Exercises and Problems

This page is a problem index for Core Formalism. It collects the main skills readers should be able to perform after studying the volume and gives representative solved problems for self-checking.

The problems here are intentionally compact. Longer derivations and topic-specific exercises remain on their canonical teaching pages. Before using this page as an exam-style review, read the Dependency Graph of the Formalism and keep Common Checks and Sanity Tests nearby.

Skill Map

Problem group	Core skills	Preparation pages
State vectors and basis changes	normalize states, compute components, translate bases	State Vectors, Change of Basis
Born rule calculations	identify projectors, compute probabilities, handle phases	Born Rule, Probability in Different Bases
Expectation values, variances, and covariances	compute averages, spreads, correlations, and reality checks	Expectation Values, Variance and Standard Deviation, Correlations and Covariance
Commutators and uncertainty	evaluate commutators, interpret compatibility and spreads	Commutators, General Uncertainty Relations
Projective measurements	use projectors, degeneracy, and state update	Projective Measurement, State Update Rule
Time evolution	use Hamiltonians and unitary evolution	Hamiltonians, Unitary Time Evolution
Tensor products and entanglement	identify product states, compute local predictions	Tensor Products, Entangled States
Density matrices	check positivity, traces, purity, reduced states	Density Operators, Reduced Density Matrices
Postulates and conceptual questions	identify assumptions, scope, and interpretation boundaries	Minimal Postulates, What the Postulates Do Not Say

Problem Metadata

Each representative problem states:

• Level: undergraduate, graduate bridge, or conceptual review.
• Skills: compact tags such as basis-change, born-rule, density-matrix, or postulates.
• Solution status: full solution or guided solution.

When adding new problems, keep them tied to canonical pages and avoid duplicating long derivations that already live elsewhere.

Representative Problems

1. Basis Components of a Qubit

Level: undergraduate
Skills: basis-change, state-vector, normalization
Solution status: full solution

Let

$$\lvert\psi\rangle = \frac{\lvert0\rangle+i\lvert1\rangle}{\sqrt2}, \qquad \lvert+\rangle = \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}, \qquad \lvert-\rangle = \frac{\lvert0\rangle-\lvert1\rangle}{\sqrt2}.$$

Compute the components of $\lvert\psi\rangle$ in the $\{\lvert+\rangle,\lvert-\rangle\}$ basis and check normalization.

Solution

The new components are inner products with the new basis:

$$c_+ = \langle+\vert\psi\rangle = \frac{1+i}{2}, \qquad c_- = \langle-\vert\psi\rangle = \frac{1-i}{2}.$$

Normalization is preserved:

$$\lvert c_+\rvert^2+\lvert c_-\rvert^2 = \frac{2}{4}+\frac{2}{4} = 1.$$

2. Born Rule in Two Bases

Level: undergraduate
Skills: born-rule, relative-phase, basis-change
Solution status: full solution

Let

$$\lvert\psi\rangle = \frac{\sqrt3}{2}\lvert0\rangle + \frac{e^{i\phi}}{2}\lvert1\rangle.$$

Find the probabilities for measuring $\lvert0\rangle$ or $\lvert1\rangle$, and the probability for the $+$ outcome in the $\{\lvert+\rangle,\lvert-\rangle\}$ basis.

Solution

In the $\{\lvert0\rangle,\lvert1\rangle\}$ basis,

$$p(0) = \frac{3}{4}, \qquad p(1) = \frac{1}{4}.$$

For the $+$ outcome,

$$p(+) = \lvert\langle+\vert\psi\rangle\rvert^2 = \frac{1}{2} \left\lvert \frac{\sqrt3}{2} + \frac{e^{i\phi}}{2} \right\rvert^2.$$

Expanding the squared magnitude gives

$$p(+) = \frac{1}{2} + \frac{\sqrt3}{4}\cos\phi.$$

The relative phase does not affect the $\lvert0\rangle,\lvert1\rangle$ probabilities, but it does affect interference in the $\lvert+\rangle,\lvert-\rangle$ basis.

