6B (Quantum Mechanics)

ρ = ∣ ψ ∣^{2}, J = \frac{ℏ}{2 mi} (ψ^{*} \nabla ψ - ψ \nabla ψ^{*}) .

The Schrödinger equation and its conjugate, for a real potential, are

i ℏ ψ_{t} = - \frac{ℏ ^{2}}{2 m} \nabla^{2} ψ + V ψ, - i ℏ ψ_{t}^{*} = - \frac{ℏ ^{2}}{2 m} \nabla^{2} ψ^{*} + V ψ^{*} .

Multiplying respectively by

ψ^{*}

and

ψ

and adding their expressions for the time derivatives cancels the potential terms. Hence

ρ_{t} = \frac{i ℏ}{2 m} (ψ^{*} \nabla^{2} ψ - ψ \nabla^{2} ψ^{*}) = - \nabla \cdot J .

The gradient cross-terms cancel when expanding the divergence. Therefore

ρ_{t} + \nabla \cdot J = 0.

This continuity equation expresses local conservation of probability. Integrating over a region says that its probability decreases by the outward flux of the probability current.

b

A

⟨ A ⟩_{ψ} = \int_{R^{3}} ψ^{*} (x) (A ψ) (x) d^{3} x,

Δ_{ψ} A = [\int_{R^{3}} ψ^{*} (A - ⟨ A ⟩_{ψ})^{2} ψ d^{3} x]^{1/2} .

Equivalently the squared uncertainty is

∥ (A - ⟨ A ⟩_{ψ}) ψ ∥^{2}

, which is visibly nonnegative. Since the expectation value of a self-adjoint operator is real, expansion of the square and

\int ∣ ψ ∣^{2} = 1

give

Δ_{ψ} A = ⟨ A^{2} ⟩_{ψ} - ⟨ A ⟩_{ψ}^{2} .

For a state not normalized to one, divide each defining expectation integral by

\int ∣ ψ ∣^{2}

before using this formula.

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