Return of Combined Potential

A particle of mass m moves in one dimension according to the potential

(i)

where V₀ and are both constants.

a) Show that the wave function must vanish at x = 0 so that a particle on the right of the origin never gets to the left.

b) Use variational methods to estimate the energy of the ground state.

SOLUTION

a) The potential V(x) contains a term which diverges as O(x^-2) as x → 0.

The only way integrals such as ∫ dx ѱ² V(x) are well defined at the origin is if this divergence is canceled by factors in ѱ. In particular, we must have ѱ ~ x at small x. This shows that the wave function must vanish at x = 0. This means that a particle on the right of the origin stays there.

b) The bound state must be in the region x > 0 since only here is the potential V(x) attractive. The trial wave function is

(1)

where the variational parameter is α. We evaluate the three integrals in (A.3.1)–(A.3.4), where the variable s = x/b:

(2)

(3)

(4)

(5)

(6)

The minimum energy is obtained by setting to zero the derivative of E(α) with respect to α. This gives the optimal value α₀ and the minimum energy E(α₀):

(7)

(8)

(9)