3. Expectation Value and Variance

Level: undergraduate
Skills: expectation-value, variance, pauli-matrices
Solution status: full solution

For the same state as Problem 2, compute $\langle\sigma_z\rangle$ and $\operatorname{Var}(\sigma_z)$.

Solution

Since $\sigma_z\lvert0\rangle=\lvert0\rangle$ and $\sigma_z\lvert1\rangle=-\lvert1\rangle$,

$$\langle\sigma_z\rangle = p(0)-p(1) = \frac{3}{4}-\frac{1}{4} = \frac{1}{2}.$$

Also $\sigma_z^2=I$, so $\langle\sigma_z^2\rangle=1$. Therefore

$$\operatorname{Var}(\sigma_z) = \langle\sigma_z^2\rangle-\langle\sigma_z\rangle^2 = 1-\frac{1}{4} = \frac{3}{4}.$$

4. Commutator and Uncertainty Check

Level: undergraduate
Skills: commutator, uncertainty, operator-algebra
Solution status: full solution

Use $[\sigma_x,\sigma_y]=2i\sigma_z$. In the state $\lvert0\rangle$, compute the Robertson lower bound for $\Delta\sigma_x\,\Delta\sigma_y$, and compare it with the actual product.

Solution

The Robertson bound gives

$$\Delta\sigma_x\,\Delta\sigma_y \ge \frac{1}{2} \left\lvert \langle[\sigma_x,\sigma_y]\rangle \right\rvert = \frac{1}{2} \left\lvert 2i\langle\sigma_z\rangle \right\rvert.$$

In $\lvert0\rangle$, $\langle\sigma_z\rangle=1$, so the lower bound is $1$.

Since $\langle\sigma_x\rangle=\langle\sigma_y\rangle=0$ and $\sigma_x^2=\sigma_y^2=I$,

$$\Delta\sigma_x = 1, \qquad \Delta\sigma_y = 1.$$

The product is $1$, so the bound is saturated.

5. Degenerate Projective Measurement

Level: undergraduate
Skills: projectors, measurement-update, degeneracy
Solution status: full solution

Let $\{\lvert1\rangle,\lvert2\rangle,\lvert3\rangle\}$ be an orthonormal basis. A measurement has outcome $a$ on the subspace spanned by $\lvert1\rangle,\lvert2\rangle$ and outcome $b$ on the subspace spanned by $\lvert3\rangle$. For

$$\lvert\psi\rangle = \frac{\lvert1\rangle+\lvert2\rangle+\sqrt2\,\lvert3\rangle}{2},$$

find $p(a)$ and the selective post-measurement state after observing $a$.

Solution

The projector for outcome $a$ is

$$P_a = \lvert1\rangle\langle1\rvert + \lvert2\rangle\langle2\rvert.$$

Applying it gives

$$P_a\lvert\psi\rangle = \frac{\lvert1\rangle+\lvert2\rangle}{2}.$$

Thus

$$p(a) = \langle\psi\vert P_a\vert\psi\rangle = \frac{1}{2}.$$

The normalized selective state is

$$\lvert\psi_a\rangle = \frac{P_a\lvert\psi\rangle}{\sqrt{p(a)}} = \frac{\lvert1\rangle+\lvert2\rangle}{\sqrt2}.$$

The relative coherence inside the degenerate outcome subspace is preserved by the ideal Lüders update.

6. Two-Level Time Evolution

Level: undergraduate
Skills: unitary-evolution, hamiltonian, born-rule
Solution status: full solution

Let

$$H = \frac{\hbar\omega}{2}\sigma_z, \qquad \lvert\psi(0)\rangle = \lvert+\rangle.$$

Compute the probability of finding the system in $\lvert+\rangle$ at time $t$.

