Moving Particles and Wave Packet Propagation
A Computer Animated Supplement

Giles Henderson
Eastern Illinois University, Charleston, IL 61920

Authors and educators have frequently been frustrated by the inability of equations and graphs to properly convey the temporal qualities of dynamic systems. Previous journal articles have used time-dependent wave functions to calculate and describe quantum trajectories of both free particles (1) and particles bound by a force field (2-6). Here we wish to supplement these previous articles with computer animated motions as described by both classical and quantum methods.

Free Particles

Animation 1.

Ref 1 described the use of wave packets to obtain quantum trajectories of a two dimensional pulsed particle beam. Each particle is described classically by its momentum. Quantum descriptions are obtained from two dimensional de Broglie traveling waves, which are solutions to the time-dependent Schrödinger equation for a free particle. Figure 2 of the original manuscript disclosed the spatial properties of these wave functions. Computer animations allow us to visualize both their spatial and temporal behavior.

A pulse of particles with some distribution of velocities and momenta is described by a wave packet that is composed of the sum of the wave functions of each individual particle, e.g. a sum of time-dependent functions of the type portrayed in Animation 1. The position probability of the particle pulse at a specified time is then calculated from the complex square of the superposition wave packet. The probability amplitude and its evolution in time is determined by the various constructive and destructive interactions of the wave packet components. An animation (Animation 2) in which the time evolution of the two-dimensional wave packet is compared to the classical motion of a particle pulse complements Figure 3 of the original manuscript.

Animation 2.

The classical trajectory reveals a spreading of the pulse and a decay in the particle number density consistent with the momentum spectrum. The quantum probability surface exhibits a corresponding delocalization as evident from the decay in peak probability amplitude and the increase in the width of the probability surface in the x-direction. This temporal behavior is a natural consequence of the dephasing of the individual wave functions as time advances. The animation confirms that the group velocity of the quantum wave packet coincides with the classical mean velocity of the particle pulse.

The original manuscript explored the effect of a change in slit width on the diffraction of the component matter waves and on the wave packet structure. Here we can animate these effects (Animation 3).

This behavior is perhaps contrary to naive intuition but is consistent with and a consequence of the Heisenberg Principle: in our attempt to restrict the uncertainty in the y position of the particles by narrowing the slit width, we cause an increase in the velocity and momentum components in the y direction.

Animation 3.

Bound Particles

1. Harmonic oscillator with no external perturbations

Chemistry students are familiar with the eigenstates of the simple harmonic oscillator in which one or more particles are moving in a Hooke's law force field. However, the quantum dynamics of this simple system have seldom been considered by undergraduates. This situation reflects the abstractness of the mathematical formalism, and, at least in the past, the level of computational efforts required to examine time-dependent quantum trajectories.

In 1990, Tanner (2) described the use of a split operator fast Fourier transform grid method to numerically solve the time-dependent Schrödinger equation. He used this formalism along with digital numerical methods to calculate the quantum trajectory of a harmonic oscillator. More recently (3) a spreadsheet template has been described in which a propagation operator was employed to calculate the time evolution of wave packets. These authors also chose a simple harmonic oscillator to illustrate their method in which the position probability of a harmonic oscillator is described by a normalized Gaussian function. The corresponding wave packet is then defined by the square root of the Gaussian position probability function:

and can be expressed as a linear combination of the harmonic oscillator eigenstates:

Since the harmonic oscillator eigenstates are defined by an orthonormal basis set, we obtain the composition of the wave packet by simply projecting the harmonic oscillator basis functions onto the Gaussian wave packet:

Figure 1 depicts the components of a Gaussian wave packet at time zero using Tanner's (2) initial conditions:

1. The mass of the particle is 1 g/mol.
2. The initial wave packet is centered at x(0) = 0.5 au (1 au = 0.529 Å).
3. The RMS spread in coordinate space = 0.14 au.
4. The trajectory occurs at an energy of 6.5e-20 joules.

Figure 1.

The time evolution of the wave packet can be animated by evaluating G'(x,t) at an appropriate time interval and sequencing these frames in a animation (Animation 4).

Animation 4.

The animation compliments Figure 2 of ref 2 and Figure 5 of ref 3. In this example, the most probable quantum trajectory is depicted by the maximum amplitude of the Gaussian wave packet, which correlates precisely with the classical motion depicted by the particle in the bottom of the animation window. Both models describe a simple sinusoidal motion of a single frequency. The animation also illustrates how the width of the Gaussian wave packet changes with the momentum of the particle as described previously (2, 3).

2. Resonance perturbations: spectroscopic transitions

In the limit of low pressure and intense resonance radiation, collision induced radiationless processes are minimized and transition dynamics are dominated by stimulated absorption or emission. Time-dependent superposition functions have been employed to describe spectroscopic transitions in atoms and molecules under these conditions (4-7). These methods have been used to produce computed animations of the oscillating electric charge density of a hydrogen atom as it evolves from one quantum level to another upon irradiation with resonance radiation (7).

Animation 5.

The harmonic oscillator wave functions, equation 6 in ref 4, can be used to calculate the evolution of bond length probability as defined by equation 4 in ref 4.

Equation 6 in ref 4 can be evaluated at regularly-spaced time increments to generate "animation frames" for this process. In order to scale the animation to a reasonable duration, we deliberately exaggerated the resonance radiation power to reduce the number of vibrations per Rabi cycle.

This animation depicts the evolution of the bond length probability function for a stimulated transition between v=2 and v=1. It complements Figure 4 in ref 4 and Figures 1 and 2 in ref 5. The time-dependent mixing coefficients, equations 8 and 9 of ref 5 undergo sinusoidal oscillations at the Rabi period (8). The resulting periodic stimulated emission-absorption cycles are known as transient nutations (9).

References

1. Henderson, G. L. J. Chem. Educ. 1993, 70, 972-976.
2. Tanner, J. J. J. Chem. Educ. 1990, 67, 917-921.
3. Hansen, J. C.; Kouri, D. J.; Hoffman, D. K. J. Chem. Educ. 1997, 74, 335-342.
4. Henderson, G. J. Chem. Educ. 1979, 56, 631-635.
5. Henderson, G. Am. J. Phys. 1980, 48, 604-611.
6. Henderson, G. J. Chem. Educ. 1990, 67, 392-398.
7. Henderson, G.; Rittenhouse, R. C.; Wright, J. C.; Holmes, J. L. "How a Photon is Created or Absorbed".
8. Rabi, I. Phys. Rev. 1937, 51, 652.
9. Hocker, G. B.; Tang, C. L. Phys. Rev. 1969, 184, 356.

Abstract
	J. Chem. Educ. 2000, 77, 134.