Subsections


7.11 Almost Classical Motion

This section examines the motion of a particle in the presence of a single external force. Just like in the previous section, it will be assumed that the initial position and momentum are narrowed down sufficiently that the particle is restricted to a relatively small, coherent, region. Solutions of this type are called “wave packets.”

In addition, for the examples in this section the forces vary slowly enough that they are approximately constant over the spatial extent of the wave packet. Hence, according to Ehrenfest's theorem, section 7.2.1, the wave packet should move according to the classical Newtonian equations.

The examples in this section were obtained on a computer, and should be numerically exact. Details about how they were computed can be found in addendum {A.27}, if you want to understand them better, or create some yourself.

There is an easy general way to find approximate energy eigenfunctions and eigenvalues applicable under the conditions used in this section. It is called the WKB method. Addendum {A.28} has a description.


7.11.1 Motion through free space

First consider the trivial case that there are no forces; a particle in free space. This will provide the basis against which the motion with forces in the next subsections can be compared to.

Classically, a particle in free space moves at a constant velocity. In quantum mechanics, the wave packet does too; figure 7.15 shows it at two different times.

Figure 7.15: A particle in free space.
 
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If you step back far enough that the wave packet in the figures begins to resemble just a dot, you have classical motion. The blue point indicates the position of maximum wave function magnitude, as a visual anchor. It provides a reasonable approximation to the expectation value of position whenever the wave packet contour is more or less symmetric. A closer examination shows that the wave packet is actually changing a bit in size in addition to translating.


7.11.2 Accelerated motion

Figure 7.16 shows the motion when the potential energy (shown in green) ramps down starting from the middle of the plotted range. Physically this corresponds to a constant accelerating force beyond that point. A classical point particle would move at constant speed until it encounters the ramp, after which it would start accelerating at a constant rate. The quantum mechanical solution shows a corresponding acceleration of the wave packet, but in addition the wave packet stretches a lot.

Figure 7.16: An accelerating particle.
 
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7.11.3 Decelerated motion

Figure 7.17 shows the motion when the potential energy (shown in green) ramps up starting from the center of the plotting range. Physically this corresponds to a constant decelerating force beyond that point. A classical point particle would move at constant speed until it encounters the ramp, after which it would start decelerating until it runs out of kinetic energy; then it would be turned back, returning to where it came from.

The quantum mechanical solution shows a corresponding reflection of the wave packet back to where it came from. The black dot on the potential energy line shows the “turning point” where the potential energy becomes equal to the nominal energy of the wave packet. That is the point where classically the particle runs out of kinetic energy and is turned back.

Figure 7.17: A decelerating particle.
 
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7.11.4 The harmonic oscillator

The harmonic oscillator describes a particle caught in a force field that prevents it from escaping in either direction. In all three previous examples the particle could at least escape towards the far left. The harmonic oscillator was the first real quantum system that was solved, in chapter 4.1, but only now, near the end of part I, can the classical picture of a particle oscillating back and forward actually be created.

There are some mathematical differences from the previous cases, because the energy levels of the harmonic oscillator are discrete, unlike those of the particles that are able to escape. But if the energy levels are far enough above the ground state, localized wave packets similar to the ones in free space may be formed, {A.27}. The animation in figure 7.18 gives the motion of a wave packet whose nominal energy is hundred times the ground state energy.

Figure 7.18: Unsteady solution for the harmonic oscillator. The third picture shows the maximum distance from the nominal position that the wave packet reaches.
 
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The wave packet performs a periodic oscillation back and forth just like a classical point particle would. In addition, it oscillates at the correct classical frequency $\omega$. Finally, the point of maximum wave function, shown in blue, fairly closely obeys the classical limits of motion, shown as black dots.

Curiously, the wave function does not return to the same values after one period: it has changed sign after one period and it takes two periods for the wave function to return to the same values. It is because the sign of the wave function cannot be observed physically that classically the particle oscillates at frequency $\omega$, and not at $\frac12\omega$ like the wave function does.


Key Points
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When the forces change slowly enough on quantum scales, wave packets move just like classical particles do.

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Examined in detail, wave packets may also change shape over time.