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.
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Notes
The boundary conditions select from all eigenstates of H (0) the ones satisfying the appropriate physical constraints.
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.
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).
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.
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.
Recall that the determinant of a matrix is the product of the eigenvalues.
To define au operator, it is sufficient to define its matrix elements between a complete set of states.
To be notationally exact, the last sum should be written as We omit this complication.
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.
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.
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.
As we will see in Chapter 5, perturbations due to electromagnetic radiation take this form.
We take the parameter ω ≥ 0 to avoid ambigtiity. Negative frequencies are given explicitly by —ω.
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.
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.
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 DXIII in Quantum. Mechanics, by C. Cohen-Tannoudji, B. Diu, and F. Laloë (Wiley, New York, 1977).
Feynmann diagrams.
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.
A central potential V depends only on the magnitude r of the coordinate vector r, not the direction.
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.
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.
For example, G. Arfken, Mathematical Methods for Physicists, third edition, (Academic Press, New York, 1985).
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.
We estimate the strength of an operator A by taking the expectation value in an appropriate eigenstate.
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.
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.
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 ms.
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).
The coefficients in (1.249) assume a particular set of phase conventions. They may be found in Cohen-Tannoudji, et. al., Complement Ax.
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.
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 (2l + l)(2s + l)-fold degenerate.
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 αm = (αmn j*) and therefore.
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.
When we explicitly consider the electron spin we must work with so-called double groups, in which the symmetry operations T are more complicated.
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.
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.
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Loehr, J.P. (1998). Quantum-Mechanical Preliminaries. In: Physics of Strained Quantum Well Lasers. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-5673-2_1
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