 Solution

ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2022/paper-2/section-b/21/i/solution

One consistent five-postulate formulation is the following; different textbooks group the same content differently.

State. A pure state is a normalized vector $∣ ψ ⟩$ in a complex Hilbert space, with $⟨ ψ ∣ ψ ⟩ = 1$ ; vectors differing only by an overall phase represent the same state. For a spinless particle in position representation, $ψ (r) = ⟨ r ∣ ψ ⟩$ is its wavefunction.
Observables. A physical observable A is represented by a self-adjoint operator $\hat{A}$ . In a Cartesian position representation, $\overset{x}{^} = x$ and $\overset{p}{^}_{x} = - i ℏ \partial_{x}$ , with $ℏ = h / (2 π)$ .
Outcomes and probabilities. Allowed ideal measurement outcomes are spectral values $a$ of $\hat{A}$ . The Born rule gives $P (a) = ⟨ ψ ∣ P_{a} ∣ ψ ⟩$ , where $P_{a}$ projects onto the corresponding eigenspace; continuous spectra require the analogous spectral probability measure. The expectation is $⟨ \hat{A} ⟩ = ⟨ ψ ∣ \hat{A} ∣ ψ ⟩$ .
Ideal measurement update. Conditional on a nonzero-probability outcome $a$ , an ideal projective measurement leaves $P_{a} ∣ ψ ⟩ / ⟨ ψ ∣ P_{a} ∣ ψ ⟩$ . For a nondegenerate result this is its eigenstate. This is the ideal projection rule, not a description of every general measurement apparatus.
Time evolution. Between measurements a closed system evolves by the Schrödinger equation, $i ℏ \partial_{t} ∣ ψ (t)⟩ = \hat{H} ∣ ψ (t)⟩$ , with Hamiltonian $\hat{H}$ . For time-independent $\hat{H}$ , this gives unitary evolution $e^{- i \hat{H} t /ℏ}$ .

∣ ψ ∣^{2}

, and its flux is the probability current. These distinguish the wave amplitude from the measurable particle flow in the following scattering problem.

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