Problem set 92

1. Given $[x_i,P_j]=i\hbar\delta_{ij}$, $i,j=1,2,3$. $[x_1,P_2^2]$ is:
1. 0
2. $i\hbar P_2$
3. $2x_1$
4. $2P_2$

\begin{align*} [x_1,P_2^2]&=P_2[x_1,P_2]+[x_1,P_2]P_2\\ &=0 \end{align*}

Hence, answer is (A)

2. Which of the following is an eigen state of square of linear momentum operator $P_x^2$?
1. $Ax^2$
2. $A\left(\sin{kx}+\cos{kx}\right)$
3. $Ae^{-\alpha x^2}$
4. $A\sin^2{kx}$

$P_x=-i\hbar\frac{\partial}{\partial x}$. Hnece, $P_x^2=-\hbar^2\frac{\partial^2}{\partial x^2}$.
Clearly, \begin{align*} {\scriptstyle P_x^2\left(A\left(\sin{kx}+\cos{kx}\right)\right)}&={\scriptstyle-\hbar^2\frac{\partial^2}{\partial x^2}\left(A\left(\sin{kx}+\cos{kx}\right)\right)}\\ &=k^2A\left(\sin{kx}+\cos{kx}\right) \end{align*}

Hence, answer is (B)

3. The electron in a hydrogen atom is in a superposition state described by the wavefunction $\psi(\vec r)=A\left[4\psi_{100}(\vec r)-2\psi_{211}(\vec r)+\sqrt{6}\psi_{210}(\vec r)-\sqrt{10}\psi_{21-1}(\vec r)\right]$, $\psi_{nlm}(\vec r)$ normalized wave function. The value of normalization constant, $A$, is:
1. $\frac{1}{3}$
2. $\frac{1}{6}$
3. $6$
4. $36$

Normalization condition is $\int\psi^*(\vec r)\psi(\vec r)dr=1$. As $\psi_{nlm}(\vec r)$ are normalized wave functions, we have, $\int\psi^*_{nlm}(\vec r)\psi_{n'l'm'}(\vec r)dr=\delta_{n'n}\delta_{l'l}\delta_{m'm}$ $$\int\psi^*(\vec r)\psi(\vec r)dr=|A|^2\left(16+4+6+10\right)=1$$ $$A=\frac{1}{6}$$

Hence, answer is (B)

4. Two coherent light sources of intensities I and 9I are used in an interference experiment. The resultant intensity at points where the waves from the two sources with phase difference $\pi$ is :
1. 16I
2. 9I
3. 4I
4. zero

Resultant intensity is given by $$I_R=I_1+I_2+2\sqrt{I_1I_2}\cos\delta$$ $$I_R=I+9I+2\sqrt{9I}\cos\pi$$ $$I_R=10I-6I=4I$$

Hence, answer is (C)

5. Non-relativistic hydrogen atom spectrum is proportional to $-1/n^2$. The degeneracy of $n^{th}$ level is:
1. $n$
2. $2n+1$
3. $n^2$
4. $1/n^2$

The energy levels in the hydrogen atom depend only on the principal quantum number $n$. For given values of $n$ and $l$, the $( 2 l + 1 )$, states with $m_{l}=-l \rightarrow l$ are degenerate. The degree of degeneracy of the energy level $E_n$ is therefore : $\sum _{l=0}^{n-1}(2l+1)=n^{2}$, which is doubled if the spin degeneracy is included.

Hence, answer is (C)

Problem set 91

1. Which of the following equations signifies the conservative nature of the electric field $\vec E$?
1. $\vec\nabla\cdot\vec E(\vec r)=\frac{\rho(\vec r)}{\epsilon_0}$
2. $\vec\nabla\times\vec E(\vec r)=\vec 0$
3. $\vec\nabla\times\vec E(\vec r,t)=\frac{-\partial\vec B(\vec r,t)}{\partial t}$
4. $\epsilon_0\mu_0\frac{\partial\vec E(\vec r,t)}{\partial t}=\vec\nabla\times\vec B(\vec r,t)-\mu_0\vec J(\vec r,t)$

A field is said to be conservative if it can be expressed as a gradient of scalar potential $V$. The equation $\vec\nabla\times\vec E(\vec r)=\vec 0$ implies that $\vec E(\vec r)$ can be expressed as $\vec E(\vec r)=-\vec\nabla V$.

Hence, answer is (B)

2. Plane electromagnetic wave is propagating through a perfect dielectric material of refractive index $\frac{3}{2}$. The phase difference between the fields $\vec E$ and $\vec B$ associated with the wave passing through the material is
1. Zero
2. $\pi$
3. $\frac{3}{2}\pi$
4. any non-zero value between $-\pi$ and $\pi$

For a perfect dielectric the phase difference between $\vec E$ and $\vec B$ is zero.

Hence, answer is (A)

3. An electromagnetic wave is propagating in a dielectric medium of permittivity $\epsilon$ and permeability $\mu$ having an electric field vector $\vec E$ associated with the wave. The associated magnetic field $\vec H$ is
1. Parallel to $\vec E$ with magnitude $E\sqrt{\mu/\epsilon}$
2. Parallel to $\vec E$ with magnitude $E\sqrt{\epsilon/\mu}$
3. Perpendicular to $\vec E$ with magnitude $E\sqrt{\mu/\epsilon}$
4. Perpendicular to $\vec E$ with magnitude $E\sqrt{\epsilon/\mu}$

Perpendicular to $\vec E$ with magnitude $E\sqrt{\epsilon/\mu}$

Hence, answer is (D)

4. Power radiated by a point charge moving with constant acceleration of magnitude $\alpha$ is proportional to
1. $\alpha$
2. $\alpha^2$
3. $\alpha^{-1}$
4. $\alpha^{-2}$

The power radiated by a point charge is given by Larmor formula as $$P=\frac{\mu_0q^2\alpha^2}{6\pi c} $$

Hence, answer is (B)

5. The output of a laser has a bandwidth of $1.2\times10^{14}$ Hz. The coherence length $l_c$ of the output radiation is
1. 3.6 mm
2. 50 $\mu$m
3. 2.5 $\mu$m
4. 1.5 cm

Coherence length of laser is given by $$l_c=\frac{c}{\Delta\nu}$$ $$l_c=\frac{3\times10^8}{1.2\times10^{14}}=2.5 \mu m$$

Hence, answer is (C)