Thermodynamics, Statistical Physics, and Quantum Mechanics

256 SOLUTIONS

We take the limit that of the function on the left, and this must

equal – since we assumed that the potential vanishes at infinity. Thus,

we find that

The energy is negative, which signifies a bound state. The potential

can be deduced from (S.5.12.1) since everything else in this expression is

known:

This potential energy has a bound state which can be found analytically,

and the eigenfunction is the function given at the beginning of the problem.

5.13 Combined Potential (Tennessee)

Let the dimensionless distance be The kinetic energy has the scale
factor In terms of thesevariables we write Schrödinger’s
equation as

Our experience with the hydrogen atom, in one or three dimensions, is that

potentials which are combinations of and are solved by exponentials

times a polynomial in The polynomial is required to prevent the

particle from getting too close to the origin where there is a large repulsive

potential from the term. Since we do not yet know which power of

to use in a polynomial, we try

where and need to be found, whileA is a normalization constant.
Thisform is inserted into the Hamiltonian. First we present the second
derivative from the kinetic energy and then the entire Hamiltonian: