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where Xn l are roots of the spherical Bessel functions with n being the number of the root and l being the order of the function.

This formulation can be adapted to the case of a rectangular quantum box where the envelope function can be written as

and thus we obtain three equations in order to calculate the X, Y, and Z new envelope functions for each direction,

h2 d2Y A'B(y) 2mA,B(y) dy2 = en,yYnA,B(y) h2 d2ZfB(z)

which represent three uncoupled equations corresponding to three isolated single quantum wells. If we consider that the dimensions of the rectangular quantum box are a, b, and c, according to x, y, and 2 directions, respectively, then, taking into account the infinite barrier potential between A and B materials, we obtain the solutions

where l, m, and p are quantum numbers taking positive nonzero integers.

In this case energy values are expressed as follows:

^ _ n2h2 ¡2 n2 h2 2 n2h2 2 '¡■m'p = 2m*A(x)a2 + 2m*A(y)b2 m + 2m* (z)c2p

The theory of the quasi-particle states used previously to study the nanostructures is suitable for quasi-particles like electrons and holes; however, in order to be used for the excitons we need to make some modifications.

2.6. Idealized Confined Systems when Symmetry of the Confinement Region Does Not Have Regular Form and the Schrödinger Equation is Nonseparable

When the Schrödinger equation for a given quantum system is nonseparable in nature, then some approximation techniques must be used to study its solution. In a similar situation for a confined quantum system, the variational method might constitute an economical and physically appealing approach. It has been shown that the variational method is useful for studying this class of systems when their symmetries are compatible with those of the confining boundaries.

We will only consider the application of the variational method in the case of a symmetric quantum system confined by impenetrable potentials, as a simple example of a more general situation.

Let us assume a quantum system confined within a domain D, for which the Hamiltonian can be written as

where H0 is the Hamiltonian for the free system, and

is the confining potential.

The solution of Eq. (96) must satisfy ft(q) = 0 q e 3D

2.6.1. General Description of the Method

For the moment, let us assume that, for a given system, the time-independent Schrodinger equation can be written as

where H = -(1/2)V2 + V(q) is the associated Hamiltonian for this system, {q} is the set of coordinates on which V depends, and h = m = 1.

An approximate solution of Eq. (91) may be obtained by replacing by a trial wavefunction xn, which possesses a similar behavior at the "origin" as well as asymptotically at infinity. An estimate of the energy of the system is then obtained by minimizing the functional

with the additional restriction that lin / x„hx, ndr = 8n,n>

As is shown in any quantum mechanics textbook, one can find that en = min

That is, a poor guess of xn leads to a poor estimate of en, which means that we have not properly included our qualitative knowledge of the system under study.

In practical calculations, the functions xn depend upon the coordinates, say {q1' q2 '...'qs}, and on a set of unknown parameters {a-^a^'...'ak} such that en = en{a1' a2'...' ak}. In order to determine the set {ak} one must solve the system of equations

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