The WKB approximation

The program n4.txt is written in QuickBasic. It uses the WKB approximation to find the energies of the bound states in the one-dimensional potential shown below. The numerical integration uses Simpson's rule. For comparison, the program n3a.txt uses the same potential and integrates the time-independent Schroedinger equation numerically to find the energies of the bound states.

Explanation:

The program n4.txt uses the WKB approximation to find the energies of the bound states of an electron in a potential V(x)=½kx²-10eV for |x|£ 5Å, and V(x)=0 for |x|>5Å, with k=(20eV)/(5Å)².

The WKB approximation requires that

.

For the chosen potential, the integral can be evaluated analytically. Let E'=E+10eV. Then

4,
x_max=2E'/k,
,
.

These energies are the eigenenergies of the harmonic oscillator with V(x)=½kx²-10eV for all x. The WKB approximation for bound states does not distinguish between this potential and the harmonic oscillator potential V(x)=½kx²-10eV for all x when finding eigenenergies below 0eV.

The program n4.txt evaluates the integral numerically using Simpson's rule.

It assumes that V(x)=V₀ fnv(x), where fnv(x)_min=-1.

where

.

The program first calculates the integral s for a very small value of E/V₀. It then calculates the number of bound states it will look for by truncating s/p -0.5+1. The program guesses that the value of E/V₀ for the lowest bound state is just slightly larger than -1. It changes the energy in small steps until s-0.5p is minimized. This yields the energy of the ground state, E₀/V₀. For the next excited state the program guesses E/V₀ to lie just above E₀/V₀, i.e. just above the first value found. It changes the energy in small steps until s-(3/2)p is minimized, etc….

The chosen potential supports 4 bound states. The table below compares the results of the numerical integration with analytical results.

The program n3a.txt finds the bound states for the same potential by numerically integrating the time-independent Schroedinger equation. For a description of the program click here. The potential in n3.txt was changed to yield n3a.txt. The energies found by this program are given below.

For the potential chosen here, the WKB approximation becomes less accurate as E increases, because it does not take into account that V(x)=0 for |x|>5Å. Near the edge of the well, the linear approximation that is used to derive the WKB approximation is valid only over a small range of x.