Perturbation effect on harmonic osclillator

[fahmed6]

Im confused about taking out the first and second order perturbation effect on a 1d harmonic oscillator.I get to the integration part but don't know where to go from there.

for example if a term of ax^3 is added to the hamiltonian of the harmonic oscillator this is how i start.if i want to take out the result for the ground state i take the square of the wave function for the ground state, multiply it by ax^3 and integrate it.

But now i don't know what the integration limits should be.

Also if i want to find a general solution for all energy states then how should i put in the hermiye polynials as the are a function of x too and I am integating wrt x too.

Thanx

 

[malawi_glenn]

have you tried x from -inf to +inf ?

And perhaps posting this in Home work forums?

But don't make a new one, a moderator will move this.

 

[meopemuk]

You should be careful with perturbation ax^3 and integration limits. This potential tends to $- \infty$ at large negative x. So the Hamiltonian becomes not positive-definite with various complications that follow.

Eugene.

 

[fahmed6]

what about the hermite polynimials.wat to do with them if i want a general solution.

 

[blechman]

Actually, far and away the best way to do perturbation theory of the harmonic oscillator is to use raising and lowering operators in the energy basis:

$$a|n\rangle=\sqrt{n}|n-1\rangle$$
$$a^\dag|n\rangle=\sqrt{n+1}|n+1\rangle$$
$$x\propto(a+a^\dag)$$

and from the last equation, you can do any polynomial perturbation very easily (don't forget that they don't commute!). This also avoids the topological effects mentioned by meopemuk.

 

[fahmed6]

Thanks blechman.you r right.using raising and lowering operators is far more convenient and it solved my pblm.thanks a lot.

 

[blechman]

Glad I can help. Let me just close by saying for the benefit of anyone else reading this post: NEVER use the position/momentum space basis for a quantum SHO calculation! It is always easier to use the energy basis, or sometimes the coherent state basis, but never the position/momentum space basis. By following this rule of thumb, you will never have to do an integral, and you can always return to position space basis at the very end of the calculation if you are required to do so.