Student Quick Reference

This is the quick survival sheet for an undergraduate quantum mechanics problem set or exam review. It is not a formula dump. It is a compact reminder of the objects to name, formulas to use carefully, checks to run, and pages to open when a problem stops being routine.

For the fuller workflow, use How to Solve Problems. For diagnosis after a wrong turn, use Common Mistakes.

The Core Loop

Every problem should pass through this loop, even if some steps are quick.

1. Identify the physical system and approximation regime.
2. Identify the Hilbert space and representation.
3. Identify the Hamiltonian and its domain or boundary conditions.
4. Identify the observable or measurement being asked about.
5. Choose a basis that simplifies the state, Hamiltonian, or observable.
6. Solve, then convert amplitudes into probabilities, expectation values, spectra, or time scales.
7. Check dimensions, normalization, limits, symmetries, and interpretation.

If you cannot name the state space, Hamiltonian, observable, and basis, do not start algebra yet.

Objects to Name

Object	Question to answer	Typical failure
State	Is it a ket, wavefunction, density operator, or coefficient column?	Treating a representation as the state itself.
Hilbert space	Is it $L^2(\mathbb R)$, $L^2([0,L])$, $\mathbb C^2$, a tensor product, or something else?	Using the wrong normalization or basis size.
Hamiltonian	What dynamics or energy model is assumed?	Copying $\hat p^2/(2m)+V$ into a problem where spin, fields, or driving matter.
Domain	What boundary, regularity, or matching conditions are part of the operator?	Reusing a spectrum after changing walls or intervals.
Observable	What measurement or quantity is requested?	Computing a state when the question asks for a probability or expectation value.
Basis	Which basis makes the calculation simple?	Mixing coefficients from one basis with operators from another.
Convention	Are units, Fourier signs, spin basis, or tensor ordering declared?	Losing a sign, $\hbar$, or basis permutation.
Approximation	What small parameter or idealization is being used?	Treating an approximation as exact.

Formulas You Actually Need

Use these as reminders with assumptions attached. For a larger list, see Core Formulas Index.

Need	Formula	Warning
Pure-state normalization	$\langle\psi\vert\psi\rangle=1$	Nonzero scalar multiples represent the same ray only after normalization.
Position normalization	$\int\lvert\psi(x)\rvert^2\,dx=1$	The measure changes in other coordinates.
Born rule, basis state	$P(a_n)=\lvert\langle a_n\vert\psi\rangle\rvert^2$	Assumes normalized states and a specified measurement basis.
Born rule, projector	$P(P_a)=\langle\psi\vert P_a\vert\psi\rangle$	Use projectors for degeneracy or subspace outcomes.
Expectation value	$\langle A\rangle=\langle\psi\vert A\vert\psi\rangle$	The state and operator must be in the same representation.
Variance	$(\Delta A)^2=\langle A^2\rangle-\langle A\rangle^2$	Requires finite moments and a self-adjoint observable.
Schrödinger equation	$i\hbar\,d\lvert\psi\rangle/dt=H\lvert\psi\rangle$	Closed-system form; open systems need density-operator dynamics.
Time-independent evolution	$\lvert\psi(t)\rangle=\sum_n c_ne^{-iE_nt/\hbar}\lvert E_n\rangle$	Works after expanding in energy eigenstates.
Commutator	$[A,B]=AB-BA$	Operator order matters.
Uncertainty bound	$\Delta A\,\Delta B\ge\frac12\lvert\langle[A,B]\rangle\rvert$	This is a lower bound, not a measurement-error formula.
Fourier pair	$\phi(p)=(2\pi\hbar)^{-1/2}\int e^{-ipx/\hbar}\psi(x)\,dx$	Check sign and normalization conventions.
Tensor product	$\lvert ij\rangle=\lvert i\rangle_A\otimes\lvert j\rangle_B$	Ordering must be declared.
Density operator	$\rho=\sum_k p_k\lvert\psi_k\rangle\langle\psi_k\rvert$	Ensemble decompositions are not unique.
Trace rule	$\langle A\rangle=\operatorname{Tr}(\rho A)$	Essential for mixed states and subsystems.

Model Recognition

Clue in the problem	Think of	First page to open
No potential, free motion, wave packets	Free particle	Free Particle
Hard walls or fixed interval	Infinite square well or box	Infinite Square Well
Quadratic potential or small oscillations	Harmonic oscillator	Quantum Harmonic Oscillator
Step, barrier, tunneling, reflection	One-dimensional scattering	Rectangular Barrier Tunneling
Spherical symmetry or $V(r)$	Central potential	Radial Schrödinger Equation
Coulomb attraction and orbitals	Hydrogen atom	Hydrogen Atom
Two basis states, avoided crossing, qubit	Two-level system	Two-Level Systems
Spin measurement or magnetic field	Spin-half system	Spin-1/2 as a Canonical System
Two parts or entanglement	Tensor-product system	Tensor Products
Small perturbation to solvable model	Perturbation theory	Nondegenerate Perturbation Theory
Slow variation or large quantum numbers	WKB or semiclassical method	WKB Approximation

For a broader table, use Most-Used Hamiltonians and Common Hamiltonians.

