Guided reading
Appendix 3.A develops the basis-change machinery used throughout matrix mechanics. A change of basis is a rotation of the vector-space description. It changes the components of vectors and the matrices representing operators, but not the physical predictions.
Unitary transformations
A unitary operator \(\hat U\) connects two representations of the same vector:
\[|n\rangle_{a}=\hat U|n\rangle_{b}.\]
Unitarity means
\[\hat U^{-1}=\hat U^\dagger,
\qquad \hat U^\dagger\hat U=\hat U\hat U^\dagger=\hat 1.\]
This condition preserves inner products. Since probabilities are squared inner products, a unitary basis change cannot alter measurable probabilities.
Operator transformation
If the basis changes, an operator's matrix representation must also change. The appendix writes the relation as
\[\hat A_b=\hat U^\dagger\hat A_a\hat U,
\qquad \hat A_a=\hat U\hat A_b\hat U^\dagger.\]
The physical operator is the same; only its coordinate expression changes. This is exactly like rotating axes in ordinary vector analysis, except now the vectors live in Hilbert space.
Invariant expectation value
Transform the state and operator together. The expectation value stays fixed:
\[\langle\Psi'|\hat A'|\Psi'\rangle=\langle\Psi|\hat A|\Psi\rangle.\]
This is why basis changes are allowed. They simplify calculations without changing what the apparatus would measure.
Example 3.9: z basis to x basis
The relations between \(z\)- and \(x\)-spin bases can be collected into the unitary matrix
\[\hat U=\frac{1}{\sqrt2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}.\]
This matrix maps the \(z\)-basis representation into the \(x\)-basis representation:
\[|\pm\rangle_x=\hat U|\pm\rangle.\]
Because \(\hat U\) is unitary, the inverse transformation is \(\hat U^\dagger\).
Diagonalizing \(\hat\sigma_x\)
In the \(z\)-basis, \(\hat\sigma_x\) is off-diagonal:
\[\left(\hat\sigma_x\right)_z=\begin{pmatrix}0&1\\1&0\end{pmatrix}.\]
Changing to the \(x\)-basis gives
\[\left(\hat\sigma_x\right)_x=\hat U\left(\hat\sigma_x\right)_z\hat U^\dagger=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.\]
The same observable becomes diagonal in its own eigenbasis. This is the matrix form of orienting the Stern-Gerlach device along the axis whose spin component is being measured.
What changes and what does not
Components, matrices and labels may change when the basis changes. Inner products, probabilities and expectation values do not. This is the operational meaning of equivalence between bases.
A basis change is a better coordinate system, not a new physical system.
How to transform without changing physics
A basis change is useful only if it preserves the inner products that become probabilities. That is why the transformation must be unitary:
\[\langle\phi'|\psi'\rangle=\langle\phi|\hat U^\dagger\hat U|\psi\rangle=\langle\phi|\psi\rangle.\]
States and operators must be transformed consistently. If a state vector changes coordinates, the operator matrix must change coordinates too, so that expectation values remain invariant.
Exercises should focus on this invariant structure: different matrices can represent the same physical observable in different bases.
Exercise-ready boundary
This page is designed to support short guided exercises on: How states and operators transform between bases while probabilities and expectation values remain invariant.
- Use from this page: the definitions, physical setup, highlighted equations and conceptual links needed to start a first calculation or explanation.
- Keep in the book: complete derivations, extended historical discussion, worked solutions and the full textbook narrative remain in the original chapter and linked book resources.
- Good exercise balance: ask the student to identify assumptions, apply one relation, and interpret the result physically, without requiring material not introduced on this page.
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: How states and operators transform between bases while probabilities and expectation values remain invariant.
- Conceptual check: state what the main result says physically before using it algebraically.
- Equation: \[|n\rangle_{a}=\hat U|n\rangle_{b}.\]
- Equation: \[\hat U^{-1}=\hat U^\dagger, \qquad \hat U^\dagger\hat U=\hat U\hat U^\dagger=\hat 1.\]
- Equation: \[\hat A_b=\hat U^\dagger\hat A_a\hat U, \qquad \hat A_a=\hat U\hat A_b\hat U^\dagger.\]
- 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 3 of Mario Reis, Quantum Mechanics, Elsevier, 2026. Book text and figures are copyright © 2026 Elsevier Inc. Cropped figures are shown here with source indication for educational review; consult the original book for the complete presentation.