WKB approximation

Problem:

Use the WKB approximation to derive the energy levels of a particle of mass m confined to the one-dimensional potential U(x) = F|x|.

Solution:

Problem:

A particle of mass m moves in a one-dimensional potential energy function U(x) given by
U(x) = ∞         for x < 0,
U(x) = ax        for x > 0.
The energy E = p2/(2m) + ax must be positive, so the particle moves between x = 0 and x = a-1E, the classical turning points.
Use the Bohr-Sommerfeld quantization rule,
∮pdx =  (n + γ)h,
where n = 0, 1. 2, … and 0 < γ < 1 is a constant, to determine the energy levels En.

Solution

Problem:

A particle of a given energy E > 0 is confined by a potential energy function which is given by U(x) = c|x|.
(a)  Describe in detail, quantitatively and qualitatively as best as you can, what an excited state wave function looks like for region E > U(x) and region E < U(x).
(b)  Introduce the parity operator P and prove that each eigenstate with an eigenenergy E is also an eigenstate of the parity operator P.
(c)  With the energy diagram ordered accordingly to the magnitude of the eigenenergy E, show that the ground state (lowest eigenenergy state) is an even parity state, followed by the next excited state being an odd parity state.  All subsequent excited states possess even or odd parity alternatively.

Solution:

Problem:

Let U(x) = ∞ for x < 0, U(x) = ½mω2x2 for x > 0.  Use the WKB approximation to find the energy levels of a particle of mass m in this potential.  Compare the WKB energies with the exact energies for this potential.

Solution: