Problem set 52

1. Which one of the following relations is true for Pauli matrices $\sigma_x$, $\sigma_y$ and $\sigma_z$
1. $\sigma_x\sigma_y=\sigma_y\sigma_x$
2. $\sigma_x\sigma_y=\sigma_z$
3. $\sigma_x\sigma_y=i\sigma_z$
4. $\sigma_x\sigma_y=-\sigma_y\sigma_x$

Pauli spin matrices are $\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}$, and $\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ $$\begin{align*} \sigma_x\sigma_y&=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}0&-i\\i&0\end{pmatrix}\\ &=\begin{pmatrix}i&0\\0&-i\end{pmatrix}\\ &=i\sigma_z \end{align*}$$

Hence, answer is (C)

2. The free energy of a photon gas enclosed in a volume $V$ is given by $F=-\frac{1}{3}aVT^4$, where $a$ is constant and $T$ is the temperature of the gas. The chemical potential of the photon gas is
1. 0
2. $\frac{4}{3}aVT^4$
3. $\frac{1}{3}aT^4$
4. $aVT^4$

Chemical potential is given by $$\mu=\frac{\partial F}{\partial N}$$ $$\mu=\frac{\partial (-\frac{1}{3}aVT^4)}{\partial N}=0$$

Hence, answer is (A)

3. The wavefunctions of two identical particles in states $n$ and $s$ are given by $\phi_n(r_1)$ and $\phi_s(r_2)$, respectively. The particles obey Maxwell-Boltzmann statistics. The state of the combined two-particle system is expressed as
1. $\phi_n(r_1)+\phi_s(r_2)$
2. $\frac{1}{\sqrt{2}}\left[\phi_n(r_1)\phi_s(r_2)+\phi_n(r_2)\phi_s(r_1)\right]$
3. $\frac{1}{\sqrt{2}}\left[\phi_n(r_1)\phi_s(r_2)-\phi_n(r_2)\phi_s(r_1)\right]$
4. $\phi_n(r_1)\phi_s(r_2)$

In Maxwell-Boltzmann statistics particles are distinguishable, hence, $$\Phi=\phi_n(r_1)\phi_s(r_2)$$ or $$\Phi=\phi_n(r_2)\phi_s(r_1)$$

Hence, answer is (D)

4. The target of an X-ray tube is subjected to an excitation voltage $V$. The wavelength of the emitted X-rays is proportional to
1. $1/\sqrt{V}$
2. $\sqrt{V}$
3. $1/V$
4. $V$

Energy of electrons is $$E_k=eV=h\nu$$ $$\lambda=\frac{hc}{eV}$$

Hence, answer is (C)

5. Which one of the following is NOT a correct statement about semiconductors?
1. The electrons and holes have different mobilities in a semiconductor
2. In $n$-type semiconductor, the Fermi level lies closer to the conduction band edge
3. Silicon is a direct band gap semiconductor
4. Silicon has diamond structure

Silicon is a indirect band gap semiconductor

Hence, answer is (C)