Solution

The evolution operator is

$$U(t) = e^{-iHt/\hbar} = e^{-i\omega t\sigma_z/2}.$$

Since $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$,

$$\lvert\psi(t)\rangle = \frac{e^{-i\omega t/2}\lvert0\rangle + e^{i\omega t/2}\lvert1\rangle}{\sqrt2}.$$

The return amplitude is

$$\langle+\vert\psi(t)\rangle = \frac{e^{-i\omega t/2}+e^{i\omega t/2}}{2} = \cos\frac{\omega t}{2}.$$

Therefore

$$p_+(t) = \cos^2\frac{\omega t}{2}.$$

7. Product State or Entangled State

Level: undergraduate
Skills: tensor-product, product-state, entanglement
Solution status: full solution

Classify the two states:

$$\lvert\psi\rangle = \frac{\lvert00\rangle+\lvert01\rangle}{\sqrt2}, \qquad \lvert\Phi^+\rangle = \frac{\lvert00\rangle+\lvert11\rangle}{\sqrt2}.$$

Which one factors into subsystem states?

Solution

The first state factors:

$$\lvert\psi\rangle = \lvert0\rangle \otimes \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2} = \lvert0\rangle\otimes\lvert+\rangle.$$

It is a product state. The Bell state $\lvert\Phi^+\rangle$ does not factor into a state of $A$ times a state of $B$; it has Schmidt rank $2$ and is entangled.

8. Density-Matrix Positivity

Level: undergraduate
Skills: density-matrix, positivity, purity
Solution status: full solution

For real $r$, consider

$$\rho(r) = \frac{1}{2} \begin{pmatrix} 1 & r \\ r & 1 \end{pmatrix}.$$

For which values of $r$ is this a valid density matrix? When is it pure?

Solution

The matrix is Hermitian and has trace one for every real $r$. Its eigenvectors are the $\lvert+\rangle$ and $\lvert-\rangle$ states, with eigenvalues

$$\lambda_\pm = \frac{1\pm r}{2}.$$

Positivity requires both eigenvalues to be nonnegative, so

$$-1 \le r \le 1.$$

The state is pure when one eigenvalue is $1$ and the other is $0$, which occurs at $r=1$ or $r=-1$. Equivalently, $\operatorname{Tr}\rho^2=1$ only for $\lvert r\rvert=1$.

9. Postulates and Scope

Level: conceptual review
Skills: postulates, measurement-context, scope
Solution status: full solution

Someone writes: “The state $\lvert\psi\rangle$ tells us what value an observable already has before measurement.” Identify what is wrong and replace the sentence with a formalism-safe version.

Solution

The statement treats the state as assigning pre-existing sharp values to all observables. The standard formalism instead uses the state together with a specified measurement to assign probabilities.

A safer version is:

Given a state and a specified measurement, the Born rule assigns probabilities to possible outcomes. If the state is an eigenstate of the measured observable, one outcome can be sharp; otherwise the formalism generally gives a probability distribution.

This answer separates the state, the observable or measurement, and the probability rule. It also avoids adding an interpretation that is not part of the minimal postulates.

Building a Review Set

For a balanced Core Formalism review, choose:

1. one state-normalization or basis-change problem;
2. one Born-rule problem in a non-measurement basis;
3. one expectation-value and variance problem;
4. one commutator or uncertainty problem;
5. one projective-measurement update problem;
6. one unitary time-evolution problem;
7. one tensor-product or entanglement problem;
8. one density-operator or partial-trace problem;
9. one conceptual postulates question.

After solving, run the Common Checks and Sanity Tests on every answer. If an answer fails a check, use Common Mistakes in the Formalism to diagnose the likely failure mode.

Cross-Links

• Dependency Graph of the Formalism
• Common Checks and Sanity Tests
• Common Mistakes in the Formalism
• Glossary for Core Formalism
• Representation Translation Table
• Symbol Map

References

• 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.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.