Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 34C / b / ii

Let

Π = p - q A

H = \frac{1}{2 m} [(p_{x} + \frac{qB}{2} y)^{2} + (p_{y} - \frac{qB}{2} x)^{2} + p_{z}^{2}] = \frac{p _{x}^{2} + p _{y}^{2} + p _{z}^{2}}{2 m} + \frac{q ^{2} B ^{2}}{8 m} (x^{2} + y^{2}) - \frac{qB}{2 m} L_{z} .

Assume

qB > 0

and zero longitudinal momentum, as in the displayed states. Since

[Π_{x}, Π_{y}] = i ℏ qB, a = \frac{Π _{x} + i Π _{y}}{2ℏ qB},

we have

[a, a^{†}] = 1

and

H = ℏ ω_{B} (a^{†} a + \frac{1}{2}) .

For

w = x + i y

Π_{x} + i Π_{y} = - 2 i ℏ (\partial_{\overset{w}{ˉ}} + \frac{qB}{4ℏ} w),

which annihilates every

ψ = f (w) e^{- qB ∣ w ∣^{2} / (4ℏ)}

with

f

holomorphic. Hence all these states have

E_{0} = \frac{1}{2} ℏ ω_{B} .

E_{n} = ℏ ω_{B} (n + 1/2)

for

n = 0, 1, \dots

; the arbitrary holomorphic function exhibits the degeneracy of the lowest level. For

qB < 0

, the normalizable lowest-level states are anti-holomorphic instead.

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