\[
\textbf{Mathematical Physics – MCQs with Answers}
\]
\[
\textbf{Q1: The Fourier transform of the Dirac delta function, \(\delta(x)\), is:}
\]
\[
\text{(A) } 1 \quad \text{(B) } e^{ikx} \quad \text{(C) } \delta(k) \quad \text{(D) } \frac{1}{2\pi}
\]
\[
\textbf{Answer: (A) } 1
\]
\[
\textbf{Q2: The Wronskian of two solutions \(y_1(x)\) and \(y_2(x)\) of a second-order linear differential equation is:}
\]
\[
\text{(A) Zero everywhere} \quad \text{(B) A constant if the coefficient of \(y’\) is zero} \quad \text{(C) Always a non-zero function} \quad \text{(D) Related to the Green’s function}
\]
\[
\textbf{Answer: (B) A constant if the coefficient of \(y’\) is zero}
\]
\[
\textbf{Q3: The eigenfunctions of the one-dimensional harmonic oscillator satisfy the differential equation:}
\]
\[
\text{(A) } \frac{d^2\psi}{dx^2} + (E – x^2)\psi = 0 \quad \text{(B) } \frac{d^2\psi}{dx^2} + (x^2 – E)\psi = 0 \quad \text{(C) } \frac{d^2\psi}{dx^2} + x\psi = 0 \quad \text{(D) } \frac{d^2\psi}{dx^2} + (E^2 – x)\psi = 0
\]
\[
\textbf{Answer: (B) } \frac{d^2\psi}{dx^2} + (x^2 – E)\psi = 0
\]
\[
\textbf{Q4: The Green’s function for the Helmholtz equation, \((\nabla^2 + k^2)G(\mathbf{r}, \mathbf{r}’) = -\delta(\mathbf{r} – \mathbf{r}’)\), in three dimensions is:}
\]
\[
\text{(A) } G(\mathbf{r}, \mathbf{r}’) = \frac{e^{-ik|\mathbf{r} – \mathbf{r}’|}}{4\pi|\mathbf{r} – \mathbf{r}’|} \quad
\text{(B) } G(\mathbf{r}, \mathbf{r}’) = \frac{e^{ik|\mathbf{r} – \mathbf{r}’|}}{4\pi|\mathbf{r} – \mathbf{r}’|} \quad
\text{(C) } G(\mathbf{r}, \mathbf{r}’) = \frac{1}{|\mathbf{r} – \mathbf{r}’|} \quad
\text{(D) } G(\mathbf{r}, \mathbf{r}’) = |\mathbf{r} – \mathbf{r}’|
\]
\[
\textbf{Answer: (A) } G(\mathbf{r}, \mathbf{r}’) = \frac{e^{-ik|\mathbf{r} – \mathbf{r}’|}}{4\pi|\mathbf{r} – \mathbf{r}’|}
\]
\[
\textbf{Q5: The Frobenius method is used to solve:}
\]
\[
\text{(A) Nonlinear differential equations} \quad
\text{(B) Differential equations with regular singular points} \quad
\text{(C) Homogeneous partial differential equations} \quad
\text{(D) Integral equations}
\]
\[
\textbf{Answer: (B) Differential equations with regular singular points}
\]