This program allows you to study bound state problems in Quantum Mechanics.
The particle is constrained to move between two infinite walls at x = -L nm and x = +L nm. The infinite walls do not really restrict your choice of potential. You can move them further away from the region of interest by making L larger.
The units of energy are electron Volts (eV), the units of length are nm and the mass of the particle is that of an electron.
Schrodinger's equation in one dimension is
Sprite: Schro
The wavefunction must be zero at the edges of the well because the potential is infinite there.
The program finds the eigenvalue E and eigenfunction y(x) in the following way. It first guesses a value for E, and solves the differential equation numerically, starting at the left end x = - L with y=0.
If the energy is exactly equal to an eigenvalue, the wavefunction will also be 0 at the right end.
If the guess is not correct, the program chooses a different value for the energy and tries again. After a few iterations, it obtains the eigenvalue correct to six significant figures.
Heading: The initial problem
When the program starts, the bounding walls are set by L = 0.1 nm. The potential is 100eV in the left half, x<0, and 0 in the right half, x>0. This configuration corresponds roughly to an electron confined to a diatomic molecule.
Heading: Change W
W is the strength of the potential: V(x) = W*F(x). This option allows you to change the strength of the potential without changing its shape.
If the values of the potential V(x) that you have chosen are so large that the algorithm for solving the differential equation breaks down, the program will give you a warning and reduce the value of W.
Heading: Change F
F(x) is the shape of the potential. You can choose from a variety of pre-defined potentials or type in your own functional form in the top line. Any acceptable BASIC function of X is allowed.
When making step functions, it is useful to remember that, in BASIC, the quantity TRUE takes the value -1 and FALSE takes the value 0.
For example, V(X)=10*(X<0) represents
V(X) = -10, X<0
V(X) = 0, X>0
If you type in an unacceptable function, the program will complain and revert to using the previous function.
Heading: Change Z
Z is the number of zeroes in the wavefunction, excluding the two at the left and right ends of the potential. A value Z=0 gives the ground state, while Z=1, 2 ... give states of increasing excitation energy.
Heading: Change L
The distance between the infinite bounding walls of the well is 2L, measured in nanometres. This option allows you to change its value.
