Operators, Eigenvalues and Measurement

An observable is represented by a self-adjoint operator. In the position representation, position sends psi(x) to x psi(x). The Hamiltonian H is the energy operator. The time-dependent Schrodinger equation, iℏ ∂psi/∂t = H psi, governs evolution. For time-independent H, stationary energy states solve H psi = E psi, as with the oscillator levels you met. Their probability densities stay fixed while their phases evolve.

An ideal projective measurement returns a value in the operator's spectrum, which may be discrete or continuous. For a normalized superposition of orthonormal eigenstates with distinct discrete eigenvalues, each probability is its coefficient's squared magnitude. When finite, the expectation value is the probability-weighted mean. For continuous outcomes, use a probability density instead. A mean need not be an allowed result.

In one dimension, canonical momentum acts as -iℏ ∂psi/∂x in the position representation. Here i² = -1 and ℏ is the reduced Planck constant. A plane wave exp(ikx) returns ℏk times itself. On the full line, it is a generalized eigenfunction, not a normalizable state.

Position and momentum satisfy Δx Δp >= ℏ/2, where the deltas are standard deviations in the prepared state. Thus both spreads cannot be zero. This bound comes from [x,p] = iℏ I, not detector imperfections. Here [x,p] = xp - px compares the two operator orders; I leaves every state unchanged.

An ideal discrete projective measurement projects the state into the measured eigenspace. An immediate repeat gives the same result if the state is unchanged. Fresh preparations sample the original probabilities. A discrete result excludes preparations assigning it zero probability, but does not identify the earlier state uniquely: both an eigenstate and a superposition can yield the energy eigenvalue E1.