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Perturbation Theory (Non-Degenerate)

by jhosamelly
Tags: nondegenerate, perturbation, theory
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jhosamelly
#1
Nov23-12, 08:21 AM
P: 126
If I have V(x)=[itex]\frac{1}{2}[/itex]m[itex]\omega^{2}[/itex]x[itex]^{2}[/itex] (1+ [itex]\frac{x^{2}}{L^{2}}[/itex])

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?
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juampeace
#2
Nov25-12, 05:34 AM
P: 1
[itex]H= H_{0} + H_{p} [/itex]

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

hope i made myself clear ( ;
jhosamelly
#3
Nov26-12, 12:35 AM
P: 126
so does this mean my hamiltonian would be [itex]H= \hbarω(n + \frac{1}{2}) + \frac{1}{2L^{2}}mω^2 x^4 [/itex] ?

andrien
#4
Nov26-12, 06:58 AM
P: 1,020
Perturbation Theory (Non-Degenerate)

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
jhosamelly
#5
Nov26-12, 07:02 AM
P: 126
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.
andrien
#6
Nov26-12, 07:09 AM
P: 1,020
http://hyperphysics.phy-astr.gsu.edu...tum/hosc5.html
jhosamelly
#7
Nov26-12, 07:11 AM
P: 126
Is this the same for an anharmonic oscillator? That is the problem about.
andrien
#8
Nov26-12, 07:18 AM
P: 1,020
No,you use unpertubed harmonic oscillator wave function for calculation.


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