Worked Examples Index

This index points to the worked calculations, guided estimates, and exercise entry points already distributed across the Wave Mechanics and Model Systems volume. It is organized by problem-solving task rather than by sidebar location.

Use How to Solve a Wave-Mechanics Problem for the general workflow, then use Problem-Solving Patterns to recognize recurring moves before finding a concrete example below.

For a structured practice path, use Exercise Sets after choosing the technique you want to work on.

How Examples Are Classified

A useful worked example should make the hidden choices visible: coordinate system, Hilbert space, boundary conditions, normalization convention, observable, approximation, and limiting checks. A calculation that only quotes a final formula is not a worked example for this index.

The categories below match the main recurring tasks:

• normalization and measures;
• expectation values and observables;
• boundary conditions and matching;
• eigenvalue problems;
• scattering coefficients and currents;
• wave-packet evolution;
• radial and angular equations;
• magnetic fields and gauge choices.

Example Map

Task	Good Starting Pages	What the Reader Should Practice
Normalization and measures	Wavefunctions and Probability Density, Normalization Conventions, Infinite Square Well	Normalize square-integrable states, track coordinate measures, and distinguish density from probability
Expectation values and observables	Expectation Values, Number States, Two-State Hamiltonians	Compute averages from states, identify the correct operator, and check dimensions
Boundary conditions and matching	Boundary Conditions, Boundary Conditions Table, Finite Square Well, Delta Function Potential	Decide when $\psi$ vanishes, when $\psi'$ is continuous, and when a derivative jump is required
Eigenvalue problems	Time-Independent Schrödinger Equation, Spectra and Eigenfunctions Table, Quantum Harmonic Oscillator, Three-Dimensional Box	Turn boundary conditions into allowed energies and test the resulting eigenfunctions
Scattering coefficients	Potential Step, Reflection and Transmission Coefficients, Rectangular Barrier Tunneling, Transfer Matrix Method	Match regions, compute currents, and avoid confusing amplitudes with probabilities
Wave-packet evolution	Gaussian Wave Packets, Wave Packet Spreading, Group Velocity and Phase Velocity, Free-Particle Propagator	Follow a localized state in time and compare envelope motion, phase motion, and spreading
Radial equations	Radial Schrödinger Equation, Hydrogen Atom, Degeneracy of the Hydrogen Atom	Use $u(r)=rR(r)$, impose regularity at $r=0$, and separate radial from angular structure
Angular probability	Separation of Variables, Rigid Rotor, Spherical Harmonics	Interpret angular wavefunctions, degeneracy, and probability on the sphere
Magnetic fields and gauges	Minimal Coupling in Wave Mechanics, Landau Levels, Canonical Plots Gallery	Separate gauge-dependent wavefunctions from gauge-invariant energies, currents, and degeneracies

First-Pass Routes

For a first course in wave mechanics, start with normalization, infinite-well quantization, the finite-well matching problem, Gaussian spreading, and oscillator eigenstates. This route builds the habits that later pages reuse.

For scattering and tunneling, begin with the potential step, then read reflection and transmission coefficients before using the rectangular barrier or transfer matrix method. The main transferable skill is computing probabilities from current.

For three-dimensional and atomic problems, begin with separation of variables, then the three-dimensional box, then the radial Schrödinger equation, then hydrogen. The recurring mistake is forgetting the measure: $dx$ in one dimension, $d^3r$ in Cartesian coordinates, and $r^2dr\,d\Omega$ in spherical coordinates.

For oscillator and finite-dimensional models, read the harmonic oscillator page, then the ladder-operator first encounter, number states, coherent states, and two-state Hamiltonians. This route prepares the bridge to field modes, quantum optics, and qubits without duplicating those later volumes.

What to Record When Solving

When writing a worked solution, record enough context that the answer can be checked later:

• the Hamiltonian and the domain;
• the normalization convention;
• all boundary or matching conditions;
• the physical meaning of any coefficients;
• a dimension check;
• a limiting-case check;
• the page where the canonical derivation lives.

This record prevents a common failure mode: a solution that is algebraically correct for a different physical problem.

Common Mistakes

• Looking for an example by model name when the real need is a technique, such as matching or flux normalization.
• Copying a formula from an eigenvalue example into a scattering problem.
• Using a one-dimensional normalization habit in spherical coordinates.
• Treating a plotted wavefunction as normalized without checking the caption or convention.
• Forgetting that many examples are first encounters and deliberately defer advanced machinery to later volumes.

Where This Is Used

• Exercise Sets turns this index into curated practice ladders.
• Problem-Solving Patterns explains the recurring moves behind the examples listed here.
• Numerical Notebooks Index tracks reproducible computational versions of selected examples.
• Visualization Gallery records committed visual assets and the checks required before adding more.
• Common Hamiltonians helps identify which model an example belongs to.
• Spectra and Eigenfunctions Table provides quick formula checks after working a problem.
• Boundary Conditions Table helps choose the right endpoint or matching condition.
• Canonical Plots Gallery gives visual checks for qualitative behavior.

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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.

Exercises

1. You need to solve a delta-function potential problem and check whether the derivative is continuous. Which pages in this index should you read first?

Solution

Start with Boundary Conditions for the conceptual rule, then Boundary Conditions Table for the quick matching condition, and then Delta Function Potential for the canonical model page. The derivative is not continuous across the delta interaction; it has a jump proportional to $\psi(0)$.

2. A numerical wave-packet plot shows a transmitted pulse after a rectangular barrier. What two checks should you make before interpreting the transmitted height as a probability?

Solution

First check the normalization convention and whether the plotted quantity is $\psi$ or $\lvert\psi\rvert^2$. Second check probability current: transmission probabilities in scattering are flux ratios, so amplitude height alone is not enough when velocities or normalization conventions differ.