By E. P. O'Reilly
ISBN-10: 0789739623
ISBN-13: 9780789739629
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Additional info for Quantum Theory of Solids
Sample text
First, it is found that in many cases each electron effectively sees a similar average potential as all the others, so that instead of having to solve something like a 1023 body problem, we can use an ‘independent electron approximation’, and calculate the energy spectrum using the one-electron Schrödinger equation introduced in Chapter 1. While we may not know the exact form of this average potential we expect that it should be closely related to the isolated atomic potentials of the atoms which form the solid.
5 Using the variational wavefunction ψ(x) = αφL (x) + βφR (x), where φL (x) and φR (x) are the isolated quantum well ground state wavefunctions defined in eq. 12), calculate each of the integrals I, II and IV in eq. 18) for the double square well potential. Hence show that the variational method also predicts that the splitting between the ground and first excited state energy varies as 2Ce−κb for two weakly coupled square quantum wells. 5)!! 7 We can write the wavefunctions for an s-state and for three p states on an isolated atom as φs (r) = fs (r), φz (r) = fp (r) cos θ, φx (r) = fp (r) sin θ cos φ and φy (r) = fp (r) sin θ sin φ, where (r, θ, φ) are spherical polar coordinates centred on the atomic nucleus, and fs (r) and fp (r) describe the radial variation of the s and p wavefunctions.
It is often useful to plot a diagram of the energies Enk as a function of the wavevector k, which is then referred to as the band structure of the given solid. 2(a) shows the band structure for an electron in free space, which is described by the parabola E = 2 k2 /(2m). The free electron band structure is modified in several ways in a periodic solid. In particular, the wavevector k associated with a given energy state is no longer uniquely defined. This can be shown by considering a 1-D periodic structure, with unit cell of length L.
Quantum Theory of Solids by E. P. O'Reilly
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