Time evolution of wave function I'm currently working on this problem. Given the Hamiltonian $\hat{H}$ = $\mu B \hat{L}_y$ and the wave function $\psi(x,y,z) = C\exp(-r/a)x $ at $t=0$, find the state of the system at time $t$.
My idea is to use the time evolution operator, but the wave function doesn't seem to be an eigenfunction of $\hat{L}_y$.
Any help on how to get started would be much appreciated.
 A: First decide if you want to attack the problem in
cartesian ($x,y,z$) or spherical ($r,\theta,\phi)$ coordinates.
I am not sure what to prefer.
Anyway, your initial wave function is
$$\psi_0(\vec{r})=Ce^{-r/a}x=Ce^{-r/a}r\sin\theta\cos\phi$$
The $L_y$ operator in cartesian and in spherical coordinates is
$$\begin{align}
L_y&=-i\hbar\left(
z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}
\right) \\
&=i\hbar\left(
-\cos\phi\frac{\partial}{\partial\theta}
+\cot\theta\sin\phi\frac{\partial}{\partial\phi}
\right)
\end{align}$$
Then my route of attack would be to write $\psi_0(\vec{r})$
as a linear combination of $L_y$-eigenfunctions.
Unfortunately, off-hand we don't know the $L_y$-eigenfunctions.
But we know the $L_z$-eigenfunctions, they are the
spherical harmonics multiplied by an arbitrary function of $r$.
Because of your special given $\psi_0(\vec{r})$
we will only need the ones with $l=1$.
$$\begin{align}
&\text{eigenvalue }L_z=+\hbar:
 &f(r)Y_1^{+1}(\theta,\phi) &\propto &f(r)\sin\theta e^{+i\phi}
 &=& f(r)\frac{x+iy}{r} \\
&\text{eigenvalue }L_z=0:
 &f(r)Y_1^0(\theta,\phi)    &\propto &f(r)\cos\theta
 &=& f(r)\frac{z}{r} \\
&\text{eigenvalue }L_z=-\hbar:
 &f(r)Y_1^{-1}(\theta,\phi) &\propto &f(r)\sin\theta e^{-i\phi}
 &=& f(r)\frac{x-iy}{r}
\end{align}$$
The $L_y$-eigenfunctions will be certain linear combinations
of these. I leave the task of finding them to you.
Then write your $\psi_0(\vec{r})$ as a linear combination of these.
To get the time-dependent solution $\psi(t,\vec{r})$
insert appropriate $e^{i\Omega t}$ factors for each of its terms.
A: Usually in these cases you need to manipulate your wavefunction in order to write it as combinations of spherical harmonics which are eigenfucntions of the angular momentum operator and then you can apply the operator $L_y$. You can find these functions on internet easily. Be quite with the fact that the operator you need to apply is oriented on $y$ so you have to reinterpret the variables of the spherical harmonics (that usually are eigenfucntions of $L_z$).
