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.

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