A general math theory on the quantum harmonic oscillator?

[tim_lou]

looking at the quantum mechanical harmonic oscillator, one has the differential equation in the form:
$$\frac{d^2\psi}{du^2}+(\alpha-u^2)\psi=0$$

when a person who doesn't know any physics sees the equation, he will try a serial solution for psi, and he will find a solution with some recursive relations on the coefficients.

letting psi be
$$\sum_i a_i y^i$$

one gets the recursive relations:
$$a_{i+2}=\frac{\alpha a_i-a_{i-2}}{(i+2)(i+1)}$$

but this gets no where near the usual solutions we want in physics.

When the solution method is presented in a physics book, one sees the tag
"trying a solution in the form"
$$f(x)e^{-y^2}$$
and get another second order differential equations. At first glance, it seems that nothing is gained using this substitution, but it turns out that the series must terminated if psi is to vanishes at both infinities and we get a nice family of solutions (in closed forms too).

What exactly is the motivation behind this? is this just a elegant guess? or is there some general mathematical theory dealing with equations like these (with vanishing psi at both infinities)?

I just would like to get more information on this matter... and perhaps someone can explain how the first method fail and if anything can be done to patch things up so that the first method yield the same result with the substitution method.

 

[smallphi]

After the initial failure with the series they try to evaluate the behavior of the solution for u-> +/- infinity. For such values of u, you can ignore alpha in the differential equation. If you make Mathematica solve the differential equation without alpha (try it), you will get a general solution with two integration constants. The first constant multiplies a Bessel function that goes to infinity when u-> +/- infinity. That part of the solution blows up at infinity and is dropped since it is not integrable. The second integration constant multiplies a Hermite polinomial times Exp(-u^2 / 2). That part of the solution goes to zero for u-> +/- infinity and is physically acceptable.

The Exp(-u^2 / 2) is an infinite series in u giving the general behavior of the solution at infinite u. There is always hope that isolating the general behavior, you will get simpler series for the remainder which is exactly what happens in this case - that's why the series left is finite.

 

[Hurkyl]

A more lowbrow observation is "degree" counting: because of the x^2 term on psi, you need the second derivative of psi to have degree two greater than psi. The simplest way I can think of to do that is to have psi look something like e^f(x), where f is a quadratic polynomial.

Incidentally, you can split the equation into the sum of two equations:
one has alpha but not x^2
one has x^2 but not alpha

 

[tim_lou]

thanks for the help. Indeed the substitution of e^x2 gets rid of the (x^2-alpha) term. Plus, canceling that term makes the coefficients of each derivatives have the same order, giving a nicer recursion. The motivation becomes clear now.