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For convenience we will omit the band index n in the following treatment.

The upper signs correspond to the interior region of quantum dot and lower signs correspond to the exterior. The solutions of these equations are the envelope functions in A and B materials and can be written as fA,B(p) = { fA(p)Yi,m(0, ç) = aji(ttAr)Yi m(Q, ç)

where jl(aAr) is the spherical Bessel function of the first kind, k(aBr) is the modified spherical Bessel function of the third kind, and a, b are constants which are determined with the normalization condition of the wavefunction. The new boundary condition in the r = r0 interface of quantum dot can be expressed as aj i(aAr0) = bk ¡(a11^) and

Then, the eigenstates are determined solving the following transcendental equation:

The energy states en l m of the system are characterized by three quantum numbers, namely, the principal number n, the orbital number , and the magnetic number m. The orbital quantum number determines the angular momentum value L:

The magnetic quantum number determines the Lz component parallel to the z axis:

Every state with a certain l value is (2 l + 1) degenerate according to 2 l + 1 values of m. The states corresponding to different l values are usually denoted as s, d, /, and g states and so forth in alphabetical order. For example, states with zero angular momentum (l = 0) are referred to as s states, those with l = 1 are denoted as p states, and so on. The parity of states corresponds to the parity of the l value, because the radial function is not sensitive to inversion (r remains the same after inversion) and spherical function after inversion transforms as follows:

The specific values of energy are determined by the VA'B(r) function. Consider a simple case corresponding to one spherically symmetric potential well with an infinite barrier; that is,

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