Measurement, Collapse, and Spectral Expansion

The Big Picture

Imagine you have a quantum state, which we represent as a vector. Now imagine an observable, such as energy or position, which is represented by a Hermitian operator. The spectral theorem tells us that the eigenvectors of this operator form a complete set of directions in our vector space. This means we can describe any possible state as a mixture, or superposition, of these specific eigenvectors. When we measure the observable, we are essentially asking which of these directions the state is pointing towards. The probability of finding a specific value is determined by how much of the state vector lies along the corresponding eigenvector. If we measure and get a specific result, the state vector instantly "snaps" to align perfectly with that eigenvector. This process is known as the collapse of the wavefunction. It is the fundamental rule that connects the smooth, deterministic evolution of the wavefunction with the discrete, probabilistic nature of measurement outcomes.

Spectral Expansion

Let \(\hat{A}\) be a Hermitian operator representing an observable with a discrete, non-degenerate spectrum \(\{a_n\}\). The eigenvectors \(|a_n\rangle\) satisfy the eigenvalue equation \(\hat{A}|a_n\rangle = a_n|a_n\rangle\) and form an orthonormal basis. This means \(\langle a_m | a_n \rangle = \delta_{mn}\). Because they form a complete basis, any normalised state \(|\psi\rangle\) can be expanded as a linear combination of these eigenvectors.

The coefficients \(c_n\) are found by taking the inner product of the state with each eigenvector. Using the orthonormality condition, we project the state onto the basis vectors:

This expansion is the mathematical foundation of the statistical interpretation. The coefficients \(c_n\) are complex numbers, but their squared magnitudes have direct physical significance.

Probability and Collapse

According to the Born rule, the probability \(P(a_n)\) of measuring the value \(a_n\) is the square of the magnitude of the coefficient \(c_n\). This is often referred to as the Born probability rule.

Since the state is normalised, the sum of all probabilities must equal one. This is a direct consequence of the completeness of the basis:

Crucially, the measurement process is not passive. If the measurement yields the outcome \(a_n\), the system is left in the corresponding eigenstate \(|a_n\rangle\). This is the projection postulate, or state collapse. Immediately after the measurement, the state is no longer the superposition \(|\psi\rangle\), but the pure eigenstate associated with the observed value.

This collapse is instantaneous and non-unitary, distinguishing it from the smooth time evolution governed by the Schrödinger equation. It represents a fundamental discontinuity in the state of the system triggered by the interaction with the measuring apparatus.

Expectation Values

While individual measurements yield discrete eigenvalues, the expectation value \(\langle \hat{A} \rangle\) gives the average result of many repeated measurements on identically prepared systems. Using the spectral expansion, we can express this expectation value in terms of the probabilities and eigenvalues.

This formula shows that the expectation value is a weighted average of the possible outcomes, where the weights are the probabilities of those outcomes. It provides a bridge between the quantum formalism and classical statistical mechanics, where averages play a central role in describing macroscopic behaviour.