Relativistic quantum harmonic oscillator

[lonewolf5999]

The question is as follows:

Suppose that, in a particular oscillator, the angular frequency w is so large that its kinetic energy is comparable to mc². Obtain the relativistic expression for the energy, E_n of the state of quantum number n.

I don't know how to begin solving this question. I know the expressions for the energy states of a quantum harmonic oscillator, and relativistic mechanics. How do I combine the two together? Do I simply append the expression for rest-mass energy, E=m₀c² to the expression for the energy of the quantum harmonic oscillator, E_n= (n + 1/2) (hw/2pi) ?

 

[nickjer]

Not exactly sure how accurate the question wants the answer. But one method you can do is find the correction term in relativistic kinetic energy, O(p⁴). Then treat this as a perturbation and solve for the first order perturbation energy term and add it in.

 

[lonewolf5999]

Hmm. The course I'm doing is an introductory quantum mechanics course, so I haven't covered anything on perturbations. The perturbation theory article on Wikipedia seems a little too complex to be covered in one sitting, so I'll try to look up perturbations elsewhere. But since I haven't covered perturbations, I don't think that this question requires the use of that. Is there any simpler method of arriving at the answer, and if so, how do I begin my solution?

Thanks for the reply.

 

[lonewolf5999]

I apologize for double posting, but is there anybody else who could help me with this problem?