psi(x) = A psi_l(x) + B psi_r(x), -(a)/2 <= x <= (a)/2.
Now Bloch's theorem asserts that psi can be chosen to satisfy
psi(x + a) = e^(ika) psi(x),
for suitable k. Differentiating (8.68) we also find that psi' = d(psi)/dx satisfies
psi'(x + a) = e^(ika) psi'(x).
(a) By imposing the conditions (8.68) and (8.69) at x = -(a)/2, and using (8.65) to (8.67), show that the energy of the Bloch electron is related to its wave vector k by:
coska = (t^2 - r^2)/(2t) e^(iKa) + (1)/(2t) e^(-iKa), epsi = (hbar^2 K^2)/(2m).
Verify that this gives the right answer in the free electron case (v-) = (0).
Note: in this problem K is a continuous variable and has nothing to do with the reciprocal lattice.
A special case of the general theorem that there are n independent solutions to an nth-order linear differential equation.
Chapter 8 Electron Levels in a Periodic Potential
Equation (8.70) is more informative when one supplies a little more information about the transmission and reflection coefficients. We write the complex number t in terms of its magnitude and phase:
t = |t| e^(is).
The real number delta is known as the phase shift, since it specifies the change in phase of the transmitted wave relative to the incident one. Electron conservation requires that the probability of transmission plus the probability of reflection be unity:
1 = |t|^2 + |r|^2.
This, and some other useful information, can be proved as follows. Let phi_1 and phi_2 be any two solutions to the one-barrier Schrödinger equation with the same energy:
-(hbar^2)/(2m) phi_i'' + r phi_i = (hbar^2 K^2)/(2m) phi_i, i = 1,2.
Define w(phi_1, phi_2) (the "Wronskian") by
w(phi_1, phi_2) = phi_1'(x) phi_2(x) - phi_1(x) phi_2'(x).
(b) Prove that w is independent of x by deducing from (8.73) that its derivative
(c) Prove (8.72) by evaluating w(psi_l, psi_i**) for x <= -(a)/2 and x >= (a)/2, noting that because is real psi_l** will be a solution to the same Schrodinger equation as psi_1.
(d) By evaluating w(psi_1, psi_r**) prove that r t** is pure imaginary, so r must have the form
r = +-i |r| e^(is)
where delta is the same as in (8.71).
(e) Show as a consequence of (8.70), (8.72), and (8.75) that the energy and wave vector of the Bloch electron are related by
(cos(Ka + delta))/|t| = coska, epsi = (hbar^2 K^2)/(2m).
Since |t| is always less than one, but approaches unity for large K (the barrier becomes increasingly less effective as the incident energy grows), the left side of (8.76) plotted against K has the structure depicted in Figure 8.6. For a given k, the allowed values of K (and hence the allowed energies {(: epsi(k) = hbar^2 K^2; 2m)} are given by the intersection of the curve in Figure 8.6 with the horizontal line of height cos(ka). Note that values of K in the neighborhood of those satisfying
Ka + delta = n pi
give |cos(Ka + delta)|/|t| > 1, and are therefore not allowed for any k. The corresponding regions of energy are the energy gaps. If delta is a bounded function of K (as is generally the case), then there will be infinitely many regions of forbidden energy, and also infinitely many regions of allowed energies for each value of k.
(f) Suppose the barrier is very weak (so that |t| ~ 1, |r| ~ 0, delta ~ 0). Show that the energy gaps are then very narrow, the width of the gap containing K = n(pi)/a being
(x) = A(x) + B(x)
