 Solution

ID: past-exam-of-the-general-relativity-and-cosmology-and-condensed-matter-physics-paper-of-part-b-of-the-physics-course-of-the-university-of-oxford/2011/section-vi/7/i/solution

Let

u = V_{10} < 0

and

E_{X} = \frac{ℏ ^{2} π ^{2}}{2 m _{e} a ^{2}} .

X = (π / a, 0)

the two degenerate plane waves have momenta

(π / a, 0)

and

(- π / a, 0)

u

. Thus

E_{X} - E u u E_{X} - E = 0, E_{X, 1} = E_{X} + u, E_{X, 2} = E_{X} - u .

Their symmetric and antisymmetric combinations are the lower and upper states respectively. The local band gap is

2∣ u ∣

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