Probability Workflow

When the problem asks for a probability:

1. Name the measurement.
2. Write the eigenstate, projector, or POVM effect for the outcome.
3. Put the state and measurement operator in the same basis.
4. Apply the Born rule.
5. Check that all exhaustive outcome probabilities sum or integrate to one.

For a discrete orthonormal basis $\{\lvert a_n\rangle\}$,

$$\lvert\psi\rangle = \sum_n c_n\lvert a_n\rangle, \qquad c_n=\langle a_n\vert\psi\rangle, \qquad P(a_n)=\lvert c_n\rvert^2.$$

For a continuous position measurement in one dimension,

$$P(a\le x\le b) = \int_a^b\lvert\psi(x)\rvert^2\,dx.$$

The density $\lvert\psi(x)\rvert^2$ is not the probability of the exact point $x$.

Time-Evolution Workflow

For a time-independent closed system:

1. Solve or identify $H\lvert E_n\rangle=E_n\lvert E_n\rangle$.
2. Expand the initial state as $\lvert\psi(0)\rangle=\sum_n c_n\lvert E_n\rangle$.
3. Attach phases $e^{-iE_nt/\hbar}$.
4. Transform to the measurement basis if needed.
5. Compute probabilities or expectation values.

For a time-dependent Hamiltonian, this shortcut usually fails. Use Time-Dependent Hamiltonians and approximation tools when the drive or perturbation matters.

Sanity Checks

Check	What to ask
Dimensions	Does every term in an equation have the same units? Are phases dimensionless?
Normalization	Does the state have norm one, box normalization, or delta normalization?
Probability	Do probabilities lie between 0 and 1 and sum or integrate to one?
Reality	Are expectation values of observables real?
Positivity	Is a variance or probability accidentally negative?
Boundary conditions	Does the wavefunction obey the stated wall, matching, or regularity condition?
Limiting cases	Does the answer simplify correctly when a coupling, field, or perturbation goes to zero?
Symmetry	Are conserved quantities and degeneracies consistent with the Hamiltonian?
Phase	Would a global phase incorrectly change the answer?
Basis	Were states, operators, and components translated together?

The formalism-specific version is Common Checks and Sanity Tests.

When Stuck

If a calculation has become long and unclear, stop and classify the failure.

Symptom	Likely issue	Recovery
The result has wrong units	Missing $\hbar$, mass, length, charge, or frequency scale	Rebuild the formula by dimensions before continuing.
Probabilities do not sum to one	State, basis, or density is misnormalized	Normalize first; then recompute.
A spectrum looks continuous in a box	Boundary conditions were not imposed	Solve the eigenvalue problem with the domain included.
Spin probabilities look basis independent	Measurement basis was not changed correctly	Diagonalize the measured spin component or use projectors.
A tensor-product matrix acts on the wrong qubit	Subsystem order is ambiguous	Restore identity factors and basis ordering.
Perturbation theory gives a huge correction	Small parameter is not small or degeneracy was ignored	Switch to degenerate perturbation theory or another method.
Two references disagree by a sign	Convention mismatch	Use Common Convention Translations.

Common Pitfalls

• Starting with a memorized formula before naming the system.
• Treating a Hamiltonian expression as complete without boundary conditions.
• Forgetting that eigenstates of one observable are usually not eigenstates of another.
• Using $\lvert\psi(x)\rvert^2$ as a probability rather than a density.
• Confusing a probability amplitude with a probability.
• Dropping a relative phase that affects interference.
• Forgetting that a global phase is physically irrelevant.
• Comparing tensor-product coefficient vectors without checking basis order.
• Treating every two-level system as already diagonal in the chosen basis.
• Assuming a first-order perturbation formula works in a degenerate subspace.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• J. S. Townsend, A Modern Approach to Quantum Mechanics, 2nd ed., University Science Books, 2012.

Exercises

1. A normalized state is $\lvert\psi\rangle=(\lvert0\rangle+i\lvert1\rangle)/\sqrt2$. What are the probabilities for measuring in the $\{\lvert0\rangle,\lvert1\rangle\}$ basis?

Solution

The coefficients have squared magnitudes $1/2$ and $1/2$. Therefore $P(0)=1/2$ and $P(1)=1/2$.

2. A particle in an infinite well has a proposed eigenfunction that does not vanish at one wall. What should you do before computing energies?

Solution

Reject or modify the proposed function so that it satisfies the boundary conditions. The hard-wall boundary conditions are part of the Hamiltonian domain, and energies computed from functions outside the domain are not valid eigenvalues of the well problem.

3. A perturbation correction is the same size as the unperturbed level spacing. What warning should that trigger?

Solution

The perturbative expansion may not be controlled. Check for degeneracy or near-degeneracy, identify the small parameter, and consider degenerate perturbation theory, a variational method, or numerical diagonalization.