Index
Chapter 2 · Item 2.10
Expectation values and collapse
The measurement postulate and statistical averages
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Guided reading

The measurement postulate separates preparation from outcome. Before measurement, the state may be a superposition. After a definite result, the state is updated to the eigenstate compatible with that result.

Measurement postulate

When a measurement of observable \(\hat A\) gives the eigenvalue \(a_m\), the state immediately after the measurement is the corresponding eigenstate \(\psi_m\), apart from normalization and phase.

\[\Psi=\sum_n C_n\psi_n\quad \xrightarrow{\mathrm{measure}\ a_m}\quad \psi_m.\]

This is the collapse rule. It connects a probabilistic state before measurement to a definite state after a recorded result.

Expectation value as ensemble average

If many identical preparations are measured, the average result is

\[\langle \hat A\rangle=\sum_n |C_n|^2 a_n.\]

This average is not necessarily one of the possible individual outcomes. It is the weighted mean over repeated measurements.

Integral form

In the position representation, the expectation value is computed by inserting the operator between \(\Psi^*\) and \(\Psi\):

\[\langle \hat A\rangle=\int \Psi^*(x,t)\,\hat A\,\Psi(x,t)\,dx.\]

For position, \(\hat A=\hat x=x\). For momentum, \(\hat A=\hat p=-i\hbar\,d/dx\).

Energy measurement in the well

If the well state is \(\Psi=(\psi_1+\psi_2)/\sqrt{2}\), a measurement can return \(E_1\) or \(E_2\) with equal probability. If the result is \(E_1\), the state after measurement is \(\psi_1\).

The same state can therefore be a superposition before measurement and a single eigenstate after a definite outcome.
Ensemble average versus single outcome

The expectation value is not usually the value seen in one measurement. It is the weighted average over many identically prepared systems:

\[\langle \hat A\rangle=\sum_n a_nP_n.\]

If one run gives the value \(a_m\), the post-measurement state is the corresponding eigenstate \(\psi_m\) in the ideal projective model. This state update is what later measurements see.

Good exercises should keep these ideas separate: probabilities predict outcome frequencies, expectation values summarize ensembles, and collapse describes preparation after a definite result.

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  • Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
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Practice anchors

Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.

  • Focus: Measurement postulate: expectation values are ensemble averages, and after a definite result the state collapses to the measured eigenstate.
  • Conceptual check: state what the main result says physically before using it algebraically.
  • Equation: \[\Psi=\sum_n C_n\psi_n\quad \xrightarrow{\mathrm{measure}\ a_m}\quad \psi_m.\]
  • Equation: \[\langle \hat A\rangle=\sum_n |C_n|^2 a_n.\]
  • Equation: \[\langle \hat A\rangle=\int \Psi^*(x,t)\,\hat A\,\Psi(x,t)\,dx.\]
  • Boundary: use this page for setup and first-step reasoning; cite the book for longer derivations, complete experimental history or solved-problem detail.
  • Typical task: derive, interpret, or apply the relation above to a simple case without introducing topics outside this page.
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Source note: Original auxiliary summary for this book-app, based on Chapter 2 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc. No original book figure is reproduced on this page.