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This is supposed to be a question for high school seniors who've had instruction in introductory concepts of special relativity and non-relativistic QM. According to my TA, he isn't too certain if it can be done within these confines but nonetheless I've attempted the problem... Please let me know if I'm doing it wrong (quite sure I am); or if I'm on the right track, how to reducerin terms ofn. Thanks in advance!

A particle of mass m vibrates as a harmonic oscillator with angular frequency [tex]\omega[/tex]. For this harmonic oscillator, the general expression for the energy [tex]E_n[/tex] of the state of quantum numbernis

[tex]E_n = (n - \frac{1}{2})\hbar\omega[/tex]

Suppose that the angular frequency [tex]\omega[/tex] is so large that the kinetic energy of the particle is comparable to [tex]mc^2[/tex]. Obtain the relativistic expression for the energy [tex]E_n[/tex] of the state of quantum numbern.

Relevant equations and the attempt at a solution

Considering the relativistic kinetic energy [tex]E_k[/tex] of the particle,

[tex]E_k = (mc^2)(\gamma - 1)[/tex]

If [tex]E_k \approx mc^2[/tex]

then [tex]\gamma - 1 \approx 1[/tex]

[tex]\therefore \gamma \approx 2[/tex]

Expressing angular frequency in terms of linear velocity in [tex]\gamma[/tex]

[tex]\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}[/tex]

[tex]\omega = \frac{c}{\sqrt{2}r}[/tex]

whereris the radius of the oscillation

[tex]\therefore E_n = \frac{(n - \frac{1}{2}) \hbar c }{\sqrt{2}r}[/tex]

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# Homework Help: Energy of harmonic oscillator

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