Anharmonic Oscillator

Use variational methods in one dimension to estimate the ground state energy of a particle of mass m in a potential V(x) = Ax⁴.

SOLUTION

Many possible trial functions can be chosen for the variational calculation. Choices such as exp (-α|x|) are poor since they have an undesirable cusp at the origin. Instead, the best choice is a Gaussian:

(1)

where the potential in the problem is

(2)

We evaluate the three integrals in (A.3.1)–(A.3.4).

(3)

(4)

(5)

(6)

We have used (A.3.1) to derive the last expression. Now we find the minimum energy for this choice of trial function by taking the derivative with respect to the variational parameter α. Denote by α₀ the value at this minimum:

(7)

(8)

(9)

This result for E₀ is higher than the exact eigenvalue.