Relativistic Oscillator

Consider a spineless particle in a one-dimensional harmonic oscillator potential:

a) Calculate leading relativistic corrections to the ground state to first order in perturbation theory.

b) Consider an anharmonic classical oscillator with

For what values of α will the leading corrections be the same as in (a)?

SOLUTION

a) The classical Hamiltonian is given by H₀ = p²/2m + V(x), whereas the relativistic Hamiltonian may be expanded as follows:

(1)

The perturbation to the classical Hamiltonian is therefore

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

(2)

where x, p are operators. Defining new operators Q, P,

and noting the commutation relations

we may rewrite (2) as

(3)

Introducing the standard creation and annihilation operators:

(4)

we find that

Using these results, we may express the first-order energy shift ∆₀ as

(5)

where

The expansion of ⟨0|P⁴|0⟩ is simplified by the fact that a|0⟩ = 0, so

(6)

Finally, we obtain

Second solution: Instead of using operator algebra, we can find a wave function a(p) in the momentum representation, where

(6)

The Hamiltonian then is

(7)

The Schrodinger equation for a(p) becomes

(8)

This equation has exactly the same form as the standard oscillator Schrodinger equation:

(9)

We then obtain for the momentum probability distribution for the ground state:

(10)

Therefore

(11)

where A ≡ -1/8m³c². Using the old “differentiate with respect to an innocent parameter method” of simplifying an integral, we may rewrite ∆₀ as

(12)

where we substituted (10) into (11) and let ζ ≡ 1/mωh. Finally,

(13)

as found in the first solution.

b) The first-order energy shift from ax³ would be zero (no diagonal elements in the Q³ matrix). The leading correction would be the second-order shift as defined by the formula

where ∑' means sum over m ≠ n = 0. From (3) and (4), we have

So,

As for any second-order correction to the ground state, it is negative. To make this expression equal to the one in part (a), we require that