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For students learning this material in my Quantum Physics course, there has often been confusion between the above formulae, and those for a well which goes from 0 < x < a, where the solution always has the sine form, and the normalisation constant is (2/a)1/2. Factors of 4 easily creep into the energy expression for the same reason.
The sin-cos form resulting from a symmetrically placed well is more instructive, because it emphasises the difference between solutions with even parity (cosine-like) and those with odd parity (sine-like). Because of this, the physics is much more easily extended to unbound and bound states in finite wells (e.g. Gasiorowicz, 2nd edition: fig. 5.2 page 78 and fig. 5.10 page 90; 3rd edition: fig 4.2, page 69 and fig 4.7, page 77; or Liboff, 3rd edition fig. 7.27 page 244 and fig. 8.1, page 290). Chemists typically don't need convincing on this point, because they meet states which are labelled g (gerade, even parity) and u (ungerade, odd parity) at an early stage of their undergraduate careers.
In a finite well, the wavefunctions are not zero in the classically forbidden regions, but they decay exponentially as
Thus the less strongly bound states have extensive wavefunctions; there are many aspects of solids which can be understood qualitatively on this basis alone. For example, when many wells are put close together, valence electrons interact strongly and form bands, whereas core electrons largely retain their atomic-like character. The subtle problems in electronic structure arise when different atomic states give rise to overlapping bands, as in the sp3 bands in diamond structure semiconductors, or the overlap of broad s and narrow d bands in the transition metals.
Thus the allowed k-values are equally spaced, with k = 2pn/L, and so each state occupies a volume element of k-space, dk = (2p/L). This result generalises easily to 2D and 3D, provided we use the vector k, or its components kx, ky, kz, as is most easily seen by using periodic boundary conditions exp(ik.r) = 1, with r = iLx + jLy + kLz. The volume element of k-space in 3D is now dk = (8p3/LxLy Lz) = (2p)3/V, where V is the volume of the box. The corresponding formula for 2D is clearly (2p)2/A, with A the area of the 2D box. Sometimes these formulae are written as (2p)d/Vd, where the d-dimensional volume Vd = L, A or V for d = 1, 2 or 3.
This k-space volume element contains two electrons, when spin is taken in account, but beware that different authors introduce this point at different stages of the argument. Note also, in passing, that we have L = nl, not nl/2, which we might have deduced from the standing wave solutions sin() and cos(); these trigonometric forms just differ by a phase factor when periodic boundary conditions are used. Moreover, these formula are used to describe the density of states (DOS) per unit energy, called D(E) by Sutton or g(E) by Liboff, where for free electrons, E = (hk)2/2m, so that the allowed values of nx, ny, nz are all positive. This gives much scope for mistakes of factors of 2, 4, 8, so considerable numerical care needs to be taken at each stage of the argument.
Differentiating E = (hk)2/2m gives dE = h2kdk/m in one dimension (1D). The number of states this corresponds to, excluding spin but including negative values of k, is dN = 2dk/(2p/L). So the number of states per unit energy interval in 1D is D(E) = dN/dE = (L/p).(m/h 2k) = 0.5(E1E)-1/2. In the last equality, the form given by Liboff, E1 is the ground state energy of an electron in a box of width L, i.e. E1 = h2/(8mL2). But the most important result is that D(E) is proportional to E-1/2, and so peaks at low energies (Sutton figure 7.2 page 138, or Liboff section 8.8, page 349-50 in the third edition).
In the case of 2D and 3D, we need to include the degeneracy caused by different directions of k having the same energy, and to apply the derivatives and signs consistently. For example, Liboff's derivations use n, which is always positive, but only a quadrant in 2D (factor 1/4) or an octant in 3D (factor 1/8) is used in the summation; Sutton's derivation using k has no such restriction. The results are simply expressed in terms of E1 in the following Table. In 2D D(E) has no energy dependence, and in 3D D(E) is proportional to E1/2.
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These differences in dimensionality are important for discussions of electron states in low dimensional structures, which form a large class of solid state devices. Some of the subtleties can be explored, at this or a later stage, via problem 2.4.3.
When the lattice potential has non-zero Fourier components, energy gaps open up in the electron energy spectrum at particular k-values, and may find their way into gaps in the density of states D(E). Whether or not there are gaps at the Fermi energy forms the basis of the elementary classification of solids into metals, semiconductors and insulators. Metals have no gap at EF, insulators a large gap, and most interestingly semiconductors have an intermediate size gap. It is this feature which allows semiconductors to form the basis of devices. By manipulating the Fermi level via temperature, light or defects, the electronic structures of semiconductors can be externally controlled, in ways which are not in general possible for either metals or insulators.
