This demonstration is meant to show, by means of a computer program, one of the fundamental principles of quantum mechanics: how the requirement that a wave function be "well behaved" leads to quantization of a physically observable quantity such as the energy.

A wave function is a solution to a differential equation, Schrodinger's equation,

where H is the Hamiltonian operator, E is a simple number, associated with the energy, and y(x) is the function that satisfies the equation. So far, there is no quantization of the energy taking place. E can take on any value at all. However, the function y may or may not be a "well behaved" wave function.

For a function to be well behaved it must have certain properties. One of them is easy to understand. We can calculate the probability of finding a physical system in a particular state from the wave function. An example of such a probability is how likely is the electron of a hydrogen atom to be within 0.3 Angstrom of the nucleus. For this to make sense, we require that the wave function be "quadratically integrable". That is, the sum of probabilities derived from the wave function can't add up to infinity. You can see that there might be problems with a wave function which itself approaches infinity because the probabilites could add up to infinity. Such a wave function would not be acceptable.

Many of the solutions to the Schrodinger equation will diverge to infinity - unless the value of E is carefully chosen to prevent that from happening. The set of values for E which yield wave functions that do not diverge to infinity are the "allowed" values. At this point, quantization of the energy has been achieved. More commonly, the quantization is introduced during an analytic solution of the problem, frequently by requiring that a series truncate..

The program which may be started on the next page is applied to several different cases. Each problem involves a single particle subjected to a different potential discussed below. The Noumerov method which is used to solve the differential equation can be applied to many kinds of problems.The mathematical method, discussed in texts on numerical methods, does not involve any quantum mechanical concepts.

The different cases presented for you to explore are:

Polynomial

A particle subject to a potential described by a polynomial. The potential energy is given by a function A+ B x +C x² + D x³ + E x⁴. (This includes the special case of the harmonic oscillator, C x².) You choose the values for the constants. If B and D are both zero then the potential is called symmetric and it turns out that the wave functions are either even (such as the cosine function) or odd (such as the sine function). This greatly simplifies the search for an acceptable solution. Now you can then choose different values for the energy, E, and examine the resulting wave function to see if your choice provides a well behaved wave function, i.e., one that does not diverge to infinity.

If either B and/or D are non-zero then the situation becomes more difficult and is called an asymmetric potential. Not only do you have to supply a value for the energy but you have to supply some shaping to the function y(x) as well. You provide both a value for the function at x=0, y(0), and the value of the slope at x = 0, y'(0). Remember, that for an acceptable function the value of y(x) must not diverge to infinity at either + x or - x. For the polynomial case the energies, E, are positive.

Morse

A Morse potential is frequently used to describe an interatomic potential. You will be asked to supply a parametric value, gamma, which characterizes the potential. The search for a well behaved solution is similar to the asymmetric case above. You will have to supply the energy, initial value for the Noumerov process, y(0), and the initial slope, y'(0). Once again the energies are positive.

Hydrogen Like Atom (one electron systems)

You can solve a portion of the problem for the energies of an atom containing only one electron. At this stage in the process you can pick a value for the l quantum number, 0,1,2.... Anticipating the solution, it may be helpful to remember that the allowed values for the principal quantum number, n, are related to l by n>l. The energies obtained in this excercise are negative for the bound states.

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