Perturbation Theory (Non-Degenerate)

1. Nov 23, 2012

jhosamelly

If I have V(x)=$\frac{1}{2}$m$\omega^{2}$x$^{2}$ (1+ $\frac{x^{2}}{L^{2}}$)

How do I start to solve for the hamiltonian Ho, the ground state wave function ?? Calculate for the energy of the quantum ground state using first order perturbation theory?

2. Nov 25, 2012

juampeace

$H= H_{0} + H_{p}$

So basically, you have an aditional term, $H_{p} = \frac{1}{2L^{2}}mω^2 x^4$, that perturbates your hamiltonian.
You already know the solution for the harmonic oscillator, $H= H_{0} = \hbarω(n + \frac{1}{2})$, so you just have to find the corrections for the $H_{p}$.

hope i made myself clear ( ;

3. Nov 26, 2012

jhosamelly

so does this mean my hamiltonian would be $H= \hbarω(n + \frac{1}{2}) + \frac{1}{2L^{2}}mω^2 x^4$ ?

4. Nov 26, 2012

andrien

Don't you know the ground state wave function of unperturbed oscillator.you can see them elsewhere and then just evaluate(with normalized eigenfunctions)
<E>=∫ψ0*(Hp0

5. Nov 26, 2012

jhosamelly

I actually dont know the wave function.. That's also my prob... if i only know the wave function I'll be able to solve this.

Last edited: Nov 26, 2012
6. Nov 26, 2012

andrien

7. Nov 26, 2012

jhosamelly

Is this the same for an anharmonic oscillator? That is the problem about.

8. Nov 26, 2012

andrien

No,you use unpertubed harmonic oscillator wave function for calculation.