Schrodinger's Equation

Constantly Changing Potentials

Schrodinger's time-independent equation is:

Schrodinger's equation

Using this equation, the wavefunction for a particle inside a "box" with walls made of a potential can be found. For example, this includes the box with infinite walls: a region with potential 0, with infinite potential at the edges of its width. The differential equation can be solved for this particular problem to yield a sinusoidal wavefunction for the particle's motion, with the function equal to zero outside the box. The square of the wavefunction is the probability, which can thus be found. The wavefunction also quantizes the availiable energies of the particle, since the it must conform to six boundary conditions. The wavefunction must be finite, single-valued, and continuous, and its derivative with respect to position must also be finite, single-valued and continuous. These boundary conditions neatly specify the availiable energies, and provide the inital conditions necessary to solve the differential equation.

Infinite Box Diagram

The process on the particle in the infinite box can be changed by altering one or both of the walls to finite walls. The walls must still have a potential higher than the energy of the particle, for the particle to be classically bound and energy to be quantized. When the wall is lowered, even though classicaly the particle would have zero probability of being found in the region, there exists an exponentially decreasing probability of the particle being found outside the wall. This leads to some conclusions: a constant potential which is below the total energy of the particle leads to a sinusoidal wavefunction, and a constant potential higher than the total energy of the particle yields a exponentially decreasing wavefunction.

Finite Box Diagram

This conclusion is justified because Schrodinger's equation can be reduced for any constant potential to one of the following differentials.

The differential with the second derivative of the wavefunction equal to negative a constant squared times the wavefunction is solved as a sinusoidal equation. The differential with the second derivative of the wavefunction equal to positive a constant squared times the wavefunction is solved as an exponential, which must be decreasing for the wavefunction to be finite at large values of x.

Therefore we know the basic solution to wavefunctions consisting of constant potentials. To fully solve the functions it is also necessary to normalize the wavefunction, so that the total probability of the particle being at any one location (the integral of the wavefunction squared over the entire length) is equal to one. Yet how does one solve an equation with a constantly changing potential? It might seem feasible at first to simply plug in the equation for the potential in terms of x, and solve the differential equation. This is in fact possible for the harmonic oscillator, where the potential is a function of x squared. Unfortunately, for most potentials, the analytical method of solving the differential is anywhere from horrible to impossible to solve. Each differential equation is completely independent, and requires new methods in order to solve. The conventional way of solving these difficult differentials is by computer; it is relatively easy to write a short computer program which can take the differential equation and find the necessary energies and plot the wavefunction. This is done by imputing starting points and changing the differential equation to a difference equation, using small changes in x and manually working out the wave function all along an interval. The wavefunction only converges on the axis for proper, quantized energies, and diverges upward or downward to infinity otherwise. It is thus possible to try different energies, closing in on the proper value, to find the first few proper energies.

While it is thus possible to use a computer to solve all possible potentials, it would be preferable to be able to approximate other potentials by looking at them. To solve this problem we look at the "step" well. Such a well is almost exactly like the finite well, except that its potential at the bottom of the well changes discontinuously, like a step. We know that each of the wavefunctions at the bottom of the well must be sinusoidal. One manner of expressing the function is: Asin(kx + b), where A is the amplitude, k is a constant which is determined by the potential, which is shown in one of the equations above, x is the position, and b is a phase constant. The wavefunction must be continuous. The derivative of the wavefunction is kAcos(kx + b), which must be continuous. Dividing the derivative by k and squaring it, and adding the wavefunction squared cancels out the sin and cos, leaving A2 = (psi2 + [d(psi)/dx]2/k2) Since the wave function and its derivative are continuous, at values of x infinitesimally close to the location where the potential changes, the wavefunction and its derivative are the sane on both sides. However, since the potential increases discontinuously at that location, k decreases at that location. Since k changes, but not the wavefunction or its derivative, A must also change. Since k decreases, A increases at the step upwards. This means that the amplitude of the wavefunction increases with higher potential, and since the kx in sin(kx + b) decreases, the wavelength of the sinusoidal wavefunction also increases. Together this means that at discontinuous step increases in potential, the wavefunction increases in amplitude and wavelength. This provides for qualitative analysis of almost any potential. Any continuously changing potential, like a ramp or triangle, can be approximated by a series of steps. The smaller the width in the steps, the greater the accuracy. From this approximation, it is clear that a linearly changing well bottom will have a constantly changing "wavelength" and amplitude, based on the principles earlier. By substituting V=cx for V in the equation for k, we could obtain an equation for k based on x, which can be substituted in the equations for amplitude and "wavelength" to find an approximation for their erratic changes. A2 = (psi2 + [d(psi)/dx]2/[2m/h2*(E-cx)]), and the wavelength is proportional to 1/k, or h/sqrt[(2m)(E-cx)].

Bibliography

French, A.P. and Edwin F. Taylor An Introduction to Quantum Physics. W.W. Norton & Company: 1978, New York, pp. 105-182. Beiser, Arthur Concepts of Modern Physics. McGraw-Hill: 1987, New York, pp. 165-198, 569-573.