This is represented by an infinite square well. It is the simplest bound state problem in Quantum Mechanics. It corresponds to an electron confined to a linear molecule.
Within the molecule, the potential is low because the electron is close to positive charges. Beyond the ends of the molecule, the potential is high because the electron is far from any of the atoms.
The width of the well depends on the number of atoms N in the chain. The value of L, half the width of the well, is approximately
Heading: L = N/20 (nm)
We represent the potential by a well in which the V(x) is zero inside and infinity outside the well.
Choose F=1, W=0, Z=0 and L=0.1. The wavefunction is a cosine function whose wavelength is twice the width of the well. Increase Z to 1. What happens to the wavelength? What happens when Z=2, 3, 10?
For various values of Z, compare the energies for W=0 with the energies for W=1, 2, 20 and -30. What happens to the energies? What happens to the wavefunctions?
Heading: Varying L
Choose F=1, W=0, Z=0 and vary L. Use L=0.1 (a diatomic molecule), L=0.2 (a linear molecule with 4 identical atoms) and L=1 (20 atoms). How does the energy depend on L? How do the wavefunctions change?
Heading: The Potential step
Choose F= -(X<0), W=2, Z=0 and L=0.1. Study the shape of the potential function. It is a step function of value +2 in the left half of the well and 0 in the right half.
The energy is about 1eV greater than it was when V=0 but the wavefunction is not noticeably altered.
Change W to 5. The energy increases, but the wavefunction is larger in the right half of the well. Increase W to 10, then to 15, then to 20. Notice that the particle becomes more and more confined to the right half of the well.
When W=2000, the probability of finding the particle in the left half of the well is very small. Remember that the probability of finding the particle at a point is proportional to the square of the wavefunction.
The energy is 4 times the value it had in the original well because the uncertainty in momentum is doubled when the width of the well is halved, and so the energy is multiplied by 4. The potential energy is still 0 in th right half.
Heading: The finite square well.
Choose L=0.1, Z=0, W=2000 and F=ABS(X)<0.02.
Then V(x) = -W for |x|<0.02
= 0 for |x|>0.02
Study the form of the potential well. Compute the wavefunctions of the ground state and the first two excited states (Z=1 and Z=2).
For Z=3, the energy is positive and you can see clearly that the wavelength is much shorter inside the well than it is outside the well.
One might expect that the probability of finding the particle in the region of low potential is large, but this is not true. Why?
Heading: The Harmonic Oscillator
Choose L=1, F = X^2 , W = 200 and Z=0.
Find the energy of this state (write it down) and sketch the wavefunction. Repeat the process for Z = 1, 2, 3 and 4 .
What do you notice about the differences in energy between successive states?What do you notice about the parities of the wavefunctions?
Heading: A two-centre problem
Choose L=0.1, W=1000, Z=0 and F=ABS(ABS(X)-0.05)<0.02.
First, study the form of the potential. It has two square wells centred on x=+0.05 and x=-0.05. Each has width 0.04 nm. The ground-state wavefunction has two identical peaks centred on the two wells. Note the energy.
Find the energy for Z=1. How does it compare with the previous value?
They are nearly equal because the two peaks in the wavefunction are almost independent.
Compare the energies for Z=2 and Z=3.
Heading: Perturbation Theory
This program can be used to study perturbation theory in quantum mechanics. A perturbation is a small additional term in the Hamiltonian.
As an example, consider an electron in a hydrogen molecule, formed from two hydrogen atoms (L=0.1nm). The original potential energy is V(x)=0 within the well and the energy of the ground state is E'=9.4096.
Change the potential slightly by adding a potential step to give
V(x) = W , x < 0
= 0 , x > 0
Then perturbation theory predicts that the new energy of the ground state will be E' +0.5 W. (If you have studied perturbation theory in quantum mechanics, you will be able to prove this result quite easily.)
In physical terms, this change corresponds to replacing one of the hydrogen atoms in the hydrogen molecule with a different type of atom.
Use the program to draw a graph of the energy of the ground state as a function of W, from W = -20 to +20. Use steps of 5 initially, and fill in parts of your graph with steps of 1 if you think it necessary.
Draw the line E = E' + 0.5 W on your graph and compare the two results.
Repeat the procedure for Z=1 and Z=2 and describe your results.
Heading: A challenge
Obtain an analytical solution of the problem used to illustrate perturbation theory. Show that for small values of W, your result for the energy is in agreement with the result from perturbation theory: E = E' + 0.5 W.
