 Solution

ID: past-exam-of-the-condensed-matter-physics-course-of-the-university-of-oxford/2022/2/b/ii/solution

At every

y

and every

ϵ > 0

4 y^{2} + ϵ^{2} > 2∣ y ∣,

so the lower energy band lies strictly below its zero-coupling counterpart. Its state counting in a Bloch band and complete occupation are unchanged. Subtracting the two energy densities directly gives

U (ϵ) - U (0) = - \frac{E _{ZB}}{a} \int_{- 1}^{1} [4 y^{2} + ϵ^{2} - 2∣ y ∣] d y < 0.

Thus

U (ϵ) < U (0) (ϵ > 0) .

Equality holds at

ϵ = 0

. This is the downward displacement of the filled lower energy band in the sketch. The upward displacement of the empty upper energy band contributes nothing to the zero-temperature electronic energy density.

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