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.

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 in Position Space, 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 densities and currents, and spectra in equivalent static gauges

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.

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.

References

• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.