1D Groundstate wavefunction always even for even potential?

[Wavelet]

Hi!
I have calculated various eigenstate wavefunctions for a one-dimensional system of a particle in a potential. The potential is an even function.
All the wavefunctions have become either even or odd functions which I understand why. The ground-state wavefunction has always been even, is this always the case and if so why? If not does anyone know of a system with odd ground-state wavefunction?

Thanks in advance,
Wavelet

 

[Avodyne]

For a hamiltonian of the form H = p^2/2m + V(x) in one dimension, the ground state wave function has no nodes (that is, is never zero). So, if it must be odd or even (as it must if V is even), then it has to be even.

I don't have time now to type in a proof that the ground state wave function has no nodes, but it shouldn't be hard to find one on the web somewhere.

 

[Wavelet]

Thanks!

 

[90mer]

Hi Avodyne,

Good whatever time of day it is where you are.

I'm searching for a rigorous proof of the fact that for any even potential the ground state wave function is always even. Non-relativistic 1 dimensional case proof would be fine for me. Landau Lifschitz, volume #3 the next paragraph after (21.8) equation, produces a proof (if it can be called a proof), but it's not enough for example for V(x)=A*x^4-B*x^2 case.

So could you direct me in this search? A book or web-site?

With best regards,
Anton