Square potentials, bound states

Problem:

Consider the square potential well shown in the figure below.

image

(a)  Find the most general solution Φ(x) of the eigenvalue equation HΦ(x) = EΦ(x), (E < 0), in regions 1, 2, and 3 and apply boundary conditions.
(b)  Solve the equation that results from part (a) graphically, and find the conditions under which even and odd solutions exist.

Solution:

Problem:

The one-dimensional square well shown in the figure rises to infinity at x = 0 and has a range “a” and a depth U0.  Find the bound state solutions.
image

  Solution:

Problem:

Consider a particle with mass m in a potential well in one dimension as shown.  The potential energy U is zero between x = 0 and x = a, U0 between x = a and x = 2a, and infinite otherwise.
(a)  Assume E = U0.  Find the smallest value of U0 for which that state exists.
(b)  Sketch the wave function and the probability density in the whole well for this state.

image

Solution: