Past exam of the chemistry A course of the University of Cambridge / 2022 / Paper 1 / A1 / a

Solution

With the physics convention

⟨ Φ∣Ψ ⟩ = \int Φ^{*} Ψ d τ

, the adjoint operator is defined by

⟨ Φ∣ \hat{A} Ψ ⟩ = ⟨ \hat{A}^{†} Φ∣Ψ ⟩ .

\hat{A}

is a Hermitian operator on the common domain, then

\hat{A}^{†} Φ = \hat{A} Φ

. Conjugate symmetry of the inner product consequently gives

⟨ Φ∣ \hat{A} Ψ ⟩ = ⟨ \hat{A} Φ∣Ψ ⟩ = ⟨ Ψ∣ \hat{A} Φ ⟩^{*},

which is the required integral identity.

The boundary conditions must make the boundary form from integration by parts vanish for every pair of allowed functions. For a Schrödinger kinetic operator in one dimension, this means

[Φ^{*} Ψ^{'} - (Φ^{'})^{*} Ψ]_{boundary} = 0.

Sufficient examples are decay of the functions and derivatives at spatial infinity, homogeneous hard-wall conditions

Φ = Ψ = 0

at finite endpoints, or matching periodic conditions on both functions and their derivatives. The exact boundary form depends on

\hat{A}

; square integrability alone does not replace the domain requirement.

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