I'm having a problem understanding solutions of differential equation in QM:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\psi''(z)+\frac{p}{z}\psi(z)+k^2\psi(z)=0[/tex] (1)

I usualy use Fourbenious method, and in this case I get a 3 coefficients recursion relation which is really messy.

So I do it like this:

for really large z second term expires, and I have a simple H.O.equation:

[tex]\psi''(z)+k^2\psi(z)=0[/tex] (2)

with a solution:

[tex]\psi(z)=Aexp(ikz)[/tex] (3)

Now, the solution of (1) must be:

[tex]\psi(z)=Aexp(ikz)*f(z)[/tex] (4)

Where f(z) is some function of z. Supstituting (4) in (1) I get the equation for f(z):

[tex]f''(z)+2kif'(z)+\frac{p}{z}f(z)=0[/tex]

If I use Fourbenious method here, I get a nice recursion relation and the solution is something like this (hypergeometric function):

[tex]f(z)=Cz(1-\frac{2ki+p}{2!}z+\frac{(2ki+p)(4ki+p)}{2!3!}z^2-....(-1)^{n+1}\frac{(2ki+p)(4ki+p)...(2kin+p)}{n!(n+1)!}z^n)[/tex]

There is also a second solution, but no need for me to write it down becoase the same problem arrises.

I checked with mathematica 5.0 and the solution is OK. Is this OK? I mean, every usual problem in QM, HO, hydrogen atom, square well... have real space part and imaginary time part. But my space part has an imaginery and real part. Have I done something wrong? Or is it a coincidence that these standard problems have just real part of space part of wave function.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Interpretation of solution in QM

**Physics Forums | Science Articles, Homework Help, Discussion**