20 Mathematical Physics Quiz Questions and Answers

1. Question: What is the divergence of the vector field \(\mathbf{F} = (x, y, z)\) in Cartesian coordinates?
Options:
A) 0
B) 1
C) 2
D) 3
Answer: D) 3
Explanation: The divergence is given by \(\nabla \cdot \mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3\).

2. Question: Which of the following is the correct form of the Laplacian operator in three dimensions?
Options:
A) \(\nabla^2 f = \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}\)
B) \(\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\)
C) \(\nabla^2 f = \frac{\partial f}{\partial x} \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}\)
D) \(\nabla^2 f = \frac{\partial^2 f}{\partial x} + \frac{\partial^2 f}{\partial y} + \frac{\partial^2 f}{\partial z}\)
Answer: B) \(\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\)
Explanation: The Laplacian is the sum of the second partial derivatives of the function with respect to each coordinate.

3. Question: For a simple harmonic oscillator, the differential equation is \(m \frac{d^2 x}{dt^2} + kx = 0\). What is the general solution?
Options:
A) \(x(t) = A \cos(\omega t) + B \sin(\omega t)\), where \(\omega = \sqrt{k/m}\)
B) \(x(t) = A e^{\omega t} + B e^{-\omega t}\), where \(\omega = \sqrt{k/m}\)
C) \(x(t) = A t \cos(\omega t) + B t \sin(\omega t)\), where \(\omega = \sqrt{k/m}\)
D) \(x(t) = A \cos(\omega t) – B \sin(\omega t)\), where \(\omega = \sqrt{m/k}\)
Answer: A) \(x(t) = A \cos(\omega t) + B \sin(\omega t)\), where \(\omega = \sqrt{k/m}\)
Explanation: The equation is a second-order linear differential equation with constant coefficients, and its characteristic equation leads to oscillatory solutions with angular frequency \(\omega = \sqrt{k/m}\).

4. Question: In quantum mechanics, what does the eigenvalue of the Hamiltonian operator represent?
Options:
A) Momentum
B) Energy
C) Position
D) Angular momentum
Answer: B) Energy
Explanation: The Hamiltonian operator corresponds to the total energy of the system, and its eigenvalues are the possible energy levels.

5. Question: Which theorem states that the line integral of a vector field around a closed path is equal to the surface integral of the curl of that field through the surface bounded by the path?
Options:
A) Green’s theorem
B) Stokes’ theorem
C) Divergence theorem
D) Gauss’s theorem
Answer: B) Stokes’ theorem
Explanation: Stokes’ theorem relates the circulation of a vector field to the flux of its curl, mathematically \(\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\).

6. Question: For the wave equation \(\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}\), what is the general solution?
Options:
A) \(u(x,t) = f(x – ct) + g(x + ct)\)
B) \(u(x,t) = f(x) + g(t)\)
C) \(u(x,t) = e^{kx – \omega t}\)
D) \(u(x,t) = \sin(kx – \omega t)\)
Answer: A) \(u(x,t) = f(x – ct) + g(x + ct)\)
Explanation: The general solution consists of two arbitrary functions representing waves traveling in opposite directions.

7. Question: What is the curl of the vector field \(\mathbf{F} = (y, -x, 0)\) in two dimensions?
Options:
A) 0
B) 2
C) -2
D) (0, 0, 2)
Answer: D) (0, 0, 2)
Explanation: In three dimensions, \(\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 1 & 0 \\ 1 & -1 & 0 \end{vmatrix} = (0 \cdot 0 – 0 \cdot (-1)) \mathbf{i} – (0 \cdot 0 – 0 \cdot 1) \mathbf{j} + (0 \cdot (-1) – 1 \cdot 1) \mathbf{k} = (0, 0, -2 – 1 \times 1? Wait, correction: actually, it’s (0, 0, 2)\), as calculated properly.

8. Question: In linear algebra for physics, what do the eigenvectors of a matrix represent in the context of quantum mechanics?
Options:
A) Observable values
B) State vectors
C) Basis states
D) Operators
Answer: C) Basis states
Explanation: Eigenvectors of an operator correspond to the basis states for that observable in the quantum state space.

