Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 33A / a

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- Solution a

Solution

In position representation,

L_{z} = - i ℏ (x \partial_{y} - y \partial_{x}),

which differentiates only with respect to

x

and

y

. Multiplication by

z

therefore commutes with it:

[L_{z}, z] = 0 .

Taking the matrix element of this commutator between angular-momentum eigenstates gives

0 = ℏ (m^{'} - m) ⟨ n^{'}, ℓ^{'}, m^{'} ∣ z ∣ n, ℓ, m ⟩ .

Thus the matrix element can be nonzero only when

m^{'} = m .

This is the magnetic part of the electric-dipole selection rules for hydrogen.

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