Self-Diagnostic Quiz

This quiz is a routing tool. It is not an entrance exam and it is not meant to be solved under pressure. Use it to find weak spots before entering the main formalism, wave mechanics, symmetry, density-matrix, or QFT-bridge pages.

Work without notes first. Then check the solutions, mark each question as solid, shaky, or review needed, and follow the review links at the end.

Core Questions

1. Normalize the vector $v=(1,i,0)\in\mathbb C^3$.

2. Is the matrix

$$A= \begin{pmatrix} 0&i\\ -i&0 \end{pmatrix}$$

Hermitian? Explain.

3. If

$$\sigma_z= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}, \qquad u= \begin{pmatrix} 1\\ 0 \end{pmatrix},$$

what is $\sigma_z u$?

4. A probability density is $\rho(x)=Ax$ on $0<x<1$. Find $A$.

5. For a random variable with values $0$ and $2$ with equal probabilities, compute the expectation value and variance.

6. A function satisfies $f(0)=f(L)=0$ and solves $f''+k^2f=0$. What values of $k$ give nonzero sine solutions?

7. What does it mean, qualitatively, for a wave packet to be localized in position?

8. For a one-dimensional particle with

$$H(x,p)=\frac{p^2}{2m}+V(x),$$

write Hamilton’s equations.

9. State the canonical commutation relation between position and momentum.

10. What physical information is lost if a normalized state vector is multiplied by an overall phase $e^{i\alpha}$?

11. If two Hilbert spaces have dimensions $2$ and $3$, what is the dimension of their tensor product?

12. Is

$$\rho= \begin{pmatrix} 1/2&0\\ 0&1/2 \end{pmatrix}$$

a valid density matrix? Give the basic checks.

13. What is the difference between a probability amplitude and a probability?

14. A plane wave has phase $kx-\omega t$. Which quantity is the angular frequency?

15. State the Lorentz force law.

16. What is the relativistic energy-momentum relation?

17. In a canonical ensemble, what is the role of the partition function $Z$?

18. Why is the Hamiltonian more than just a symbol named $H$?

19. Why should a gauge transformation not change measured electromagnetic fields?

20. Why does high-energy relativistic quantum physics push beyond fixed-particle-number wave mechanics?

Answer Key

Solutions and checks

1. The norm is

$$\lVert v\rVert =\sqrt{\lvert1\rvert^2+\lvert i\rvert^2+0} =\sqrt2.$$

A normalized vector is

$$\frac{1}{\sqrt2}(1,i,0).$$

2. Yes. A matrix is Hermitian if $A^\dagger=A$. Here the off-diagonal entries are complex conjugates in the required transposed positions: the conjugate of $i$ is $-i$.

3. Direct multiplication gives

$$\sigma_z u = \begin{pmatrix} 1\\ 0 \end{pmatrix} =u.$$

So $u$ is an eigenvector with eigenvalue $+1$.

4. Normalize the density:

$$1=\int_0^1 Ax\,dx=\frac{A}{2},$$

so $A=2$.

5. The expectation value is

$$\mathbb E[X]=0\cdot\frac12+2\cdot\frac12=1.$$

The second moment is $\mathbb E[X^2]=0+4/2=2$, so

$$\operatorname{Var}(X)=\mathbb E[X^2]-\mathbb E[X]^2=1.$$

6. The nonzero sine solutions are proportional to $\sin(kx)$. The condition at $x=L$ requires

$$\sin(kL)=0,$$

so

$$k=\frac{n\pi}{L}, \qquad n=1,2,3,\ldots.$$

7. A localized wave packet is concentrated in a finite region of position space rather than spread uniformly over all space. Mathematically it requires a superposition of wave numbers; a perfectly sharp wave number is delocalized.

8. Hamilton’s equations are

$$\dot x=\frac{\partial H}{\partial p}=\frac{p}{m}, \qquad \dot p=-\frac{\partial H}{\partial x}=-V'(x).$$

9. The canonical commutation relation is

$$[\hat x,\hat p]=i\hbar.$$

10. No physical information is lost for an isolated state ray. An overall global phase does not change probabilities or expectation values. Relative phases between components, however, are physically meaningful.

11. The tensor product dimension is $2\cdot3=6$.

12. Yes. It is Hermitian, positive semidefinite, and has trace one:

$$\operatorname{Tr}(\rho)=\frac12+\frac12=1.$$

13. A probability amplitude is generally complex and can interfere with other amplitudes. A probability is a nonnegative real number, usually obtained from amplitudes by a Born-rule prescription such as taking a squared modulus.

14. The angular frequency is $\omega$.

15. The Lorentz force law is

$$\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B).$$

16. The energy-momentum relation is

$$E^2=\mathbf p^2c^2+m^2c^4.$$

17. $Z$ normalizes the canonical distribution and generates thermodynamic quantities. For discrete energies,

$$Z=\sum_i e^{-\beta E_i}.$$

18. The Hamiltonian encodes the system’s energy model and generates time evolution. In quantum mechanics it becomes the operator appearing in the Schrödinger equation.

19. Gauge transformations change potentials while leaving $\mathbf E$ and $\mathbf B$ invariant. Measured field strengths cannot depend on the arbitrary representation chosen for the potentials.

20. Relativity allows energy to become particle rest mass in suitable interactions, so particle number need not be fixed. A field-theoretic description naturally handles creation, annihilation, locality, and Lorentz symmetry.

Interpreting the Result

Mark a question solid if you could solve it without guessing and can explain each step. Mark it shaky if you got the answer but relied on pattern recognition. Mark it review needed if the concept or notation felt unfamiliar.

• Mostly solid on questions 1-14: you are ready for the early formalism and wave mechanics pages.
• Shaky on 1-6: review linear algebra, probability, complex numbers, Fourier analysis, and differential equations before pushing deeper.
• Shaky on 8, 15, 18, or 19: review classical mechanics and electromagnetism before studying Hamiltonians, magnetic coupling, or semiclassical arguments.
• Shaky on 11-13: review tensor products, density matrices, and the distinction between pure states, mixtures, and probabilities.
• Shaky on 16 or 20: review special relativity before taking the QFT-bridge route.
• Shaky on 17: review statistical mechanics before thermal states, entropy, open systems, or many-body material.

Where to Review

Use the targeted checklists rather than rereading everything:

• Linear Algebra Checklist
• Calculus and Differential Equations Checklist
• Complex Numbers and Fourier Analysis Checklist
• Probability Checklist
• Classical Mechanics Checklist
• Electromagnetism Checklist
• Statistical Mechanics Checklist
• Special Relativity Checklist
• Choose Your Path

References

• M. L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed., Wiley, 2005.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.