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For all their value, the stationary states of quantum mechanics are anathema to our everyday experience, and often present conceptual blocks to our understanding. The fullest appreciation of quantum ideas comes ultimately by examining the behavior of wave packets. A wave packet refers to a waveform that is concentrated in some well-defined region of space. In this way we can approximate in quantum terms the localization expected of a classical particle: according to the Born interpretation, a highly-concentrated wavefunction (read wave packet) is one for which the particle can be found with appreciable probability only in a very small region. Free particles are those subject to no force. For a free particle, the potential energy is everywhere constant and may be taken as zero. Even so, with few exceptions the task of solving the Schrödinger wave equation for the time development of any initial wave packet is formidable. By contrast, numerical simulation affords a viable alternative that enables visualization of quantum effects on an unprecedented scale. |
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Owing to its relative mathematical simplicity, the gaussian wave packet traditionally has been a popular way to represent the initial state of many quantum systems, though it must be confessed that the gaussian representation is often only an approximation to reality. The applet below shows a gaussian waveform used to represent a free electron that is initially localized to a region several angstroms (Å) in length (atomic dimensions). The gaussian profile for the initial wavefunction is recorded as φ(x) in the Equation View of the applet, along with the particle mass = 511 keV/c2 which we recognize as the mass of an electron. The packet is centered at the coordinate origin x = 0 and w = 1.0 Å is a measure of its width. Indeed, w may be identified with the (initial) uncertainty Δx in locating this electron. ψ(x) is the result of propagating the initial wave forward in time according to the Schrödinger equation, and coincides with φ(x) at t = 0. At this instant, the graph of ψ(x) is shown on the Graphics: [x] tab; here we can visually confirm and measure the packet width with the help of the Trace feature: right click on the graph and select Trace from the popup menu, then determine the extent of the wave packet by subtracting the coordinates of some identical feature on either side of the central peak. The assignment of width is somewhat arbitrary, since the wave never truly falls to zero. The important point is that using any sensible definition, the packet width is proportional to w. For |x| = w, the wave has fallen to exp(–1/4) = 0.7788... of its peak value. |
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Next observe the evolution of this wave packet
by clicking the Play button The waveform steadily spreads (disperses) while the peak remains fixed in place. The changing colors signal a complex-valued wavefunction. The phase of a complex number (the function value) is represented here by color which, like phase angle, repeats with a definite period. [The 'colorwheel' follows the rainbow from red to green to violet, then back to red again through magenta.] The theory for this case shows that the wave profile remains gaussian but acquires a [space-varying] phase and an ever-increasing width. As the packet spreads, the peak height diminishes so as to conserve total probability. Note that the waveform continues to spread well beyond the field of view: QMTools is able to simulate an infinite domain by applying transparent boundary conditions at the endpoints of a finite interval [see Am. J. Phys. 72, 351 (2004)]. Time is measured here in femtoseconds (fs); 1 fs = 10–15 s. Use the applet to estimate how long it takes for the initial waveform to spread uniformly throughout the entire visible range (extending from –10 Å to +10 Å). What are the consequences of this result in terms of locating the electron? Besides dispersing, a free-particle wave packet also translates with a speed equal to the group velocity of its de Broglie wave consituents. Since the waveform in the preceding applet remains centered at the origin, it cannot describe an electron with a nonzero (average) velocity. The applet below remedies this deficiency by assigning the initial wave with an overall multiplier of the form exp(iax), having real and imaginary parts cos(ax) and sin(ax), respectively. The speed of this packet is proportional to the wavenumber a (= 0.5 Å–1); in fact, the average momentum of the electron described by this packet can be shown to be <p> = ћa = (1.973 keV·Å/c)(0.5 Å–1) = 0.9865 keV/c. With the extra phase factor for nonzero momentum, the initial wave is already complex-valued, hence the colorful display below even at time t = 0. |
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Once again observe the evolution of this
wave packet by clicking the Play button |
on the bottom panel of
the applet. Complete control over the
developing waveform is afforded by the remaining
buttons on the panel: Stop 
, Reverse 
, Restart 
, Step Ahead 
, and Step
Back 
.
