Exercise Sets

These exercise sets give a structured practice path through canonical wave mechanics. They are not meant to replace the exercises on individual model pages; they gather recurring techniques and make the progression explicit.

Before starting a set, read Worked Examples Index, skim Problem-Solving Patterns, and keep How to Solve a Wave-Mechanics Problem nearby. Each problem should be solved with stated assumptions, boundary conditions, normalization convention, and at least one check.

Level Guide

Level	Role	Typical Output
Level 1	Direct calculation	Normalize, differentiate, integrate, or substitute into an equation
Level 2	Conceptual and interpretive	Explain what a formula, plot, or boundary condition means physically
Level 3	Derivation	Derive a spectrum, matching rule, current formula, or radial equation
Level 4	Computational	Set up a numerical check, convergence test, or reproducible plot
Level 5	Challenge or bridge	Connect a canonical model to a later volume without importing all later machinery

Set 1: Foundations and Normalization

1. Level 1. For the infinite square well on $0\lt x\lt L$, verify that $\psi_n(x)=\sqrt{2/L}\sin(n\pi x/L)$ is normalized.

Solution

Use

$$\int_0^L \sin^2\frac{n\pi x}{L}\,dx=\frac{L}{2}.$$

Then

$$\int_0^L \lvert\psi_n(x)\rvert^2\,dx =\frac{2}{L}\int_0^L \sin^2\frac{n\pi x}{L}\,dx =1.$$

2. Level 1. A wavefunction on the line has probability density $\rho(x)=Ae^{-2\kappa\lvert x\rvert}$ with $\kappa\gt0$. Find $A$.

3. Level 2. Explain why $\lvert\psi(x)\rvert^2$ is a probability density, not a probability, and state what must be integrated to get a probability on an interval.

4. Level 2. In spherical coordinates, explain why a radial probability density is not just $\lvert R(r)\rvert^2$.

5. Level 3. Starting from $\int dx\,\lvert x\rangle\langle x\rvert=\hat I$, derive the position-space inner product $\langle\phi\vert\psi\rangle=\int dx\,\phi^*(x)\psi(x)$.

Good companion pages: Wavefunctions and Probability Density, Normalization Conventions, Coordinate Representation.

Set 2: Boundary Conditions and Bound States

1. Level 1. For an infinite square well, show that the boundary conditions $\psi(0)=\psi(L)=0$ force $k=n\pi/L$.

2. Level 2. Explain why the finite square well does not use $\psi=0$ at the edges.

3. Level 3. Derive the derivative jump condition for $V(x)=\lambda\delta(x)$ by integrating the stationary Schrödinger equation across a small interval around the origin.

Solution

The stationary equation is

$$-\frac{\hbar^2}{2m}\psi''(x) +\lambda\delta(x)\psi(x) =E\psi(x).$$

Integrate from $-\epsilon$ to $\epsilon$:

$$-\frac{\hbar^2}{2m} \left[\psi'(\epsilon)-\psi'(-\epsilon)\right] +\lambda\psi(0) =E\int_{-\epsilon}^{\epsilon}\psi(x)\,dx.$$

The right side vanishes as $\epsilon\to0$ for a finite wavefunction. Therefore

$$\psi'(0^+)-\psi'(0^-) =\frac{2m\lambda}{\hbar^2}\psi(0).$$

4. Level 3. For a symmetric finite well, use parity to reduce the matching problem to even and odd sectors. State which matching condition is applied at the center and which is applied at the well edge.

5. Level 4. Build a finite-difference Hamiltonian for a deep finite well. Vary the outside barrier height and verify that the lowest energies approach the infinite-well values.

Good companion pages: Boundary Conditions, Boundary Conditions Table, Finite Square Well, Delta Function Potential.

Set 3: Scattering, Current, and Tunneling

1. Level 1. For a plane wave $Ae^{ikx}$ with $k\gt0$, compute the one-dimensional probability current.

Solution

Use

$$j=\frac{\hbar}{m} \operatorname{Im}\left(\psi^*\frac{d\psi}{dx}\right).$$

For $\psi=Ae^{ikx}$,

$$\psi^*\frac{d\psi}{dx} =A^*e^{-ikx}\,ikAe^{ikx} =ik\lvert A\rvert^2.$$

Thus

$$j=\frac{\hbar k}{m}\lvert A\rvert^2.$$

2. Level 2. Explain why reflection and transmission coefficients are current ratios rather than merely amplitude-squared ratios.

3. Level 3. For a potential step with $E\gt V_0$, derive $r=(k-q)/(k+q)$ and identify why $T$ includes the factor $q/k$.

4. Level 3. In the opaque-barrier approximation, $T$ is proportional to $e^{-2\kappa a}$. What happens to $T$ when the barrier width $a$ is increased by $\Delta a$?

