Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 324 / 1 / c / ii

Table of contents
- Solution ii

Solution

Write

x = g^{η}

. Applying

D_{x}

and then changing the summation variable from

k

to

ℓ = k - α η

gives

D_{x} ∣ α ⟩ ∣ χ_{β} ⟩ = \frac{1}{N} k \sum ω^{β k} ∣ α ⟩ ∣ g^{k - α η} ⟩ = ω^{α β η} ∣ α ⟩ ∣ χ_{β} ⟩ .

Hence this state is an eigenvector of

D_{x}

with eigenvalue

ω^{α β η}

.

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