# Relativistic quantum harmonic oscillator

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, En 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, En= (n + 1/2) (hw/2pi) ?

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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.

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?