Quartic in Three Dimensions

A particle of mass m is bound in three dimensions by the quartic potential V(x) = Ar⁴. Use variational methods to estimate the energy of the ground state.

SOLUTION

The potential V(r) = Ar⁴ is spherically symmetric. In this case we can write the wave function as a radial part R(r) times angular functions. We assume that the ground state is an s-wave, and the angular functions are P₀, which is a constant. So we minimize only the radial part of the wave  function and henceforth ignore angular integrals. In three dimensions the integral in spherical coordinates is d³r = 4πr² dr. The factor 4π comes from the angular integrals. So we just evaluate the r² dr part. Again we choose the trial function to be a Gaussian:

(1)

have a slightly different form in three dimensions:

(2)

(3)

(4)

(5)

Note the form of the kinetic energy integral K, which again is obtained from Rp² R by an integration by parts. Again set the derivative of E(α) equal to zero. This determines the value α₀ which minimizes the energy:

(6)

(7)