I feel a bit guilty as I haven’t gone over an example of how one might work out a problem. So here goes:
Suppose our potential function is some sort of energy well: for and elsewhere.
Note: I am too lazy to keep writing so I am going with for now.
So, we have the two Schrödinger equations with being the state vector and being one of the stationary states:
Where are the eigenvalues for
Now apply the potential for and the equations become:
Yes, I know that equation II is a consequence of equation I.
Now we use a fact from partial differential equations: the first equation is really a form of the “diffusion” or “heat” equation; it has been shown that once one takes boundary conditions into account, the equation posses a unique solution. Hence if we find a solution by any means necessary, we don’t have to worry about other solutions being out there.
So attempt a solution of the form where the first factor is a function of alone and the second is of alone.
Now put into the second equation:
Now assume and divide both sides by and do a little algebra to obtain:
are the eigenvalues for the stationary states; assume that these are positive and we obtain:
from our knowledge of elementary differential equations.
Now for we have . Our particle is in our well and we can’t have values below 0; hence . Now
We want zero at so which means .
Now let’s look at the first Schrödinger equation:
This gives the equation:
Note: in partial differential equations, it is customary to note that the left side of the equation is a function of alone and therefore independent of and that the right hand side is a function of alone and therefore independent of ; since these sides are equal they must be independent of both and and therefore constant. But in our case, we already know that . So our equation involving becomes so our differential equation becomes
which has the solution
So our solution is where .
This becomes which, written in rectangular complex coordinates is
Here are some graphs: we use and plot for and . The plot is of the real part of the stationary state vector.