# Quantum-Mechanical Preliminaries

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## Abstract

The most difficult barrier to understanding semiconductor lasers is, undoubtedly, mastering the necessary quantum mechanics. To develop useful models of semiconductor bands, we need an esoteric form of second-order degenerate perturbation theory, some basic results from angular momentum coupling theory, and a few ideas from group theory. To calculate optical emission and absorption rates, we need a firm understanding of time-dependent perturbation theory and a rudimentary understanding of the harmonic oscillator.

## Keywords

Angular Momentum Schrodinger Equation Angular Momentum Operator Degeneracy Index Perturbation Versus
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## Notes

- 1.The boundary conditions select from
*all*eigenstates of*H*^{(0)}the ones satisfying the appropriate physical constraints.Google Scholar - 2.The method of infinitesimal basis transformations is less widely-known than a related technique called Löwdin perturbation theory. This implicit method is laborious to derive, and works well only for purely degenerate problems; the method breaks down when applied to subspaces with multiple zeroth-order energies. We present it in Appendix A only to provide easier access to the k · p literature on semiconductor bandstructure. The method of infinitesimal basis transformations is superior in almost all applications.Google Scholar
- 3.We follow closely the treatment in Chapter 4 of F. W. Byron and R. W. Fuller,
*Mathematics of Classical and Quantum Physics*(Dover, New York, 1969).zbMATHGoogle Scholar - 4.Technically our notation is not quite correct, since the values of
*n*will not change as we add the perturbating terms. It is actually the*states*that change, and therefore a more accurate notation would be, etc. But this causes other headaches, so we stick with.Google Scholar - 5.As is typical in quantum mechanics, we make no attempt to determine the radius of convergence for these expansions. We simply introduce the notation and hope for the best. Note that even if
*H*depends on λ only to first order, the eigenvalues and eigenvectors may well have infinite-order corrections. Thus (1.17) and (118) make sense with λ playing either role.Google Scholar - 6.Recall that the determinant of a matrix is the product of the eigenvalues.Google Scholar
- 7.To define au operator, it is sufficient to define its matrix elements between a complete set of states.Google Scholar
- 8.To be notationally exact, the last sum should be written as We omit this complication.Google Scholar
- 9.In this section we ignore the electron spin degeneracy: including it usually multiplies the density of states by a factor of 2. In later chapters we will explicitly sum over spin indices when necessary.Google Scholar
- 10.In view of (1.120) and (1.121), wo
*define*the factorial function for arbitrary*z*via*z*! ≡ Γ(*z*— 1). Note this behaves as expected for integral*z*.Google Scholar - 11.In this section we omit the usual superscripts (0) on the zeroth-ordcr Hamiltonian and its solutions. We also label all states with a single discrete index
*i*,*k*, or*f*, which may refer to states in an approximate continuum or to a set of indices, including a degeneracy index. When we apply our results to specific cases we will take the appropriate continuum limit or sum over degeneracies as needed.Google Scholar - 12.As we will see in Chapter 5, perturbations due to electromagnetic radiation take this form.Google Scholar
- 13.We take the parameter ω ≥ 0 to avoid ambigtiity. Negative frequencies are given explicitly by —ω.Google Scholar
- 14.If we integrate (1.149) (1.150) and assume that
*c*_{i}(*t*) and*c*_{f}(*t*) vary much more slowly than the perturbation*V*(r,*t*), the rapidly oscillating terms integrate to zero over long times and we can justify the approximation (1.151)-(1.152). This is equivalent to the perturbation assumption that*V*is small.Google Scholar - l5.This assumption is purely for notatioiial convenience. We can always add the diagonal elements to
*H*, which has the effect of*shifting*all energy levels slightly.Google Scholar - 16.This assumption is tricky to justify. A mathematical argument can be made that the integral in (1.165) “remembers” only times
*t*′ very close to*t*, and thus only*c*_{i}(*t*) contributes. But the real justification is that this approximation predicts exponential decay, matching observations. For more details, see Complement D_{XIII}in*Quantum. Mechanics*, by C. Cohen-Tannoudji, B. Diu, and F. Laloë (Wiley, New York, 1977).Google Scholar - 17.Feynmann diagrams.Google Scholar
