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Looking upon these results from a computational viewpoint, it can then be seen that this 3D trial wave function has an immediate advantage over the previous, seemingly simpler, 2D case, namely that all of the integrals I1, I2, I3 and I4 have analytical expressions. Indeed, evaluation of the integrals and the minimisation of the energy is computationally much less demanding than previously. Figure 5.10 shows the change in the total energy E of the electron between the previous 2D trial wave function and the more complex 3D case presented in this section. It is clear that the energy E is lower for all donor positions across this (a typical) quantum well. The difference between the two trial wave functions is smallest when the separation between the donor and the electron is larger, i.e. when the donor is deep in the barrier (rd = 0 A) and the electron is, as always, centred in the well (z = 230 A). Thus it might be concluded that the 2D wave function is a reasonable approximation when the donor is in the barrier, although as the graph shows, for donors in the well, considerably lower energies can be obtained by using a spherical hydrogenic term. Recalling the variational principle, then the lower energies obtained imply that the 3D approximation to the wave function is a more accurate representation than the 2D case. When coupled together with the computational advantage, as mentioned above, then the argument in favour of the 3D trial wave function is clear. Figure 5.11 displays the total wave function, u = x(z)e r/t for the range of donor positions across the quantum well. It can be seen that the wave function u

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One: Every MEL of this class (or a derived class) must play this role exactly once For example, every Comment must be embedded in an IssueStateModel ZeroOne: A MEL of this class (or a derived class) may play this role no more than once This is equivalent to saying that it may be linked via this relationship with zero or one MEL ZeroMany: A MEL of this class (or a derived class) may play this role any number of times For example, each Comment can refer to any number of StateElements OneMany: Every MEL of this class (or a derived class) must play this role at least once

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Figure 5.14 Magnitude of the donor binding energy EDo for donors at the centre of GaAs wells of large width, surrounded by Ga 0.9 Al 0.1 As barriers

References We created a reference relationship earlier References are not constrained like embeddings In general, links from reference relationships can refer from any MEL to any other MEL Indeed, because links are themselves MELs, you can create links between links in order to create complex graphs of objects Bear in mind, however, that a primarily diagrammatic user interface will be used to interact with the graph at DSL runtime, so it is important to keep the design simple enough to understand through such an interface Note, however, that although it is possible to create referencing links between any kinds of MELs or links, embedding links may not target links The next relationship we create is called IssueStateTransition, which goes from IssueState to itself Figure 3-15 shows the detail of this relationship Instances of this relationship are links that represent transitions that connect predecessor states to successor states The multiplicity of both roles is ZeroMany this is the default for reference relationships This relationship has been given a domain property called Action

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resembles the one-particle wave function u for an electron without a donor present, for donors in the barrier, i.e. rd <160 A. As the donor approaches the electron wave function, i.e. nears the barrier-well interface at z = 200 A, then the electron is drawn r/ distinctly to the left towards the donor. The influence of the hydrogenic factor e r/t can be seen for the donor positions rd = 220 and 230 A within the well. In the 2D case, the total wave function was given by:

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