Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 8B (Quantum Mechanics)

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- Solution 8B

Solution

A Hermitian operator

A

satisfies

⟨ φ, A ψ ⟩ = ⟨ A φ, ψ ⟩

on its operator domain. Here

H = - \frac{ℏ ^{2}}{2 m} \frac{d ^{2}}{d x ^{2}} + V (x) .

Two integration by parts operations, together with the boundary conditions in the Hamiltonian domain, show that the kinetic term is Hermitian; multiplication by the real function

V

is Hermitian as well. Thus

H

is Hermitian.

For a time-independent operator

A

, the Schrodinger equation and its adjoint give the Ehrenfest theorem

\frac{d}{d t} ⟨ A ⟩ = \frac{i}{ℏ} ⟨[H, A]⟩ .

Using the canonical commutation relation

[x, p] = i ℏ

,

[H, x] = - \frac{i ℏ}{m} p, [H, p] = i ℏ V^{'} (x) .

Consequently

\frac{d}{d t} ⟨ x ⟩ = \frac{1}{m} ⟨ p ⟩, \frac{d}{d t} ⟨ p ⟩ = - ⟨ V^{'} (x)⟩ .

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