9. Question: For a particle in a box, the wave function is \(\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)\). What is the energy for n=1?
Options:
A) \(\frac{\pi^2 \hbar^2}{2mL^2}\)
B) \(\frac{\hbar^2}{2mL^2}\)
C) \(\frac{2\pi^2 \hbar^2}{mL^2}\)
D) \(\frac{\pi \hbar^2}{mL^2}\)
Answer: A) \(\frac{\pi^2 \hbar^2}{2mL^2}\)
Explanation: The energy levels are given by \(E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}\), so for n=1, it is \(\frac{\pi^2 \hbar^2}{2mL^2}\).

10. Question: What is the Fourier transform of a Gaussian function \(f(t) = e^{-at^2}\)?
Options:
A) \(\frac{1}{\sqrt{2a}} e^{-\frac{\omega^2}{4a}}\)
B) \(\frac{1}{a} e^{-\omega^2}\)
C) \(e^{-a \omega^2}\)
D) \(\sqrt{\frac{\pi}{a}} e^{-\frac{\omega^2}{4a}}\)
Answer: D) \(\sqrt{\frac{\pi}{a}} e^{-\frac{\omega^2}{4a}}\)
Explanation: The Fourier transform of a Gaussian is another Gaussian, derived from the integral properties.

11. Question: In special relativity, the Lorentz transformation for time is given by \(t’ = \gamma (t – \frac{vx}{c^2})\). What is \(\gamma\)?
Options:
A) \(\frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}\)
B) \(\sqrt{1 – \frac{v^2}{c^2}}\)
C) \(1 – \frac{v^2}{c^2}\)
D) \(\frac{v}{c}\).
Answer: A) \(\frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}\)
Explanation: \(\gamma\) is the Lorentz factor, accounting for time dilation and length contraction.

12. Question: For the Schrödinger equation \(i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V\psi\), what does the term \(-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2}\) represent?
Options:
A) Kinetic energy operator
B) Potential energy operator
C) Total energy operator
D) Momentum operator
Answer: A) Kinetic energy operator
Explanation: This term corresponds to the kinetic energy in the time-dependent Schrödinger equation.

13. Question: What is the determinant of the matrix representing a rotation in 2D by angle θ?
Options:
A) 1
B) 0
C) θ
D) sinθ
Answer: A) 1
Explanation: Rotation matrices are orthogonal, so their determinants are 1, preserving area and orientation.

14. Question: In tensor notation, what is the contraction of a second-rank tensor \(T_{ij}\)?
Options:
A) A scalar
B) A vector
C) Another tensor
D) A matrix
Answer: B) A vector
Explanation: Summing over one index, e.g., \(T_{ii}\), results in a vector (trace for a matrix).

15. Question: For Poisson’s equation \(\nabla^2 \phi = -\frac{\rho}{\epsilon_0}\), what does φ represent?
Options:
A) Electric potential
B) Electric field
C) Charge density
D) Magnetic field
Answer: A) Electric potential
Explanation: Poisson’s equation relates the Laplacian of the electric potential to the charge density.

16. Question: What is the solution to the differential equation \(\frac{d^2 y}{dx^2} + y = 0\)?
Options:
A) \(y = A \cos x + B \sin x\)
B) \(y = e^x + e^{-x}\)
C) \(y = A x + B\)
D) \(y = \ln x\)
Answer: A) \(y = A \cos x + B \sin x\)
Explanation: This is the harmonic oscillator equation with characteristic roots ±i, leading to sinusoidal solutions.

17. Question: In group theory for physics, what is the order of the cyclic group generated by a rotation of 90 degrees?
Options:
A) 4
B) 2
C) 3
D) 1
Answer: A) 4
Explanation: Rotations by 90 degrees repeat every 4 applications (360 degrees).

18. Question: For the vector identity \(\nabla \times (\nabla f) =\) ?
Options:
A) 0
B) \(\nabla f\)
C) \(\nabla \cdot f\)
D) f
Answer: A) 0
Explanation: The curl of a gradient is always zero for smooth scalar fields.

19. Question: What is the complex conjugate of \(z = 3 + 4i\)?
Options:
A) \(3 – 4i\)
B) \(4 + 3i\)
C) \(-3 – 4i\)
D) \(3i – 4\)
Answer: A) \(3 – 4i\)
Explanation: The complex conjugate flips the sign of the imaginary part.

20. Question: In statistical mechanics, the partition function Z for a system is related to the free energy by F = -kT ln Z. What does k represent?
Options:
A) Boltzmann constant
B) Planck constant
C) Speed of light
D) Gravitational constant
Answer: A) Boltzmann constant
Explanation: The partition function relates to the Helmholtz free energy via the Boltzmann constant in thermal equilibrium.