Question

Show that to lowest order in correction terms the relativistic (but noncavariant) Hamiltonian for the one-dimensional harmonic oscillator has the form

H = (1/2m)(p^2 + m^2 w^2 q^2) - (1/8)(p^4/m^3 c^2)

and use first order perturbation theory to calculate the lowest-order relativistic correction to the frequency of the harmonic oscillator. Express your result as a fractional change in frequency.

Answer 1

First note that relativity changes the velocity. So it will change the Kinetic energy part only.

The basic hamiltonian is

When relativity is considered the momentum changes to

The kintetic energy is

Notice that

Or

Or

Or

Hence the hamiltoinan now becomes in the lowest order in correction term

We will apply the perturbation theory to the n^th (any) state of the harmonic oscilltor.

The perturbing hamiltonian is

From the Schrodinger equaion we have

So from the expression of energy correction.

For harmonic oscillator

Now

Expanding the operator in powers and checking the non zero terms

Relativistic correction of the frequency will be then

So the fractional change in frequency