5. Level 4. Numerically propagate a Gaussian packet toward a rectangular barrier. Check that the total norm is conserved and estimate reflected plus transmitted probability after the packets separate.

Good companion pages: Probability Current, Potential Step, Reflection and Transmission Coefficients, Rectangular Barrier Tunneling.

Set 4: Oscillators and Finite-Dimensional Models

1. Level 1. If $\omega$ doubles in the harmonic oscillator, how do the level spacing $\hbar\omega$ and oscillator length $\ell=\sqrt{\hbar/(m\omega)}$ change?

2. Level 2. Explain why the harmonic-oscillator ground-state energy is not zero even though the potential minimum is zero.

3. Level 3. Starting from $\hat a\lvert0\rangle=0$, derive the Gaussian ground-state wavefunction in position space.

4. Level 3. For $\lvert\psi\rangle=(\lvert0\rangle+\sqrt3\,\lvert1\rangle)/2$, list the possible oscillator energies and their probabilities.

Solution

The coefficients are $c_0=1/2$ and $c_1=\sqrt3/2$. Therefore the possible energies are

$$E_0=\frac{1}{2}\hbar\omega, \qquad E_1=\frac{3}{2}\hbar\omega,$$

with probabilities

$$P(E_0)=\lvert c_0\rvert^2=\frac{1}{4}, \qquad P(E_1)=\lvert c_1\rvert^2=\frac{3}{4}.$$

5. Level 5. Explain why the oscillator is a bridge to field modes, while a two-level system is a bridge to qubits. Keep the answer at the level of Hilbert-space structure and spectra.

Good companion pages: Quantum Harmonic Oscillator, Ladder-Operator Solution, Number States, Two-Level Systems.

Set 5: Three Dimensions, Angular Structure, and Magnetic Fields

1. Level 1. For a three-dimensional box with equal sides $L$, find all positive integer triples $(n_x,n_y,n_z)$ with $n_x^2+n_y^2+n_z^2=6$ and state the degeneracy.

2. Level 2. Explain why $m$ degeneracy is expected in a rotationally invariant central potential, while hydrogen’s extra $\ell$ degeneracy is special to the Coulomb potential.

3. Level 3. Starting from $u(r)=rR(r)$, show how the radial normalization $\int_0^\infty\lvert u(r)\rvert^2\,dr=1$ follows from the three-dimensional normalization when the angular part is normalized.

4. Level 3. For the rigid rotor, show that the energy difference between adjacent levels is

$$E_{J+1}-E_J=2B(J+1), \qquad B=\frac{\hbar^2}{2I}.$$

5. Level 4. In Landau gauge, use guiding-center spacing to estimate the degeneracy of one Landau level in a rectangle of area $A=L_xL_y$.

Solution

For a particle with charge magnitude $\lvert q\rvert$ in a uniform magnetic field, the magnetic length is

$$\ell_B=\sqrt{\frac{\hbar}{\lvert q\rvert B}}.$$

Periodic boundary conditions in the $y$ direction give spacing $\Delta k_y=2\pi/L_y$. The guiding center spacing is therefore

$$\Delta x_0=\ell_B^2\Delta k_y =\frac{2\pi\ell_B^2}{L_y}.$$

The number of allowed centers across width $L_x$ is approximately

$$N=\frac{L_x}{\Delta x_0} =\frac{A}{2\pi\ell_B^2} =\frac{\lvert q\rvert BA}{h}.$$

Boundary corrections are ignored in this bulk estimate.

Good companion pages: Three-Dimensional Box, Radial Schrödinger Equation, Degeneracy of the Hydrogen Atom, Rigid Rotor, Landau Levels.

Study Patterns

For each set, solve one direct calculation, one interpretation question, and one derivation before moving on. If a derivation fails, write down the exact condition that blocked it: a boundary condition, a normalization convention, a missing identity, or a physical assumption.

For computational problems, record the grid, time step, boundary conditions, norm error, and comparison target. A numerical answer without these checks is not yet reproducible.

Where This Is Used

• Worked Examples Index explains how to choose examples by task.
• Problem-Solving Patterns summarizes the methods practiced across these sets.
• Numerical Notebooks Index turns selected computational exercises into reproducible artifacts.
• Benchmark Problems gives pass criteria for computational exercises that become notebooks.
• Common Hamiltonians identifies the model behind each exercise.
• Spectra and Eigenfunctions Table gives quick checks for exact spectra.
• Canonical Plots Gallery helps interpret 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.