Shallow Square Well I

A particle of mass m moving in one dimension has a potential V(x) which is a shallow square well near the origin:

(1)

where V₀ is a positive constant. Derive the eigenvalue equation for the state of lowest energy, which is a bound state (see Figure 1.1).

Figure 1.1

SOLUTION

The ground state energy E must be less than zero and greater than the bottom of the well, 0 > E > -V₀. From the expression

(1)

one can deduce the form for the eigenfunction. Denote the ground state energy E = -h²α²/2m, where α is to be determined. The eigenfunction outside the well (V = 0) has the form exp(-α|x|). Inside the well, define k² = k²₀ – α², where k²₀ = 2mV₀/h². One can show that k² is positive since E + V₀ > 0. Inside the well, the eigenfunction has the form A cos kx, so

(2)

Matching ѱ(x) and its derivative at x = a gives two expressions:

(3)

(4)

Dividing these two equations produces the eigenvalue equation

(5)

The equation given by the rightmost equals sign is an equation for the unknown k. Solving it gives the eigenvalue E.