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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 34B / a / Solution

The Born rule depends only on transition probabilities

∣ ⟨ ϕ ∣ ψ ⟩ ∣^{2}

between rays. A physical transformation must preserve these probabilities. By Wigner theorem, such a ray transformation is induced by a unitary or antiunitary operator. For ordinary continuously connected transformations one chooses the unitary branch, so

U (g)^{†} U (g) = I_{H} .

The states

U (g_{1}) U (g_{2}) ∣ ψ ⟩

and

U (g_{1} g_{2}) ∣ ψ ⟩

represent the same ray. They may therefore differ by a state-independent quantum phase:

U (g_{1}) U (g_{2}) = e^{i ϕ (g_{1}, g_{2})} U (g_{1} g_{2}) .

Thus quantum transformations form a projective unitary representation.

If

g

is a symmetry of the Hamiltonian operator

H

, it also obeys

U (g)^{†} H U (g) = H,

equivalently

[U (g), H] = 0

. For a differentiable one-parameter subgroup, write

U (s) = e^{- i s Q /ℏ}

with a self-adjoint generator

Q

. Differentiating the symmetry equation at

s = 0

yields

[Q, H] = 0.

For a time-independent

Q

, the Heisenberg picture equation is

\frac{d Q}{d t} = \frac{i}{ℏ} [H, Q] = 0.

Hence the generator is a conserved observable. Conversely, a self-adjoint conserved

Q

commutes with

H

and its unitary group generates symmetries. This is the conserved generator of a continuous quantum symmetry.

Solved by gpt-5.6-sol high.

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