A Guide to Physics Problems. Part 2. Thermodynamics, Statistical Physics, and Quantum Mechanics

5.44 Relativistic Oscillator (MIT, Moscow

Phys-Tech, Stony Brook (a))

a) The classical Hamiltonian is given by whereas the

relativistic Hamiltonian may be expanded as follows:

The perturbation to the classical Hamiltonian is therefore

First solution: For the nonrelativistic quantum harmonic oscillator, we have

where are operators.Defining newoperatorsQ, P,

and noting the commutation relations

we may rewrite (S.5.44.2) as

QUANTUM MECHANICS 293