Perturbation Theory (Non-Degenerate)

[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?

 

[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 ( ;

 

[jhosamelly]

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

 

[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*(H_p)ψ₀

 

[jhosamelly]

I actually don't know the wave function.. That's also my prob... if i only know the wave function I'll be able to solve this.

 

[andrien]

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc5.html

 

[jhosamelly]

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

 

[andrien]

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