At first the network for the one-dimensional wave equation for a single particle in Cartesian coordinates is developed in detail, then the general case. A companion paper contains results of a study made on an a.c. network analyzer of one-dimensional problems: a potential well, a double barrier, the harmonic oscillator, and the rigid rotator. The curves show good agreement, within the accuracy of the instruments, with the known eigenvalues, eigenfunctions, and "tunnel" effects.

In Cartesian coordinates, the equation is

Three types of equivalent circuits may be established.

In the diagrams to follow, unless otherwise stated, the inductors (whose reactance at the fixed frequency is denoted by X

The basic network concepts will be introduced in detail in connection with the first type of circuit (with additional comments on the second type of interpretation). The third type of model, although it offers attractive analogies, will be only cursorily treated.

If the wave equation is multiplied by ∆x and the operator p = - i ħ∂ /∂x is introduced, the equation becomes

The three energy operators will be represented in the following manner:

The sum of the two operators T and V forming the Hamiltonian operator H is represented by the interconnected network of Fig. 2a. That is, a

Finally, the sum of the Hamiltonian H and the total energy -E operators is the resultant network of Fig. 2b extending from - ∞ to + ∞.

The wave function ψ will be assumed to be represented (Fig. 3) by the differences of potential

The differences of potentials appearing between two junctions are ∆ψ = (∂ψ/ ∂x )∆x + ... where higher order terms in the Taylor series development are neglected.

The result of the operation αψ or βψ (where α and α are operators) is represented by

∆(β∆ψ) = (ħ

Since E∆xψ represents the currents flowing in the capacitors and H∆xψ = (T+ V)∆xψ those in the inductors, the equation H∆xψ = E∆xψ simply states Kirchhoff’s second law, that at each of the junctions of four coils the currents flowing into the positive resistors (inductors) are equal to the currents coming from the negative resistors (capacitors).

Both the voltages and currents appear as standing waves. When the space distribution of ψ is such that it attenuates, the network may be terminated at any point by an equivalent impedance. When ψ does not attenuate the circuit may be terminated by a short circuit at any point of zero voltage, or by an open circuit at any point of zero current. These points are easily determined on the a.c. network analyzer by trial.

It should be noted that the equivalent circuit introduces an indeterminacy between quantities measured in the vertical coils and those in the horizontal coils. If, for instance, the vertical voltage ψ is known at a certain value of x, say x

Let it be assumed as an example that the V function is a potential well (Fig. 4a). Then in the network the corresponding positive resistances assume either a constant or a zero value as shown in Fig. 4b. In practice it is sufficient to extend the network to, say, twice the width of the potential well on either side.

Let now a d.c. (or a.c.) generator be inserted anywhere in the network parallel with one of the negative resistances (inductors), as shown. If the values of all the negative resistances -E∆x (capacitors) are simultaneously varied by the same amount, it will be found that the current (reactive current) in the generator varies and at some value of E∆x becomes zero.

It should be noted that while a current (reactive current) flows in the generator the circuit does not satisfy the differential equation, since at

Now a value E of the negative resistances, at which the generator current becomes zero, represents a state at which the circuit is self-supporting and has a continuous existence of its own without the presence of the generator, as the negative resistances just supply the energy consumed by the positive resistances. (If the circuit contains inductors and capacitors, the circuit is a resonant circuit and it oscillates at its basic frequency.) E is then an eigenvalue E

When the generator current is positive the circuit draws energy from the source, and when the current is negative the circuit pumps back energy into the source. At zero generator current the circuit neither gives nor takes energy, and theoretically the generator may be removed. All values of E at which the current crosses the axes and becomes zero are eigenvalues of the equation and the corresponding voltage distribution curves are eigenfunctions. When the energy level E overflows the well, the discrete spectrum of eigenvalues changes into a continuous spectrum and the generator current is zero at all greater values of E.

When the energy E changes sign, the negative resistances become positive resistances and at no value of -E may the circuit be self-supporting (as it contains only positive resistances). That is, the equation has no negative eigenvalues.

To bring the measurements at the different energy levels to the same base, it is necessary to normalize the measured ψ values so that the new values of ψ satisfy the equation

The measured ψ functions are normalized by plotting the square of ψ. If the area under the curve is 1/N

The statistical mean of an operator α for a state ψ is defined as

Since (a∆x)ψ is a current (reactive current) flowing through an admittance, ψ*(a∆x)ψ is the power (reactive power) in a single admittance. Hence the total power in a complete set of similar admittances, namely,

represents the statistical mean of the corresponding energy operator. That is:

That is, the total power in all positive resistors is the same as the total power in all the negative resistors, or

Let the wave equation be divided by iω

In the present case:

The operand ψ is again represented by voltages and the result of the operation αψ by the same currents as in the first model. Their variation in time now is sinusoidal, with 2 π f

Instead of varying the magnitude of the capacitors, now the frequency of the generator is varied, thereby varying the admittance of the capacitors, ħω

As the currents in the horizontal inductors are (ħ

One slight disadvantage of this third model is that as the energy E changes sign, the reactance of the capacitor jω

An interesting special case occurs when the potential V is zero everywhere. The one-dimensional equivalent circuit of such a free particle is a conventional transmission line extending to infinity in both directions (Fig. 6) in which the series inductance is 2m ∆x/ ħ

It is well known that such a transmission line may maintain a standing wave at any frequency ψ=ω

The three-dimensional Schrödinger equation for a single particle is

where

In order to establish a physical model for it, it is necessary to change it to a tensor density equation.

The above equation in orthogonal curvilinear coordinates may be changed to a tensor density form by multiplying it by h

If the equation is multiplied through by ∆u

The corresponding equivalent circuit is shown in Fig. 7. For a free particle (V=0) it represents a generalization of the conventional one-dimen- sional transmission line to three dimensions.

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Copyright © Aug. 27, 2003 by U. Anders, Ph.D.

e-mail Udo Anders : udo39@t-online.de

Copyright © Aug. 27, 2003 by U. Anders, Ph.D.

e-mail Udo Anders : udo39@t-online.de

Last updated : Sept. 5, 2003 - 22:44 CET