Question

A particle of mass m is located in a unidimensional potential field U (x) whose shape is shown in Fig. 6.2, where U (0) = ∞.

Find:

(a) the equation defining the possible values of energy of the particle in the region E < U₀; reduce that equation to the form

where   Solving this equation by graphical means, demonstrate that the possible values of energy of the particle form a discontinuous spectrum;

(b) the minimum value of the quantity l²U₀ at which the first energy level appears in the region E < U₀. At what minimum value of l² U_o does the nth level appear?

Answer 1

(a) Starting from the Schrodinger equation in the regions l & II

Plotting the left and right sides of this equation we can find the points at which the straight lines cross the sine curve. The roots of the equation corresponding to the eigen values of energy E and found from the inter section points (kl)_i, for which tan (kl)_i < 0 (i.e. 2^nd & 4^th and other even quadrants). It is seen that bound states do not always exist. For the first bound state to appear (refer to the line (b) above)

(k l)_1,min=π/2

as the condition for the appearance of the first bound state. The second bound state will appear when kl is in the fourth quadrant The magnitude of the slope of the straight line must then be less than

For n bound states, it is easy to convince one self that the slope of the appropriate straight line (upper or lower) must be less than

Do not forget to note that for large n both + and - signs in the Eq. (6) contribute to solutions.