- 18.More complete derivations can be found in most quantum mechanics texts. In particular,
*Quantum Mechanics*, by C. Cohen-Tannoudji, B. Diu, and F. Laloë (Wiley, New York, 1977), addresses all the topics in this section.Google Scholar - 19.A
*central potential V*depends only on the magnitude*r*of the coordinate vector r, not the direction.Google Scholar - 20.This fundamental result of angular momentum theory is by no means obvious and is somewhat tedious to derive, but can he found in most quantum mechanics texts.Google Scholar
- 21.The most typical argument is that the wavefunction must be single-valued, forcing
*m*and consequently*l*to be an integer. Additionally, there is a more rigorous construction showing that must have half-integral eigenvalues, and therefore can only have*integral*eigenvalues.Google Scholar - 22.For example, G. Arfken,
*Mathematical Methods for Physicists*, third edition, (Academic Press, New York, 1985).Google Scholar - 23.Since the spin-orbit interaction originates from relativistic effects, it can be derived by expanding the Dirac equation in the weakly relativistic region or by adding relativistic corrections to the Schrödinger equation. See L. I. Schiff.
*Quantum Mechanics*, third edition, (Mcgraw Hill, New York. 1968), for the former approach, and Cohen-Tannoudji, Diu, and Laloë for the latter.Google Scholar - 24.We estimate the strength of an operator
*A*by taking the expectation value in an appropriate eigenstate.Google Scholar - 25.We should remark that other terms of order materialize as relativistic corrections to the Schrödinger equation, and an accurate fine-structure calculation must include them in addition to
*H*_{s.o}.Google Scholar - 26.For an excellent introduction to the (ultimately simple) idea of
*product*basis states, see Cohcn-Tannoudji, Diu, and Laloë, Chapter 2, Section F. To be careful, we should also write the operators in (1.232) to (1.235) as direct products, i.e Here 1_{L}(1_{s}) represents tlie identity operator in orbital (spin) space.Google Scholar - 27.We say that these operators form a
*Complete Set of Commuting Observables*for the direct product basis states, since their action is sufficient to determine the 4 quantum numbers*l, m, s*and m_{s}.Google Scholar - 28.Because of (1.232) and (1.234), all states in the expansion (1.246) must have the same values
*l*and*s*for the resulting states*l,s,j,m*_{j}〉 to satisfy (1.242) and (1.243).Google Scholar - 29.The coefficients in (1.249) assume a particular set of phase conventions. They may be found in Cohen-Tannoudji, et. al., Complement A
_{x}.Google Scholar - 30.A matrix
**A**is said to be unitary if**A**^{-1}=**A**†, where**A**† represents the conjugate transpose or*Hermitian adjoint*of the matrix**A**.Google Scholar - 31.Although (1.257) follows from straightforward algebra, it is cumbersome to show. We can compute the degeneracy more easily by noting that the direct product states are also cigenstates of
*H*^{(0)}. Then since*m*= —*l*,…,*l*and*m*_{s}= —*s*, …,*s*, and since is independent of*m*and*m*_{s}, each level is (2*l*+ l)(2*s*+ l)-fold degenerate.Google Scholar - 32.It is a subtle (and annoying) point that in order to transform til” matrix elements of an operator we must use the tmnspose of the matrix used to transform the basis states. This is easily shown as follows. Suppose we make the transformation, where defines the transformation matrix. Now suppose we have the matrix elements of an operator A in the In) basis, i.c. we know the matrix of numbers. Then we would like to get the matrix elements of A in the la) basis, i.c. wc want to know the matrix of numbers Then it is a simple matter to write the transformation as (rccall (
*n*_{j}α*m*= (α*mn*_{j}*) and therefore.Google Scholar - 33.Note that the
*inversion*operation r → —r can be obtained from a rotation by π and a reflection in the plane normal to the rotation axis.Google Scholar - 34.When we explicitly consider the electron spin we must work with so-called
*double groups*, in which the symmetry operations T are more complicated.Google Scholar - 35.We exclude here the possibility of
*accidental*degeneracies, where functions*not*belonging to a particular irreducible representation are degenerate with those that do belong. Such degeneracies usually reflect a deeper, overlooked, symmetry in the Hamiltonian: genuine accidents are rare.Google Scholar - 36.Complete derivations can be found in most quantum mechanics texts. In particular,
*Quantum Mechanics*, by C. Cohen-Tannoudji, B. Diu, and F. Laloë (Wiley, New York, 1977), addresses these topics.Google Scholar

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© Springer Science+Business Media New York 1998