Overview#

Quantum measurement is fundamentally different from passive observation. A measurement is a quantum operation that:

• Yields a definite outcome (an eigenvalue of the observable)

• Collapses the state irreversibly to an eigenstate

• Follows the Born rule: probability is the squared amplitude of overlap with the eigenstate

This section formalizes measurement through the measurement postulate (three axioms) and demonstrates its power by explaining the classic Stern-Gerlach experiments.

Stern-Gerlach Experiments as Motivation#

The Stern-Gerlach apparatus measures spin by splitting an atomic beam in an inhomogeneous magnetic field.

We will use the notation:

• \(\vert\uparrow\rangle, \vert\downarrow\rangle\): eigenstates of \(\hat{\sigma}^z\) (z-basis)

• \(\vert\rightarrow\rangle, \vert\leftarrow\rangle\): eigenstates of \(\hat{\sigma}^x\) (x-basis)

What Experiment 1 teaches us:

• The measurement outcome is binary: the beam always splits into exactly two components, never a continuum.

• The first apparatus doubles as a state-preparation device: filtering one branch by post-selection prepares a definite pure state.

• Repeatability: measuring the same observable twice in a row always confirms the first result — the outcome is reproducible, suggesting the existence of an underlying physical reality that persists between measurements.

What Experiment 2 teaches us:

• The intermediate X measurement washes out the previously definite Z outcome — the final Z result is now random again.

• Measurement is invasive: it is not a passive readout of a pre-existing value, but an active intervention that can alter the state.

• Non-commuting measurements change reality — inserting \(\hat{\sigma}^x\) between two \(\hat{\sigma}^z\) measurements destroys the Z-definiteness that Experiment 1 established.

Building the Mathematical Model#

Goal: Find a formula \(P(m \vert \psi, \hat{O})\) for the probability of obtaining outcome \(m\) when measuring observable \(\hat{O}\) on state \(\vert\psi\rangle\).

Positivity constraint: Since \(P(m \vert \psi, \hat{O}) \geq 0\) for all states \(\vert\psi\rangle\), we need a mathematical object that is guaranteed non-negative for any input state. A natural choice is to model \(P\) as the expectation value of some operator \(\hat{P}_{O=m}\):

This is automatically non-negative if \(\hat{P}_{O=m}\) is positive semi-definite (PSD).

Pinning down the operator: The SG experiments (Z-Z-Z) showed that repeatedly measuring the same observable on an eigenstate \(\vert O{=}m\rangle\) always returns \(m\). This means:

What PSD operator satisfies this? The projection operator \(\hat{P}_{O=m} = \vert O{=}m\rangle\langle O{=}m\vert\).

This gives us the Born rule: the probability of outcome \(m\) is the squared overlap with the corresponding eigenstate,

The Measurement Postulate#

Interpretation of collapse: After the measurement, the system is no longer in a superposition. All information about the pre-measurement state is lost (unless we knew the outcome beforehand). This is irreversible and fundamental to quantum mechanics.

Explaining the Stern-Gerlach Experiments#

The process tables above can be summarized algebraically:

• \(\vert\uparrow\rangle\) is an eigenstate of \(\hat{\sigma}^z\), so repeated Z-measurement is deterministic.

• Decomposed to \(\hat{\sigma}^x\)-basis:

  \[ \vert\uparrow\rangle = \frac{1}{\sqrt{2}}\left(\vert\rightarrow\rangle + \vert\leftarrow\rangle\right)\]

  On Z-basis state \(\vert\uparrow\rangle\), X-measurement outcomes 50-50.

  Symmetry argument: \(\vert\rightarrow\rangle\) and \(\vert\leftarrow\rangle\) looks symmetric from \(\vert\uparrow\rangle\) perspective, so they must have identical probability to be observed.

• Select \(\vert\rightarrow\rangle\), decompose to \(\hat{\sigma}^z\)-basis again:

  \[ \vert\rightarrow\rangle = \frac{1}{\sqrt{2}}\left(\vert\uparrow\rangle + \vert\downarrow\rangle\right)\]

  So selecting one X branch (e.g. \(\vert\rightarrow\rangle\)) makes the final Z outcomes 50-50. This is quantum complementarity: the intermediate X measurement erases prior Z-definiteness.

Summary#

• Measurement is fundamental: it yields a definite outcome (an eigenvalue) and irreversibly collapses the state.

• The Born rule gives the probability as the squared overlap \(|\langle O=m\vert\psi\rangle|^2\).

• The three axioms formalize the process: outcomes are eigenvalues, probabilities follow Born rule, state collapses to eigenstate.

• The Stern-Gerlach experiments directly demonstrate collapse and complementarity.

• Repeated measurement of the same observable yields the same outcome with probability 1 (because the state is now an eigenstate).

• Complementary measurements (non-commuting observables) can wipe out prior measurement information—this is not a flaw but a feature.