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.