Reference and Problem Lab

Use these references and solved problems while calculating rotations, angular-momentum couplings, transition amplitudes, or geometric phases. Start with the physical quantity you want, choose the relevant formula, and check its basis and sign conventions. A useful check should test the physics as well as the algebra: a wrong rotation direction, for example, can still preserve every norm.

Helpful background. Notation and conventions fixes rotation and phase choices. The angular-momentum and spin guides supply the explanations behind the compact references.

Find a formula or a practice problem

Calculation	Resource
Check rotation signs, phases, and basis ordering	Notation and conventions
Diagnose a conceptual or convention error	Common pitfalls
Review the main formulas	Formula sheet
Simplify angular-momentum commutators or ladder actions	Angular-momentum identities
Multiply Pauli matrices or exponentiate a qubit rotation	Pauli identities
Normalize or transform complex spherical harmonics	Spherical harmonics quick reference
Convert a product basis into coupled states	Clebsch–Gordan quick reference
Change an angular-momentum coupling scheme	Wigner symbols quick reference
Practice measurements, rotations, and precession	Spin problems
Practice orbital motion and angular-momentum addition	Angular-momentum problems
Decide which transition amplitudes must vanish	Selection-rule problems
Calculate phase, curvature, or a Chern number	Berry-phase problems
Explore calculations numerically	Computational notebooks
Connect equations to diagrams	Visualization gallery

A unitary matrix can rotate in the wrong direction

With the active convention $U_y(\theta)=e^{-i\theta\sigma_y/2}$ and basis ordered as $|+z\rangle,|-z\rangle$,

$$U_y(\theta)|+z\rangle =\cos\frac\theta2|+z\rangle+\sin\frac\theta2|-z\rangle.$$

At $\theta=\pi/2$, the Bloch vector points along $+x$, so $\langle\sigma_x\rangle=1$. Replacing $U_y$ by $U_y^\dagger$ instead gives $-x$. Both matrices are unitary, both conserve norm, and both give equal probabilities for a $z$ measurement. A measurement along $x$ distinguishes the sign error immediately.

This is why an independent observable is often a stronger check than normalization alone. For precession, compare the initial time derivative with the commutator equation. For a coupled basis, check a known highest-weight state as well as orthogonality. For a spherical harmonic, check a simple value or its nodal structure as well as its integral norm.

Coupling tables depend on the whole basis convention

The allowed total angular momenta satisfy $J=|j_1-j_2|,\ldots,j_1+j_2$ in unit steps, and dimension must be preserved:

$$(2j_1+1)(2j_2+1)=\sum_J(2J+1).$$

Nonzero coefficients require $M=m_1+m_2$. These restrictions eliminate impossible entries before any table lookup. For two spins one half, the dimensions are $4=3+1$, with triplet and singlet sectors.

Two tables may assign opposite overall signs to a coupled state. That can be a legitimate rephasing, not an error. But if the coupled states are columns of a matrix $C$, then coordinates and operators must change together: $|\psi\rangle_{\rm product}=C|\psi\rangle_{\rm coupled}$ and $O_{\rm coupled}=C^\dagger O_{\rm product}C$. Under $C\mapsto CD$ for a diagonal phase matrix $D$, use $|\psi\rangle_{\rm coupled}\mapsto D^\dagger|\psi\rangle_{\rm coupled}$ and $O_{\rm coupled}\mapsto D^\dagger O_{\rm coupled}D$.

An overall sign of one multiplet is compatible with standard positive ladder coefficients. Independently rephasing different $M$ states may also require changing those ladder-matrix signs. Compare a complete convention, not one coefficient in isolation.

Geometric phases can be computed without slow time evolution

There are two useful numerical approaches. One solves the time-dependent Schrödinger equation and removes the dynamical phase; then the time step, eigenspace leakage, and approach to the adiabatic limit all matter. The other samples instantaneous eigenvectors directly around a parameter loop. That second calculation is geometric and needs a finer parameter mesh, not a longer physical traversal time.

For normalized eigenvectors $|u_k\rangle$, $k=0,\ldots,N-1$, on an ordered closed loop, a discrete estimate in the convention $A=i\langle u|du\rangle$ is

$$\gamma_N=-\arg\prod_{k=0}^{N-1}\langle u_k|u_{k+1}\rangle, \qquad |u_N\rangle=|u_0\rangle.$$

Every arbitrary eigenvector phase cancels between neighboring overlaps. Refine the mesh until the phase converges modulo $2\pi$, keeping neighboring overlaps nonzero. A zero or very small overlap signals that the mesh or isolated-state description needs attention. For a Bloch-band loop, the endpoint states must be compared using the correct reciprocal-lattice identification.

For the lower state of $H=-\Delta\hat{\mathbf n}\cdot\boldsymbol\sigma/2$ with $\Delta>0$, a polar circle gives $-\pi(1-\cos\theta_0)$. Reversing the loop reverses this phase. These provide independent sign and orientation checks for the overlap calculation.

Reading a diagram as a quantum statement

A Bloch-sphere arrow represents Pauli expectation values. It is not the trajectory of an electron through ordinary space. A complex spherical-harmonic plot may show magnitude, phase, or probability density, each of which answers a different question. A real orbital such as $p_x$ is a specified superposition of complex $m$ states, not the same plotted function as $Y_1^1$.

Read the caption to identify the state, basis, and plotted quantity before interpreting a shape. When a calculation is involved, compare at least one analytic case and a limiting value. The visualization gallery links diagrams to the relevant explanations.

Exercises

1. A spin-rotation calculation gives equal $z$ probabilities after $U_y(\pi/2)$ acts on $|+z\rangle$. Is that enough to determine the rotation sign? Give a measurement that settles it.

Solution

No. Both $|+x\rangle=(|+z\rangle+|-z\rangle)/\sqrt2$ and $|-x\rangle=(|+z\rangle-|-z\rangle)/\sqrt2$ have equal $z$ probabilities. Measuring $\sigma_x$ gives $+1$ for the stated active rotation and $-1$ for its inverse.

2. A table rephases a coupled state by $-1$. Show why changing the basis vector alone while leaving its expansion coefficient fixed generally changes a superposition.

Solution

For $|\psi\rangle=a|e_1\rangle+b|e_2\rangle$, changing $|e_2\rangle$ to $-|e_2\rangle$ requires $b\mapsto-b$ to represent the same vector. If $b$ stays fixed, the relative phase changes. When both components are present, a measurement in a superposition basis can distinguish the states.

3. Prove that the closed overlap product is unchanged by independently replacing $|u_k\rangle$ with $e^{i\chi_k}|u_k\rangle$. Why would omitting the closing overlap spoil the conclusion?

Solution

Each overlap gains $e^{i(\chi_{k+1}-\chi_k)}$. Their product around the closed loop is one. Without the final overlap back to $u_0$, an uncanceled endpoint factor remains, so the product’s phase depends on the arbitrary endpoint choices.

References

• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• D. Vanderbilt, Berry Phases in Electronic Structure Theory, Cambridge University Press, 2018.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.