Problem set 26

1. The free energy difference between the superconducting and the normal states of a material is given by $\Delta F = F_s-F_N =\alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4$, where $\psi$ is an order parameter and $\alpha$ and $\beta$ are constants such that $\alpha > 0$ in the normal and $\alpha < 0$ in the superconducting state, while $\beta > 0$ always. The minimum value of $\Delta F$ is
1. $-\alpha^2/\beta$
2. $-\alpha^2/2\beta$
3. $-3\alpha^2/\beta$
4. $-5\alpha^2/\beta$
For $\Delta F$ to be minimum $\frac{\partial(\Delta F)}{\partial\psi}=0$ i.e. $$\begin{align*} \frac{\partial(\Delta F)}{\partial\psi}&=2\alpha|\psi|+2\beta|\psi|^3\\ &=0\Rightarrow|\psi|^2\\ &=-\frac{\alpha}{\beta} \end{align*}$$ $$\begin{align*} \Delta F&=-\alpha\times\frac{\alpha}{\beta}+\frac{\beta}{2}\frac{\alpha^2}{\beta^2}\\ &=-\frac{\alpha^2}{2\beta} \end{align*}$$ Hence, answer is (B)
2. Consider a hydrogen atom undergoing a $2P\rightarrow 1S$ transition. The lifetime $t_{sp}$ of the $2P$ state for spontaneous emission is $1.6 ns$ and the energy difference between the levels is $10.2eV$. Assuming that the refractive index of the medium $n_0 = 1$, the ratio of Einstein coefficients for stimulated and spontaneous emission $B_{21}(\omega)/A_{21}(\omega)$ is given by
1. $0.683\times10^{12} m^3J^{-1}s^{-1}$
2. $0.146\times10^{-12} Jsm^{-3}$
3. $6.83\times10^{12} m^3J^{-1}s^{-1}$
4. $1.46\times10^{-12} Jsm^{-3}$
$$\begin{align*} \frac{B_{21}(\omega)}{A_{21}(\omega)}&=\frac{\pi^2c^3}{\hbar\omega^3n_0^3}\\ &=\frac{\pi^2c^3\hbar^2}{(\Delta E)^3n_0^3}\\ &=\frac{c^3h^2}{4(\Delta E)^3n_0^3} \end{align*}$$ $\Delta E=10.2 eV$, $n_0=1$ $$\begin{eqnarray} \begin{array}{c}\frac{B_{21}(\omega)}{A_{21}(\omega)}=\frac{\left(3\times10^8\right)^3\times\left(6.63\times10^{-34}\right)^2}{\left(10.2\times1.6\times10^{-19}\right)^3\times1}\\ =0.683\times10^{12}m^3J^{-1}s^{-1}\end{array} \end{eqnarray}$$ Hence, answer is (A)
3. In the scattering of some elementary particles, the scattering cross-section is found to depend on the total energy and the fundamental constants $h$ (Planck’s constant) and $c$ (the speed of light in vacuum). Using dimensional analysis, the dependence of $\sigma$ on these quantities is given by
1. $\sqrt{\frac{hc}{E}}$
2. $\frac{hc}{E^{3/2}}$
3. $\left(\frac{hc}{E}\right)^2$
4. $\frac{hc}{E}$
Scattering cross-section is measured in units of area i.e. $m^2$ $$\frac{hc}{E}=\frac{Js\:m/s}{J}=m$$ $$\left(\frac{hc}{E}\right)^2=m^2$$ Hence, answer is (C)
4. If $y=\frac{1}{\tanh x}$, then $x$ is
1. $\ln{\left(\frac{y+1}{y-1}\right)}$
2. $\ln{\left(\frac{y-1}{y+1}\right)}$
3. $\ln{\sqrt{\frac{y-1}{y+1}}}$
4. $\ln{\sqrt{\frac{y+1}{y-1}}}$
$$\sinh{x}=\frac{e^x-e^{-x}}{2}$$ $$\cosh{x}=\frac{e^x+e^{-x}}{2}$$ $$\tanh{x}=\frac{\sinh{x}}{\cosh{x}}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$ $$y=\frac{1}{\tanh{x}}=\frac{e^x+e^{-x}}{e^x-e^{-x}}$$ $$\frac{y+1}{y-1}=\frac{\frac{e^x+e^{-x}}{e^x-e^{-x}}+1}{\frac{e^x+e^{-x}}{e^x-e^{-x}}-1}=e^{2x}$$ $$2x=\ln{\left(\frac{y+1}{y-1}\right)}$$ $$x=\ln{\sqrt{\frac{y+1}{y-1}}}$$ Hence, answer is (D)
5. The function $\frac{z}{\sin{\pi z^2}}$ of a complex variable $z$ has
1. a simple pole at $0$ and poles of order $2$ at $\pm\sqrt{n}$ for $n=1,2,3,\dots$
2. a simple pole at $0$ and poles of order $2$ at $\pm\sqrt{n}$ and $\pm i\sqrt{n}$ for $n=1,2,3,\dots$
3. poles of order $2$ at $\pm\sqrt{n}$ for $n=0,1,2,3,\dots$
4. poles of order $2$ at $\pm n$ for $n=0,1,2,3,\dots$
$$\sin{\pm n\pi}=0,\quad \text{for } n=0,1,2,\dots$$ $$z^2=\pm n$$ $$z=\pm\sqrt{n} \quad\text{or } z=\pm i\sqrt{n}$$ Hence, answer is (B)