Combined Potential

A particle of mass m is confined to x > 0 in one dimension by the potential

(i)

where V₀ and b are constants. Assuming there is a bound state, derive the exact ground state energy.

SOLUTION

Let the dimensionless distance be y = x/b. The kinetic energy has the scale factor E_b = h²/2mb². In terms of these variables we write Schrodinger’s equation as

(1)

Our experience with the hydrogen atom, in one or three dimensions, is that potentials which are combinations of y^-1 and y^-2 are solved by exponentials e^-αy times a polynomial in y. The polynomial is required to prevent the particle from getting too close to the origin where there is a large repulsive potential from the y^-2 term. Since we do not yet know which power of y to use in a polynomial, we try

(2)

where s and α need to be found, while A is a normalization constant. This form is inserted into the Hamiltonian. First we present the second derivative from the kinetic energy and then the entire Hamiltonian:

(3)

(4)

We equate terms of like powers of y:

(5)

(6)

(7)

The last equation defines s. The middle equation defines α once is known. The top equation gives the eigenvalue:

(8)

(9)

(10)