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It is now possible to associate distinct kinetic and potential energies on the left-hand side of equation (8.7), with the components Ex and Ey>z on the right-hand side, thus giving two decoupled equations, as follows:

A (m n = m p n = p) .

(7.15)

+il>(y,z)V(xty(x)+il>(x)V(y,z)il>(y,z) (y,z) = (Ex+Ey^(x)^(y,z) (8.7)

The elementary remainder computation algorithm, developed in Section 15.2, repeatedly subtracts Q from an approximation to the remainder. The efficiency can be substantially improved if, for some positive number m, Qxra is subtracted, in one go, where ra is as large as possible. Long division, which has been designed for efficient manual calculation of remainders and quotients, does just that. Figure 15.2(a) shows the computation by long division of the remainder 6 and quotient 521 on dividing 3653 by 7. (The presentation is in the format used in British schools. Other nations may use a different format.)

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In the above ip(y,z) is not acted upon by any operator in the first equation, and similarly for if}(x) in the second equation, and thus they can be divided out. In addition, as mentioned above, the potential component along the axis of the wire V(x) = 0, thus giving the final decoupled equations of motion as follows:

7)3653

For some NN types, for example the LVQ, input data is preferred to be normalized to vectors of unit length. The values Zi)P of each input parameter Zi are then normalized using (7.16) The normalization above loses information on the absolute magnitude of the input parameters, since it requires the length of all input vectors (patterns) to be the

Clearly, the first of these equations is satisfied by a plane wave of the form exp (ikxx}, thus giving the standard dispersion relationship:

35 15 14 13 7

The second of these equations of motion, equation (8.11), is merely the Schrodinger equation for the two-dimensional confinement potential characterising a quantum wire. For a general cross-sectional wire, equation (8.11) should really be solved by using a full two-dimensional solution, which lies beyond the scope of this present work. For the purpose here, special cases of the solution of equation (8.11), for the relevant commonly found geometries, will be illustrated. Such particular solutions rely upon the ability to further decouple the motion into independent components.

LEARNING)

20 500 7)3653 3500 153 140

8.3 INFINITELY DEEP RECTANGULAR WIRES Perhaps the simplest quantum wire geometry would be a rectangular cross-section surrounded by infinite barriers. This is illustrated schematically in Fig. 8.5 and can be considered to be the two-dimensional analogy to the one-dimensional confinement potential of the standard infinitely deep quantum well.

Figure 15.2 Remainder computation in decimal arithmetic (P = 3653, Q = 7 ). (a) Long division; (b) filling in details.

Within the quantum wire, the potential is zero, while outside the wire it is infinite; thus in the latter case the wave function is zero. Hence, the Schrodinger equation is only defined within the wire for the motion in the two confined y- and z-directions, i.e. equation (8.11) becomes:

same. Input patterns with parameter values of different magnitudes are normalized to the same vector, e.g. vectors ( 1,1,2,3) and ( 3,3,6,9). Z-axis normalization is an alternative approach which preserves the absolute magnitude information of input patterns. Before the normalization step, input values are scaled to the range [ 1,1]. Input values are then normalized using

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