[Also, right-clicking on the panel allows for
adjusting the frame rate, the number of
'clock ticks' between screen redraws, and the
elapsed time between 'ticks'; all affect the
speed of animation and can be tweaked (the first
two with the 'clock' running) to achieve the most
pleasing visual effect.]|
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A wave packet with even a modest degree of localization Δx is made up of many de Broglie waves taken from a relatively broad range of wavenumbers Δk. Constructing a wave packet from individual de Broglie waves is an instance of a technique known to mathematicians as Fourier transformation. The Fourier transform superposes (adds) individual plane waves of the form exp(ikx) with differing wavenumbers k and amplitudes g(k) to form a new function f(x). The profile f(x) is fixed by the choice of mixing coefficients g(k). In mathematical parlance, f(x) and g(k) constitute a Fourier transform pair, each one being the Fourier transform of the other. The theory of the Fourier transform shows that the widths of a function and its Fourier transform are inversely related, i.e., if Δx is the width of f(x) and Δk is the width of g(k), then Δx·Δk ~ 1. The applet below can be used to confirm this reciprocal relationship for the case of a gaussian waveform. The graph on the Graphics: [x] tab shows the initial wavefunction whose functional form appears in the Equation View as ψ(x); it is again a gaussian multiplied by exp(iax) and centered at x = 0, but now with reduced width w = 0.5 Å. The Fourier transform of ψ(x) is displayed on the current tab, labeled Graphics: [k], and in designated in the Equation View as ψ~(k). The appearance of the transform is unmistakably similar to the original function ψ(x); in fact, the Fourier transform of a gaussian is another gaussian! The wavefunction and its Fourier transform can be viewed simultaneously as follows: from the Graphics: [x] tab, right click anywhere in the graph and select "Display in Window" from the popup menu. Reposition | resize the new window as desired. Repeat the procedure from the Graphics: [k] tab to create a second window for the transform function. Both the function and its transform are now visible, with the axis labels clearly distinguishing the two members of this transform pair. |
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Now lets change the width of ψ(x) to see what effect this has on its Fourier transform. Switch over to Equation View and right click on the entry for w, then select "Edit parameter..." from the popup menu. Use the slider to set different values for the width (Δx) of ψ(x), and observe the effect on the width (Δk) of its transform. [The slider affects only the highlighted digits; the first digit highlighted can be moved left (right) using the up (down) arrows to the right of this field. Extra digits can be added before or after the decimal by typing directly in the text field.] Interpret your observations in terms of the reciprocity of the Fourier transform. The momentum of an individual de Broglie wave is p = ћk; it follows that a spread in wavenumbers Δk translates into an uncertainty in particle momentum Δp = ћΔk. In this way the reciprocity of the Fourier transform (Δx·Δk ~ 1) leads directly to the position-momentum Uncertainty Principle discovered by Heisenberg, Δx·Δp ~ ћ. The wavenumber a of the original wave packet also can be adjusted using the same technique used to change the width. Try different values for a to see its effect on the wavefunction and its Fourier transform. Finally, restore the original wavenumber
(a = 0.5 Å–1) and width
(w = 0.5 Å) and observe the evolution of
the Fourier transformed wave (click the Play
button |
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Gaussian wavefunctions lend themselves to studies of wave propagation because of their appealing mathematical properties, but do those special features skew our results? To put it another way, do the general conclusions reached above for a gaussian wave packet remain valid even for non-gaussian packets? To explore this question, the applet below prescribes an initial wave having a non-gaussian profile. As before, the formula for this new profile appears in the Equation View as φ(x), ψ(x) is the result of propagating the wave forward in time, and the waveform itself is plotted at t = 0 on the Graphics: [x] tab. Like its gaussian predecessor, this initial wave exhibits a single peak at x = 0, has wavenumber a = 0.5 Å–1 and width w = 0.5 Å. But the similarities end there; the shape is noticeably different and the peak, especially, is 'flattened'. Furthermore, the Fourier transform of ψ(x), displayed on the Graphics: [k] tab, is decidedly unlike the original wave. |
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Follow the preceding instructions to display the wavefunction and its Fourier transform in separate windows for easy comparison. Check the reciprocity rule for this case by changing the width of the initial wave while observing its Fourier transform. Is reciprocity valid for this transform pair? Also examine the propagation of the new waveform. Do you see any significant differences between this evolution and gaussian evolution? |
