The minimum energy is obtained by setting to zero the derivative of.
with respect to This gives the optimal value and the minimum energy
5.35 Quartic in Three Dimensions (Tennessee)
The potential is spherically symmetric. In this case we can
write the wave function as a radial part times angular functions. We
assume that the ground state is an and the angular functions are
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 The factor comes
from the angular integrals. It occurs in every integral and drops out when
we take the ratio in (A.3.1). So we just evaluate the part. Again we
choose the trial function to be a Gaussian:
The three integrals in (A.3.1)–(A.3.4) have a slightly different form in three
dimensions:
282 SOLUTIONS