R. Wayne Ohline Mexico Institute of Mining and Technology Socorro, New Mexico 87801 New
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Analog Computation for a Particle in P Finite Square Potential Well
Quantum mechanics continues to be one of the more fundamental but difficult topics in the undergraduate curriculum. The approaches developed by Corrin for the particle in the infinite square well (1) and by Tabbutt for the finite well (3,3) using the analog computer seem to be effective in demonstrating the properties of the solutions of these two quantum problems. More recently, Shirer (4) has published some interesting comments on computer solutions to these quantum problems. We have also used an analog computer as a teaching aid, and have noted some variations and modifications in the instrumental approach which may be employed with the particle in a finite square well problem. These modifications may be useful to others working in the field. NC relay Contact
M
'Horizontal' input
Figure 2. Relay driving d r d t for potential r o l l contact. The relay is operated at ih given rding of 175 ma at 6 V (all resistors in megohmd.
of ~ s ~ i l l o s c ~Orp e X-Y recorder
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element. Before the switching potential is reached by the first amplifier, the output of the second amplifier is near ground potential. When the output of the ramp-generating amplifier slightly exceeds the switching potential, the output of the second amplifier swings full negative, activating the relay. Since all initial condition contacts open simultaneously, the wall is Normally jl[l produced a t the same time relative to the start of each %itching openrelay I computation. contact potential The circuit used to power the relay is a Darlington type amplifier (5). This configuration is equivalent to a single transistor having a current gain equal to the product of the current gain of each transistor. If the Figure 1. Schematicof analog pmgrom for the porlisle in o finite square transistors shown in Figure 2 are not readily available, well (all reridon in mogohmr). some experimentation with other types will probably produce a useful relay driving circuit. Figure 1 shows the analog program. The program is Because of appreciable quantum mechanical tunnelessentially the same as that used by Tahbutt, except E approaches U , it is helpful to display the ing when that a quarter-square multiplier is not used, and an additional coefficient potentiometer, representing U(,,, wall position on the readout device, especially when fast repetitive computation is desired. This can be has been added. Since the computation is most easily easily accomplished in a four pole, double throw relay started in the center of the box, the program as shown is used for the potential wall contact. Figure 3 shows yields the antisymmetric or "odd" wave functions. To obtain the symmetric (even) functions, the initial conditions of the first two integrators are interchanged. That is, at zero time, the output of the second integrator should be a t a maximum (equal to the wave amplitude initial condition) when the output of the first integrator is zero. From the form of the differential equation it is evident that RI CI is set equal to Rz C2. The electronic "wall creating" device used in this work is shown in Figure 2. A voltage increasing linearly with time is generated by the first amplifier. The output of this amplifier is fed into the "horizontal" =20r input of a dc oscilloscope or X-Y recorder and to a Figwe 3. W a l l marking circuit wing DPDT relay. second amplifier equipped with a diode as the feedback L-
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Volume 47, Number 9, September 1970
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how such a relay can be used to provide a "pip'' on the oscilloscope screen as the potential wall contact is closed. During the time the relay is switching, a twenty volt signal is added to $. If a transistorized computer is used, the potential across the relay contacts should probably be reduced to something less than five volts. Use of Atomic Units
It is convenient to use atomic units (6) with analog solutions of the wave equation. I n these units, the unit length is 0.529 A and the unit of energy is 4.36 X 10-1' erg. For example, the student may wish to verify that when E is greater than U , the wavelength is identical with the de Broglie wavelength
Expressing the momentum in terms of computer parameters, one finds
where
Figure 4. Behavior of m m p u t e r when showing wove function at eigenvalue n = 4. The well half width is 6.0 sec (RC = unity),U
