Schrodinger's Equation: The Quantum Wave Equation

The time-dependent Schrodinger equation determines how a quantum wave in a potential well changes its shape with time. The problem is similar in many ways to the motion of a classical wave pulse on a string, and we will use the same pulse shape in the four quantum Applets as was used on the string. The Schrodinger Equation differs from Newton's law in two important ways: it involves a first time derivative instead of a second, and the square root of -1 appears explicitly in it. As a consequence, we cannot solve it using the Feynman algorithm (although we will use a half-step look ahead), and we have to keep track of real and imaginary parts of the wavefunction (the curvature in one drives the time dependence of the other). The first Applet deals with the infinite square well, and in this case the potential term drops out of the Schrodinger Equation and the situation is similar in many respects to the motion of a pulse on a string clamped at both ends. Both can be started off in any configuration we wish, both have their change in configuration driven by their curvature, and, since both obey zero boundary conditions, both can use the FFSS as an alternative means of generating the time evolution.

We can put the Schrodinger equation for an infinite square well into suitable form for numerical solution by introducing the period of the lowest energy stationary state (a single loop of a sinusoid):

The only parameters in the final form of the equation are the width L of the well and the ground state period. These parameters disappear from the numerical equation when they are used as units for dimensionless distance and time variables. In the Applet one can generate the time dependence either directly using the Schrodinger equation or indirectly using the dispersion relation and a series of stationary states. For an infinite square well, the series is the familiar FFSS, and the dispersion relation has frequency proportional to the square of the quantum number. The quantum pulse has the same sinusoidal shape as the pulse we used on the classical string, but it is normalized and multiplied by a complex exponential that introduces a phase change of either 2p across its width (or 8p if nk=4 is chosen). The complex exponential is the quantum analog of initial transverse velocity for the classical string, and causes the pulse to start moving to the right. It does not alter the initial probability density, but it increases the kinetic energy, (energies are expressed as multiples of the ground-state energy.) By default, the Applet plots the probabiity density, but you can to choose to plot the variables Yr and Yi that appear in the Schrodinger Equation instead.

For the classical string, a set of mass points on a weightless string is a physical model with its own dispersion relation. The quantum string has no corresponding physical model. The Java program divides the well into 384 segments and follows the motion for 1/8 (1/128 if nk=4) of the ground state period. The calculation uses a half-step look ahead similar to what we used in field plots: we want the curvature in the function at the mid-point of the time step we are about to take.

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The 384-segment direct solution of the Schrodinger Equation deals with the time dependence quite well. I like to watch the Yr/Yi plot, realizing that between each frame 3000 curvature calculations at 384 points on each curve are used to generate the time-dependence of the other. I find it truly remarkable that the plot in the final frame agrees reasonably well with the plot that can be obtained in a single step with a 60-term FFSS. When "Series" is chosen, the dispersion relation is plotted after the oscillations cease, and "s60", the percentage accuracy reached with a 60-term the series, is printed (99.998 %).

The square well is not typical of quantum wells. Its infinite sides require the wavefunction to go to zero so no wall penetration is possible. With no potential term in the Schrodinger Equation, the FFSS gives the stationary states and enables their energies to be found directly. The resulting disperion curve shows upward curvature, but, because the energies are integer multiples of the ground-state energy, there are resonances in the probability density pattern. If the motion is followed for 1/2 the ground-state period, the initial pulse reappears. Only the first reflection from the boundary is shown when nk=4 is selected, and the fact that the component waves travel slower than the pulse envelope shows up clearly. This is the opposite to what happens to a pulse on a classical string with a small number of beads.

The second Applet has a rounded well with 4th-power potential walls, and is a good example of the general quantum-well problem. It is difficult to start the direct solution of the Schrodinger Equation without suitable units for distance and time, so the first step is to find the lowest energy stationary state. The distance between its turning points (points where the curvature changes from toward the axis to away from the axis) provides a convenient unit for length, and its energy provides units for both energy and time.

The procedure for finding Stationary States is described in detail in the section on the time-independent Schrodinger Equation. The two potentials in the example are "bouncing ball" (1st-power) and "harmonic oscillator" (2nd-power), but the results can be generalized to nth-power: the dimensionless energy appears to the power (1+2/n) in the equation, and (1/n) in the turning point.

Procedures for finding the energies and wavefunctions of the stationary states of the 4th-power potential are included in the second Applet. Once the ground state has been found, the pulse is fitted in between its turning points, and the direct solution of the Schrodinger Equation is ready to go. Once the stationary states have been found, the expansion coefficients can be determined, the series solution is likewise ready to go, and the 4th-power potential Applet, a "typical" quantum-well Applet, is ready to run.

Java Source

Some of the parameters differ from those in the previous Applet. The separation between the ground-state turning points is made equal to 48 pixels. The number of iterations per frame of the direct solution is reduced from 3000 to 320. The series solution uses 20 terms, and that is the range of the dispersion plot. About all that can be said about the plots (beyond the fact that the direct and series solutions are in close agreement) is that the curvature in the dispersion relation really does disperse the pulse.

The next Applet deals with the 2nd-power potential well: the quantum harmonic oscillator. As the name implies, the dispersion relation is a straight line, not a curve, and although a wave pulse may change shape as it moves in the well, it will not be dispersed. The calculation proceeds in the same way as in the previous Applet, but some of the parameters are altered, and the motion of a classical harmonic oscilator (also calculated numerically) is plotted across the bottom. The display is still 384 pixels wide, but the numerical calculation extends an extra 96 pixels out each side to avoid reflections from boundaries. The number of iterations per frame in the direct calculation is reduced to 160, and the number of terms in the series solution is increased to 40. The dispersion plot covers the first 30 terms.

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The direct solution and series expansion give results that agree closely and show virtually no dispersion over one period of oscillation. The energies calculated by the two methods, 13.193 for the direct calculation and 13.180 for the series also agree closely. If you like calculus, you can try showing that an analytic calculation gives 13.240.

Basic Physics began with the Feynman algorithm and a classical harmonic oscillator, the mass-spring system shown at the bottom of the Applet. It ends here with a quantum oscillation in a harmonic well and an algorithm very similar to the Feynman algorithm. Along the way, normal modes, Fourier series, and stationary states showed us how to use a series to represent the oscillation.

After some debate, I decided to add an Applet on the V-shaped well, a symmetric form of the bouncing-ball first-power potential, more or less as a footnote. The four wells get progressively weaker as we go from square through to V, and this makes them progressively more difficult to deal with numerically. The harmonice well uses a 384-pixel plotting grid, but extends the direct solution to 576 pixels, and the series soution to 40 stationary states. The V well extends the series solution to 3072 pixels and the series solution to 240 stationary states (to get to 99.29% accuracy). When you look at the result you may well question whether the effort was worth it.

Java Source

The wave pulse is well dispersed, and not all of it ends up within the range of the plot. On a positive note, the agreement between the direct and series solutions is remarkable in view of the number of calculations involved. The dispersion relation, plotted for the first 40 terms, shows downward curvature, continuing the trend shown as the wells weaken. Downward curvature also showed up in the first three stationary states of the bouncing-ball potential, and in the normal modes of beads on a classical string.

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