I am now solving a Schrodinger equation with a magnetic field: $i \hbar U_t-(V(x)+\frac{1}{2}B(x)^2)U+\frac{\hbar^{2}}{2}\Delta U-\frac{i \hbar}{2}B(x)\cdot\nabla U=0$ where $V(x)$ is a real and smooth potential and $B(x)$ is the magnetic field which has to satisfy Gauss' law.
I am now trying to solve this equation by firstly change it into the form of the original Schrodinger equation of the form: $i \hbar U_t-(V(x))U+\frac{\hbar^{2}}{2}\Delta U=0$ with the initial condition $U(x,t_0)=U_0(x)$.
My idea is to use the coordinate method to handle the extra term $-\frac{i \hbar}{2}B(x)\cdot\nabla U$. Since it looks like an advection equation with $i \hbar U_t$ , i.e. $i \hbar U_t-\frac{i \hbar}{2}B(x)\cdot\nabla U$, I try to turn this PDE to ODE using change of coordinate method by letting $t= \tau , \xi =x-Mt$.
But I have encountered some problems:
(1) I don't know whether this is an appropriate change of coordinate.
(2) I don't know how to choose M in this case: For a simple case: $u_t+au_x=0$ , I know I should use $t=\tau, \xi =x-at$ , where I guess this transformation comes from the characteristics lines. For an analog to this simple case, I may choose $M=\frac{B(x)}{2}$. But this will induce another problem, which is that I couldn't turn the PDE into ODE as the Chain Rule involve the B(x)and its first order derivative.
Actually I don't know whether it is a good idea to solve my PDE, but I can't come up with other ideas. I have thought about using Separation of variable and the method of using Fourier Transform. But both of these methods seem complicated and it seems that I can't use Separation of variable for the unbounded potential V(x).
If I can't solve the PDE in this way, are there any other methods to solve it?