Lab 4: Numerical solutions

In this laboratory you will solve the one-dimensional, time-independent Schroedinger equation numerically and find the energy eigenvalues of an electron trapped in a finite square well, a harmonic well, and a triangular well.  You will use an Excel spreadsheet for the numerical work.

Theory:

We want to solve the eigenvalue equation

Hψ(x)  = Eψ(x),  
or  
(-ħ2/(2m))∂2ψ(x)/∂x2 + U(x)ψ(x)  = Eψ(x),

or
2ψ(x)/∂x2 - k2(x)ψ(x) = 0,

with k2(x) = (2m/ħ2)(E - U(x)).

We want to solve it in a finite region between x = -L and x = L.  We assume that in the middle of this region there exists a potential well of width a, with a/2 << L.  Because we cannot extend our numerical solution to infinity, we assume that U(x) = ∞ for x < -L and x > L.

We want to find the energies for which the wave function ψ(x) and its derivative ∂ψ(x)/∂x are continuous and the boundary conditions ψ(0) = ψ(L) = 0 are satisfied.  These are the energies of the bound states of the system.

We start by expanding ψ(x) in a Taylor series expansion.
ψ(x + ∆x) = ψ(x) + ∆x*∂ψ(x)/∂x + [(∆x)2/2]*∂2ψ(x)/∂x2 + ... .
ψ(x - ∆x) = ψ(x) - ∆x*∂ψ(x)/∂x + [(∆x)2/2]*∂2ψ(x)/∂x2 - ... .
Combining the two equations above yields
[ψ(x + x) + ψ(x - x) - 2ψ(x)]/(x)2 = ∂2ψ(x)/∂x2,
or
[ψ(x + x) + ψ(x - ∂x) -2ψ(x)]/(x)2 = -k2(x)ψ(x).

ψ(x + x) = (2 - (x)2k2(x))ψ(x) - ψ(x - x).

For our numerical work, let {xn} denote points on a grid defined in the region from x = -L to x = L, n = 0, 1, , N.  Define ψn = ψ(xn), kn = k(xn), ∆x = xn+1 - xn.  Then

ψn+1 = (2 - (x)2kn2ψn - ψn-1.

We have a recipe for finding ψn+1, given ψn and ψn-1.  Integrating the wave function using this algorithm, i.e. finding its value on the next grid point from its value at the previous two grid points, is called integrating using the Numerov method.

To solve for the bound states of the system, we pick ψ0 = 0, ψ1 = c1, where c1 is some small number.  This number c1 determines the overall normalization of the wave function.  We now calculate ψ2, ψ3, etc.  We start integrating in the classically forbidden region, where the magnitude of the wave function increases approximately exponentially.  As we reach the classically allowed region, the wave function becomes oscillatory at the classical turning point.  As we pass through the second turning point and enter again into the classically forbidden region, the integration becomes numerically unstable, because it can contain an admixture of the exponentially growing solution.  Integration into a classically forbidden region is likely to be inaccurate.  We therefore generate a second solution, picking ψN = 0, ψN-1 = c2, and calculating ψN-2, ψN-3, etc.

ψn-1 = (2 - (x)2kn2ψn - ψn+1.

For both integrations we pick the same value for the energy E.  To determine whether this energy E is an eigenvalue, we compare the results of our integrations at a matching point xm in the classically allowed region.  The constant c2 is chosen so that both integrations yield the same value ψ(xm) at the matching point and we examine the slope ∂ψ(x)/∂x near xm.  The values of E for which the slope is continuous across the matching point are the energy eigenvalues, because for these values of E the wave function is a solution to the Schroedinger equation which satisfies all boundary conditions.  We search for the eigenvalues by picking different values for the energy E.


Procedure:

Open the linked Excel workbook.  It can be used to solve the one-dimensional, time-independent Schroedinger equation for an electron trapped in various potential wells.  Sheet 1 contains the data and sheet 2 the user interface.

(a)  Examine sheet 1.

(b)  Switch to sheet 2.  Fill in the tables and  keep a log of your answers to the question in the text below.

Comment on your results.  What have you learned about the stationary states of an electron in a finite square well?

Comment on your results. What have you learned about the stationary states of an electron in a harmonic well and in a triangular well?


Use your log to prepare a report.

Name:
E-mail address:

Laboratory 4 Report