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The latter restriction can be removed and a much more general choice of the donor wave function u can be made. The motivation being generality, the problem is recast in a form which is suitable for numerical solution as described in 3, thus making it applicable to any quantum well structure, be it a double quantum well, a diffused quantum well, a graded gap quantum well, etc. With this aim, the trial wave function to be employed here is influenced by the above, but for now will be written as follows: The one-dimensional envelope function x(z) is yet to be determined and the function (z, y, z rd) is expected to be a hydrogenic type factor which is a function of the spatial coordinates and one or more variational parameters. It can be seen that forming the Schrodinger equation with the Hamiltonian of equation (5.4) and the trial wave function w of equation (5.7) will lead to a term V 2 W ; it is worthwhile to derive this now. Consider:

Exercise 11.19. Derive (11.12) from the one-point rule (11.9), the nesting rule (11.6) and the rearrangement rule (11.7). Hint: your derivation should head for using the fact that / is a bijection, i.e. that there is a function g such that for all je/ and

Hence where the subscript z implies derivatives with respect to z only. Then, forming the Schrodinger equation with the Hamiltonian of equation (5.4) gives the following:

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Artificial neural networks have been developed to model the pattern association ability of the human brain. These networks are referred to as associative memory NNs. Associative memory NNs are usually two-layer NNs, where the objective is to adjust the weights such that the network can store a set of pattern associations - without any external help from a teacher. The development of these associative memory NNs is mainly inspired from studies of the visual and auditory cortex of mammalian organisms, such as the bat. These artificial NNs are based on the fact that parts of the brain are organized such that different sensory inputs are represented by topologically ordered computational maps. The networks form a topographic map of the input patterns, where the coordinates of the neurons correspond to intrinsic features of the input patterns. An additional feature modeled with associative memory NNs is to preserve old information as new information becomes available. In contrast, supervised learning NNs have to retrain on all the information when new data becomes available; if not, supervised networks tend to focus on the new information, forgetting what the network has learned already. Unsupervised learning NNs are functions which map an input pattern to an associated target pattern, i.e. FNN : M7 -> R* (4.1) as illustrated in Figure 4.1. The single weight matrix determines the mapping from the input vector z to the output vector o.

= g.

where E is the total energy of the system. The donor binding energy ED0 is equal to the difference between this and the standard confinement energy of the electron in the heterostructure without the donor present, i.e. using the notation of 2, the ground state energy for the electron in the well without the donor would be E1, and hence

Summation is just one example of the quantifiers we want to consider. Readers already familiar with the 0 notation for continued multiplications will probably have no difficulty rewriting each of the properties of summation into a form that is applicable to multiplication. In general, it is meaningful to 'quantify' with respect to any binary operator that is associative and symmetric. As mentioned earlier, addition, multiplication, equivalence, inequivalence, minimum, maximum, conjunction, disjunction, highest common factor, and least common multiple are all examples of associative and symmetric operators and, in each case, it is meaningful (and useful) to consider the operator applied to a number of values rather than just a pair of values. Two quantifications that are particularly important in program specification are so-called universal quantification and existential quantification. Universal quantification extends conjunction to a set of booleans of arbitrary size. Just as for summation, there is a widely accepted symbol denoting universal quantification, namely the 'V ('for all') symbol. The notation ( \ / k : R : T } means the logical 'and' ('A') of all values of the boolean expression T determined by assigning to dummy k all values in the range R. In words, it reads for all (V For example, k k in the range : R it is the case that T R : T)

In order to proceed further, it is necessary to be more specific about the form of the hydrogenic factor . In a free hydrogen atom the ground state wave function would be spherically symmetric and given by:

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