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At the minimum separation, the wave functions reveal a change in character the lowest energy state switches from being localised in the left-hand well to the righthand well, and vice versa for the higher energy level. This phenomenon is referred to as an anti-crossing. 3.14 SYMMETRY AND SELECTION RULES Overlap integrals can be a very quick way of understanding or explaining complex interactions and are often instrumental in determining selection rules for carrier scattering events. The simplest overlap integral is defined as follows:

Use the conditional rule, skip rule and assignment axiom to verify that this program segment is indeed correct. n

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take corrective action by providing a set of inputs that ensures the desired response. As illustrated in Figure 20.1, a fuzzy controller consists of four main components, which are integral to the operation of the controller: Fuzzy rule base: The rule base, or knowledge base, contains the fuzzy rules that represent the knowledge and experience of a human expert of the system. These rules express a nonlinear control strategy for the system. While rules are usually obtained from human experts, and are static, strategies have been developed that adapt, or refine rules through learning using neural networks or evolutionary computing [Favilla et al. 1993, Wang and Mendel 1992]. Condition interface (fuzzifier): The fuzzifier receives the actual outputs of the system, and transforms these non-fuzzy values into membership degrees to the corresponding fuzzy sets. In addition to the system outputs, the fuzzification of input values to the system also occurs via the condition interface. Action interface (defuzzifier): The action interface defuzzifies the outcome of the inference engine to produce a non-fuzzy value to represent the actual control function to be applied to the system. Inference engine: The inference engine performs inferencing upon fuzzified inputs to produce a fuzzy output (refer to Section 19.2). As stated earlier, a fuzzy controller is basically a nonlinear control function. The nonlinearity in fuzzy controllers is caused by the fuzzification process, if nonlinear membership functions are used; the rule base, since rules express a nonlinear control strategy; the inference engine, if, for example, the min-operator is used for intersection and the max-operator is used for union; and the defuzzification process.

If un and un are eigenstates of the same Hamiltonian, whether they be wave or envelope functions, then they are orthonormal, i.e.

1 30

Hence, any scattering event which involves simply O will follow this selection rule. As will be encountered later in 9, the relative strength of an interband electron-hole recombination (both excitonic and free-carrier) is proportional to O, even though one of the eigenstates corresponds to a conduction-band electron and the other to a valence-band hole; under the envelope function approximation, they are eigenstates of different Hamiltonians. For symmetric potentials, such as a single quantum well, the states have exact parity, and thus O is non-zero, so corresponding to an allowed transition, for An=0, 2, etc. Electron electron scattering will also be discussed in 9, this is a scattering event between two initial states, resulting in two new states. The rate therefore contains an overlap between all four wave functions, as follows:

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