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Next consider G(a) as defined in equation (6.34), then:
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and so performing the differentiation and substituting r' for rj_ as above, then:
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Now let
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cosh 9, then:
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Making the further substitution, w = exp ( 0], then dO = dw/w and noting that:
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An analogous argument shows that G(a) appearing in B is exactly equal to this form. Next consider evaluation of the integral J(a) as defined in equation (6.40). With this aim, note that: This then gives the following:
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Therefore, moving from Cartesian into plane polar coordinates:
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It is standard practice in the literature [144] to expand the expressions involving (r') ~n as a power series in rT and a, and then to perform the integration numerically. This involves summing over a series of terms, each of which must be integrated over a range from 0 to 8; this procedure, however, can be avoided. Writing equation (6.63) as J(a) = J1 + J2 + J3 + J4, where Ji represents the first, second, etc, terms, respectively, then:
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Again, substituting r' for rT:
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The two remaining terms of equation (6.63) give:
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Again substituting r' for rT, then:
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Making a further scale change of
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With the final substitution of w = exp 0, then d0 = dw/w, therefore:
This last equation illustrates the advantage of the present formalism, namely that the computationally difficult integral of equation (6.63), which has hitherto been expanded into a infinite series and integrated to infinity, has been replaced with a simple integral over the range from 0 to 1. Even if the integrand had a finite number of singularities, this would still pose no problem in its evaluation. Finally consider K(a), as defined in equation (6.42), i.e.
Recalling that r2 = r2T_ + a2, then:
The form of r', i.e. r'2 = r2 B 2 a 2 , suggests the substitution r = Ba cosh 0, which gives:
Again making use of the substitution w exp 0, this then neccesitates evaluating w corresponding to 9 = cosh 1 i.e. cosh 9 = = (w+ ); this then yields the quadratic equation: Since the product of the two roots of this equation is unity, one root must correspond to exp ( 0 ) and the other to exp 0. It is readily ascertained that the exp ( 0 ) root is as follows: This follows since the limits on B are 0 and 1. Hence:
In a similar manner, the second form of ^v, with , gives the same expressions for F(a), G(a) and J(a) as above, but with the simple substitution,
1 + r 2, in place of I ft2. Only the expression for the last of the 'a' functions differs, in particular:
6.5 THE TWO-DIMENSIONAL AND THREE-DIMENSIONAL LIMITS It is always worthwhile performing convergence tests, i.e. taking the theoreticalcomputational model to established, often analytical, limits. The idea is to increase confidence in the theory and, as ever, to demonstrate that the previous theories are limits of the new. For example, classical mechanics is recovered from relativistic mechanics, in the low-velocity limit. Although not as grand an example, there exist two limits which the analysis above can be compared with. In the limit of very wide quantum wells, the exciton should look like a bulk exciton, in both its binding energy and Bohr radius. In addition in the limit of very narrow wells, the exciton should become two-dimensional in nature. The bulk, or three-dimensional limiting case of hydrogenic two-body systems, such as impurities and excitons, has been discussed and used already. A transparent treatise of this Bohr model of the hydrogen atom is given by Weidner and Sells [4]. This approach can easily be adapted to the case of an electron orbiting a positively charged central infinite mass, with the orbit restricted to a single plane, i.e. it is two-dimensional (2D). The Schrodinger equation is then written as: