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.

Set 1: Foundations and Normalization

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

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

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

Checks for the other problems

2. $1=2A\int_0^\infty e^{-2\kappa x}dx=A/\kappa$, so $A=\kappa$.
3. $P(a<x<b)=\int_a^b|\psi(x)|^2dx$; the density has units of inverse length.
4. With a normalized angular function, the shell probability is $r^2|R(r)|^2dr$ because $d^3r=r^2dr\,d\Omega$.
5. Insert the identity: $\langle\phi|I|\psi\rangle=\int dx\,\langle\phi|x\rangle\langle x|\psi\rangle=\int dx\,\phi^*\psi$.

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

Set 2: Boundary Conditions and Bound States

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

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

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. 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. 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.

Checks for the other problems

1. The first wall removes the cosine solution; the second requires $\sin(kL)=0$. Nonzero states have $k=n\pi/L$, $n\ge1$.
2. Finite walls admit decaying tails, matched continuously with their first derivatives to the interior solution.
3. At the regular center, use $\psi'(0)=0$ for even states and $\psi(0)=0$ for odd states. At the right edge, match the logarithmic derivative to $-\kappa$ from the exterior decaying solution.
4. Hold the interior potential at zero while raising the exterior barrier. Check grid refinement and domain enlargement separately from the finite-barrier limit; otherwise grid error can conceal the approach to the hard-wall energies.

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

Set 3: Scattering, Current, and Tunneling

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

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. 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. 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.

Checks for the other problems

2. A plane wave’s current is its density times velocity; different velocities require different factors even at equal amplitude.
3. Continuity gives $1+r=t$ and $k(1-r)=qt$, with $k=\sqrt{2mE}/\hbar$ and $q=\sqrt{2m(E-V_0)}/\hbar$. Thus $r=(k-q)/(k+q)$, $t=2k/(k+q)$, and $T=4kq/(k+q)^2$.
4. At fixed energy and height, $T(a+\Delta a)/T(a)\simeq e^{-2\kappa\Delta a}$ in the opaque regime.
5. Before boundary reflections occur, integrate density in the left, barrier, and right regions. Their sum must stay near one; only after central probability becomes negligible may the left and right integrals alone be identified with final reflection and transmission probabilities.

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

Set 4: Oscillators and Finite-Dimensional Models

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

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

4. 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. 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.

Checks for the other problems

1. At fixed mass, the spacing doubles and the length becomes $\ell/\sqrt2$.
2. Localizing at the potential minimum increases momentum uncertainty. Minimizing $(\Delta p)^2/(2m)+m\omega^2(\Delta x)^2/2$ subject to $\Delta x\Delta p\ge\hbar/2$ gives $\hbar\omega/2$.
3. The equation is $(x/\ell+\ell\,d/dx)\psi_0=0$, so $\psi_0=\pi^{-1/4}\ell^{-1/2}e^{-x^2/(2\ell^2)}$ after normalization.
4. An oscillator has an unbounded occupation-number ladder; an ideal two-level system has a two-dimensional Hilbert space. A physical qubit device needs an isolated, controllable pair of levels, even when additional device levels exist.

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

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

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. 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. 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. 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. 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.

Checks for the other problems

1. The triples are $(1,1,2)$, $(1,2,1)$, and $(2,1,1)$, giving degeneracy three.
2. Rotations relate the $2\ell+1$ states within a fixed angular-momentum multiplet. They do not require equal energies for different $\ell$; the Coulomb potential has an additional dynamical symmetry.
3. Substitute $|u|^2=r^2|R|^2$ in the radial integral after the angular integral has given one.
4. Subtraction gives $B[(J+1)(J+2)-J(J+1)]=2B(J+1)$.

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.

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.