# 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.

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