Linear Potential II

A particle of mass m moves in one dimension in the right half-space. It has a potential energy V(x) given by

(1)

where F is a positive real constant. Use variational methods to obtain an estimate for the ground state energy. How does the wave function behave in the limits x → 0 or x → ∞?

SOLUTION

The wave function must vanish in either limit that x → (0, ∞). Two acceptable variational trial functions are

(1)

(2)

where the prefactor x ensures that the trial function vanish at the origin. In both cases the variational parameter is α. We give the solution for the first one, although either is acceptable. It turns out that (2) gives a higher estimate for the ground state energy, so (1) is better, since the estimate of the ground-state energy is always higher than the exact value. The ground state energy is obtained by evaluating the three integrals in (A.3.1)–(A.3.4):

(3)

(4)

(5)

(6)

The optimal value of α, called α₀, is obtained by finding the minimum value of E(α):

(7)

(8)

(9)