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Just as for summation, we can enumerate a list of rules that govern the algebraic properties of universal and existential quantification. The rules have much the same shape. In this section, we list the rules for universal quantification. Only the splitting rule differs in a non-trivial way from the rules for summation. The side conditions on application of the rules will not be repeated for individual rules. As a reminder, here, once more, is the statement of the condition. Side Condition. The application of a rule is invalid if it results in the capture of free variables or release of bound variables, or it results in a variable occurring more than once in a list of dummies. The rules governing the dummies are identical to the rules for summation except for the change of quantifier. The side conditions concerning capture of free variables and/or release of bound variables remain as before. [Dummy Renaming] (Vj:R:T} = (Vk:R[j := k]:T[j := k]} . [Nesting] (Vj,js:RAS:T) = ( V j : R : ( V j s : S : T } ) . [Rearranging! ( V j , k : R : T } = ( V k , j : R : T ) . (11.20) (11.21) (11.22)

Remembering that the integrals I 1 ,I 2 , I3, and I4 are just real numbers then equation (5.23) is just a linear second-order differential equation, very much like the one met in 3. The energy E can be solved for any choice of X by expanding the derivatives in finite differences and forming an iterative shooting algorithm as before. First though for simplicity sake, it is better to simplify equation (5.23), and rewrite it as: Then, by using the finite difference expansions:

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The rules governing the range are obtained by replacing the quantifier ' ' by 'V, replacing V by V and replacing 0 (the unit of addition) by true (the unit of conjunction). The proviso on the one-point rule (e contains no occurrences of 'k') still applies. [Empty Range] ( Vfc: false :T) = true . [One-Point] (Vk:k = e : T ) = T [ k : = e ] . [Splitting] (Vk:P:T) A ( V f c : Q : T ) - ( V k : P v Q : T ) . (11.23) (11.24) (11.25)

(4-4)

Finally, gathering the terms in x(z + $z) on the left-hand side gives:

Only-if:

which is an iterative shooting equation and can be solved subject to the standard boundary conditions, x(z) > oo as z > 00. In practice, this involves choosing exponential-growth starting conditions deep inside the quantum barrier and then progressively calculating x(z) at points along the z-axis across the structure. The energy E is varied until the function x(z) satisfies the standard boundary condition, i.e. it tends to zero at the other end of the structure. For more details, see again 3. The variational aspect of the calculation arises from the value of the unknown constant A. This is varied systematically with the aim of minimising the total energy. The system coordinates are therefore represented by the minimum energy and the corresponding value of the Bohr radius A. The derivation thus far is independent of the choice of the form of the hydrogenic exponential term, exp ( r"/A), and this summarizes the versatility of this approach. In the following sections, the various choices that can be made for this 'orbital' term will be investigated. In addition, the power of this method is represented by the generality of the conduction-band-edge potential V(z). No assumptions have been made relating to the form of this function and hence the method is applicable to all forms of V(z), i.e. the formalism can be applied to any heterostructure.

Substitution: Leibniz:

where a is a positive constant, or equivalently, uki(t - 1)]

5.3 TWO-DIMENSIONAL TRIAL WAVE FUNCTION The simplest choice that can be made for the hydrogenic factor of the trial wave function for an electron confined in a quantum well structure in the presence of a positively charged donor, is one which is dependent only upon an in-plane co-ordinate, say, for example, as follows:

(e = f) A E[x := e] = (e = f ) / \ E[x := f] .

Therefore:

(e = f) = (e = f) A (E[x := e] =E[x := /]) .

(4.5)

This choice replaces the spherical symmetry perhaps expected, a priori, with circular symmetry and is often referred to as the two-dimensional (2D) form. These are an unfortunate choice of words as in fact has cylindrical symmetry, and hence is infinitely extended along the z-axis. This is another example of the word 'dimension' being interchanged with 'coordinate'. Given this specific form for the spatial coordinate r", then it is possible to calculate the integrals Ij explicitly, and hence all of the terms in the Schrodinger equation , (equation (5.23)) and the corresponding iterative shooting equation (5.28) will be known. Consider first:

The following subsections enumerate a number of theorems. That is, all the properties can be derived from the above axioms. The names given to the theorems are taken (with a few exceptions) from Cries and Schneider (1993) to which reference should be made for a very full and clear discussion of the principles with many more examples.

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