Texts: -h^2 d^2w/dx^2 + V(x)y(x) = Ey(x) 2m dx^2
(1)
Take V(x) to be a simple cosine potential:
V(x) = iGx^2/2T and G_ = -G a
(2)
G,
2
For V small, we expect y(x) to be "close" to a single deBroglie wave (which is what you get when V = 0 (free e))
y(x) = e^(ikx) (for some k = (2Ï€/L)n, n = 0, -1, 1, -2, 2...)
Try, for V small, (but not 0) just 3 terms (k, k + G, k - G)
x(ig-) = 8 + x(+n)V + xyT = (x)h (3) 1 C (k) C(k + Gi) C (k - G1)
(note: we can take C(k) = 1 since the equation is linear--convince yourself)
a) Put (2) and (3) directly into (1) and write out the resulting equation
b) Separate the above equation into terms for each separate wave term:
(-)e^(ikx) + (~)e^(i(k+G:)x) + (~)e^(i(k-G)x)
+ some other terms (which we will ignore!)
c) From b) write down the equations for the c's by setting each {~} = 0.
d) put the resulting 3 equations into the form M-C = 0, where M is a 3x3 matrix and C and 0' are column vectors with 3 elements. Write out the equation det(M) = 0 (the solution of which gives the possible energies).
