Guided reading
The probability postulate gives the operational meaning of expansion coefficients. Once the state is written in the eigenbasis of the measured observable, the squared modulus of each coefficient is the probability of the corresponding result.
Postulate of probability
If a normalized state is expanded in eigenstates of the measured observable,
\[\Psi=\sum_n C_n\psi_n,\]
then the probability of obtaining the eigenvalue associated with \(\psi_n\) is
\[P_n=|C_n|^2.\]
Why normalization becomes probability
Using orthonormality, the normalization condition becomes
\[1=\int \Psi^*\Psi\,dx=\sum_{m,n}C_m^*C_n\int\psi_m^*\psi_n\,dx=\sum_n |C_n|^2.\]
This is exactly the structure expected of probabilities: all mutually exclusive outcomes add to one.
Projection extracts the amplitude
The coefficient \(C_n\) is found by projecting the state onto the eigenstate:
\[C_n=\int \psi_n^*(x)\Psi(x)\,dx.\]
The amplitude can be complex. The probability discards the overall phase through the squared modulus \(C_n^*C_n\).
Example in the energy basis
For
\[\Psi=\frac{1}{\sqrt{3}}\psi_1+\sqrt{\frac{2}{3}}\psi_2,\]
an energy measurement gives
\[P(E_1)=\frac{1}{3},\qquad P(E_2)=\frac{2}{3}.\]
The theory predicts probabilities for individual outcomes, not which result a single run must produce.
Exercise-ready boundary
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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: Postulate of probability: expansion coefficients give outcome probabilities, Pₙ = |Cₙ|², when measuring in an eigenstate basis.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[\Psi=\sum_n C_n\psi_n,\]
- Equation: \[P_n=|C_n|^2.\]
- Equation: \[1=\int \Psi^*\Psi\,dx=\sum_{m,n}C_m^*C_n\int\psi_m^*\psi_n\,dx=\sum_n |C_n|^2.\]
- 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.
Original book and previews:
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.