misterme09]

Homework Statement

At t=0, a given wavefunction is:

$$\left\langle\theta,\phi|\psi(0)\right\rangle = \frac{\imath}{\sqrt{2}}(Y_{1,1}+Y_{1,-1})$$

Find $$\left\langle\theta,\phi|\psi(t)\right\rangle$$.

Homework Equations

$$\hat{U}(t)\left|\psi(0)\right\rangle = e^{-\imath\hat{H}t/\hbar}\left|\psi(t)\right\rangle$$

$$\hat{H}\left|\ E,l,m\right\rangle = E\left|\ E,l,m\right\rangle$$
$$\hat{L^{2}}\left|\ E,l,m\right\rangle = l(l+1)\hbar^{2}\left|\ E,l,m\right\rangle$$
$$\hat{L_{z}}\left|\ E,l,m\right\rangle = m\hbar\left|\ E,l,m\right\rangle$$

The Attempt at a Solution

I know that you can use the above operator to make time evolution of an energy eigenstate, but I can't figure out what energy to use for the two spherical harmonics in the given state at t=0.

 

[Avodyne]

It depends on what the hamiltonian is.

 

[misterme09]

Oh, okay, I should have realized this.

This is for a rigid rotator with Hamiltonian

$$\hat{H}=\hat{L^{2}}/2I$$

So this means that the energies are both:

$$E =1(1+1)\hbar^{2}/2I = \hbar^{2}/I$$

And

$$\left\langle\theta,\phi|\psi(t)\right\rangle=\frac{i}{\sqrt{2}}e^{-\imath\hbar t/I}(Y_{1,1}+Y_{1,-1})$$

 

[Avodyne]

Yep.