The time-dependent Schrodinger equation determines how a given initial wavefunction will change with time. It 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 (the curvature in one drives the time dependence of the other). A wavefunction in an infinite square well is similar in some respects to a classical string clamped at both ends. Both can be started off in a pluck or pulse configuration, both have their change in configuration driven by their curvature, and, since both obey zero boundary conditions, both can use FFSS as an alternative means of generating their time evolution. The initial complex phase of a quantum wave plays the same role as the initial transverse velocity for a wave on a string.
We can put the Schrodinger equation for an infinite square well suitable form for numerical solution by introducing the period of the lowest energy stationary state:
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 example, one has three choices for the initial wavefunction, and can generate its time dependence either directly using the Schrodinger equation or indirectly using FFSS and the dispersion relation (frequency is proportional to square of quantum number). The first two choices are the rectilinear Pluck and Pulse displacements used on the classical string. The initial wavefunction is real in both cases: this is the quantum analog of a string released from rest. The third choice, Pulse S, is a rounded version of the rectilinear Pulse, and is also real. The fourth choice, Pulse A, is Pulse S multiplied by a complex exponential that introduces a phase change of 2p across its width. The complex exponential is the quantum analog of initial transverse velocity for the classical string. It does not alter the initial probability density, but it increases the kinetic energy, and markedly changes the time dependence. (Energies are expressed as multiples of the ground-state energy.)
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 192 segments and follows the motion for 1/8 of the ground state period. The three symmetric "standing" (initially real) wavefunctions complete one probability density oscillation in this length of time. 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.
The direct numerical solution of the Schrodinger equation, even with 300 time steps between each of the plotted 640 frames, cannot be expected to give results as accurate as the FFSS over an extended period of time. The 192-segment numerical model does not reproduce an initial sharp point in the probability density, but otherwise it handles the "standing" waves quite well. What is truly remarkable is the way it deals with Pulse A, the "travelling" wave: after going through a series of seemingly random wiggles, it produces a reasonable approximation to the FFSS pattern in the final frame.
One gets a clearer picture of the way a quantum pulse propagates if one studies the motion of Pulse B, a pulse with more structure within the envelope. Pulse B has the same pulse envelope as the travelling quantum pulse used previously, but the complex exponential goes through a phase change of 8p across the width of the pulse. The motion is followed as the pulse moves to the right in an infinite square well, reflects from the boundary, and returns to the center. One can plot either the probability density or the real component of the wavefunction. The probability density plot shows that the pulse spreads in width as it moves, and the sharp oscillations introduced when it strikes the boundary damp out by the time the pulse gets back to the center of the well. The real component, although not physically observable, provides further insight into the propagation. The envelope is seen to travel faster than the structure within it, so the group velocity is said to be higher than the phase velocity. As the pulse spreads, there are more oscillations within the envelope, but the spacing between zero-crossings does not change. Over the time frame used, the FFSS solution and the direct solution of the time-dependent Schrodinger equation are in close agreement.
The next example is the quantum analog of the three-segment string applet that acted as a bridge to the material in Matrix Wave Optics. The initial wavefunction again is Pulse B (its real part is the initial displacement in classical applet). Here it is the kinetic energy that increases by a factor of 1.44 at each interface. In the quantum case the reflection probability is calculated rather than the percentage of energy reflected.
The "Series" option expands the wave pulse in terms of the eigenstates of the system. The eigenvalues are found by combining techniques from Matrix Wave Optics and Matrix Ray Optics. The wave matrix for the system is the product of the matrices for the three regions. The condition that the wavefunction go to zero at the outer boundaries is met by requiring the upper-right matrix element be zero. (This is the condition for an object-image relation in Matrix Ray Optics.) The program finds 60 eigenvalues over the range of interest (a region for which the sum of the squares of the expansion coefficients is greater than 0.9999). The eigenfunctions are then generated using the Feynman algorithm.
Again, a "film" thickness of 5 acts as an anti-reflection coating. As is the case with the classical pulse, the spacing between zero-crossings in the transmitted pulse is reduced, but unlike the classical case, the center of the transmitted pulse ends up farther from the interface than does the center of reflected pulse.
The final two examples show quantum mechanical oscillations that do not exhibit dispersion. The first is the double well whose first two states are very close together in energy. Their superposition produces a state whose oscillations are similar to classical beats. The second is the harmonic oscillator. The harmonic potential barrier reflects wave pulses without dispersion.
In the case of the double well (whose two lowest states were found in Probabliity in Quantum Mechanics, a two-term series does a very good job of locating a pulse in one half of the well. The probability density is the physically significant quantity that oscillates, but the real (blue) and imaginary (red) components of the probability amplitude provide some insight into the underlying mathematics.
In order to study the motion of a wave pulse in a harmonic potential well we have a number of additional factors to consider: (i) the Schrodinger equation has a potential energy term; (ii) we need a unit for the dimensionless distance variable (a suitable choice is the distance from the origin to the ground-state turning point); (iii) we need to select a range and resolution for the numerical calculation; and (iv) if we want to use an series expansion, we must find the stationary states of the harmonic oscillator (they are not the sine functions of the FFSS).
In the applet, the travelling pulse used on the string (Pulse A) is centered in the harmonic well, has its width matched to the distance between the ground-state turning points, and has its structure defined at the same resolution as on the string. Except for the presence of a potential energy term, the direct solution of the time-dependent Schrodinger equation proceeds in exactly the same manner as with the string. The series solution uses stationary states found by the method described in Energy Conservation and Integration (the known energy values are used in the calculation, and a wavefunction is cut off when it and its derivative have the same sign outside the well). The stationary states need only be defined for the range and resolution of the plot, but a finer resolution is needed to determine them accurately, and an extended range is needed to normalize them. The expansion coefficients can be found directly because, unlike classical normal modes, stationary states form an orthonormal set. The series uses 54 terms to bring the sum of the squares of the expansion coefficients to 99.99% of the total.
The applet displays the the real and imaginary components of the wavefunction (along with the terms in the expansion for the series) for two seconds before it shows the probability density. Energies are expressed as multiples of the ground-state energy, and the time-dependence is followed for one ground-state period. The energy is shown on a potential plot along with the turning points (in red for the ground state, in cyan for the pulse).
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 13.214 for the series, also agree closely. (The accuracy is limited in one case by the resolution of the calculation, and in the other by the number of terms in the series.) 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.