Problem set 82

1. Two parallel plate capacitors, separated by distances $x$ and $1.1x$ respectively, have a dielectric material of dielectric constant 3.0 inserted between the plates, and are connected to a battery of voltage $V$. The difference in charge on the second capacitor compared to the first is
1. +66%
2. +20%
3. -3.3%
4. -10%

Capacitance of a parallel plate capacitor is inversely proportional to the separation between the plates i.e. $C\propto\frac{1}{d}$ $$C_1\propto\frac{1}{x}$$ $$C_2\propto\frac{1}{1.1x}$$ Let us take area of plates such that $$C_1=\frac{1}{x}$$ $$C_2=\frac{1}{1.1x}$$ $$C_2-C_1=\frac{1}{1.1x}-\frac{1}{x}=-\frac{0.1}{1.1x}$$ $$\% difference=\frac{100\times\left(-\frac{0.1}{1.1x}\right)}{\frac{1}{1.1x}}$$ $$\% difference=-10\%$$

Hence, answer is (D)

2. The state of a particle of mass $m$ in a one-dimensional rigid box in the interval 0 to $L$ is given by the normalised wavefunction $\psi(x)\!=\!\!\sqrt{\frac{2}{L}}\!\!\left(\frac{3}{5}\sin{\left(\frac{2\pi x}{L}\right)}+\frac{4}{5}\sin{\left(\frac{4\pi x}{L}\right)}\!\right)$. If its energy is measured, the possible outcomes and the average value of energy are, respectively
1. $\frac{h^2}{2mL^2}$, $\frac{2h^2}{mL^2}$ and $\frac{73}{50}\frac{h^2}{mL^2}$
2. $\frac{h^2}{8mL^2}$, $\frac{h^2}{2mL^2}$ and $\frac{19}{40}\frac{h^2}{mL^2}$
3. $\frac{h^2}{2mL^2}$, $\frac{2h^2}{mL^2}$ and $\frac{19}{10}\frac{h^2}{mL^2}$
4. $\frac{h^2}{8mL^2}$, $\frac{2h^2}{mL^2}$ and $\frac{73}{200}\frac{h^2}{mL^2}$

Eigen function and energy of a particle in a 1D box are given by $$\psi_n(x)=\sqrt{\frac{2}{L}}\sin{\left(\frac{n\pi x}{L}\right)}$$ $$E_n=\frac{n^2h^2}{8mL^2}$$ Hence, $$\psi(x)=c_2\psi_2(x)+c_4\psi_4(x)$$ where, $c_2=\frac{3}{5}$ and $c_4=\frac{4}{5}$ are the expansion coefficients.

Particle may be in its one of the eigenstate, hence possible values of energies are $$E_2=\frac{2^2h^2}{8mL^2}=\frac{h^2}{2mL^2}$$ $$E_4=\frac{4^2h^2}{8mL^2}=\frac{2h^2}{mL^2}$$ Average value of energy is $$\begin{align*} < E > &= < \psi |H|\psi > \\ &=|c_2|^2E_2+|c_4|^2E_4\\ &=\frac{9}{25}\frac{h^2}{2mL^2}+\frac{16}{25}\frac{2h^2}{mL^2}\\ &=\frac{73}{50}\frac{h^2}{mL^2} \end{align*}$$

Hence, answer is (A)

3. If $\hat L_x$, $\hat L_y$ and $\hat L_z$ are the components of the angular momentum operator in three dimensions, the commutator $\left[\hat L_x, \hat L_x\hat L_y\hat L_z\right]$ may be simplified to
1. $i\hbar\hat L_x\left(\hat L_z^2-\hat L_y^2\right)$
2. $i\hbar\hat L_z\hat L_y\hat L_x$
3. $i\hbar\hat L_x\left(2\hat L_z^2-\hat L_y^2\right)$
4. 0

$$\begin{align*} \left[\hat L_x, \hat L_x\hat L_y\hat L_z\right]&= {\scriptstyle \hat L_x \left[\hat L_x, \hat L_y\hat L_z\right]+\left[\hat L_x, \hat L_x\right]\hat L_y\hat L_z}\\ &=\hat L_x \left[\hat L_x, \hat L_y\hat L_z\right]\\ &={\scriptstyle \hat L_x \left\{\hat L_y\left[\hat L_x, \hat L_z\right]+\left[\hat L_x, \hat L_y\right]\hat L_z\right\}}\\ &={\scriptstyle \hat L_x \left\{\hat L_y\left(-i\hbar \hat L_y\right)+\left(i\hbar\hat L_z\right)\hat L_z\right\}}\\ &=i\hbar\hat L_x\left(\hat L_z^2-\hat L_y^2\right) \end{align*}$$

Hence, answer is (A)

4. The eigenstates corresponding to eigen-values $E_1$ and $E_2$ of a time-independent Hamiltonian are $|1 >$ and $|2 >$ respectively. If at $t=0$, the system is in a state $|\psi(t=0) > =\sin\theta |1 > +\cos\theta |2 >$ the value of $< \psi(t)|\psi(t) >$ at time $t$ will be
1. 1
2. $(E_1\!\sin^2\theta\!+\!E_2\!\cos^2\theta)/\!\sqrt{E_1^2\!+\!E_2^2}$
3. $e^{iE_1t/\hbar}\sin\theta+e^{iE_2t/\hbar}\cos\theta$
4. $e^{-iE_1t/\hbar}\sin^2\theta+e^{-iE_2t/\hbar}\cos^2\theta$

Time evolution of quantum systems is given by Unitary Transformations, $$\begin{align*} \left|\psi(t)\right > &=\hat U\left|\psi(t=0)\right > \\ &=e^{-iHt/\hbar}\left\{\sin\theta \left|1\right > +\cos\theta \left|2\right > \right\}\\ &=\sin\theta e^{-\frac{iHt}{\hbar}}\left|1\right > +\cos\theta e^{-\frac{iHt}{\hbar}}\left|2\right > \end{align*}$$ Using property $f(\hat A)\psi_n=f(a_n)\psi_n$ we get $$\begin{align*} \left|\psi(t)\right > &=\sin\theta e^{-\frac{iE_1t}{\hbar}}\left|1\right > +\cos\theta e^{-\frac{iE_2t}{\hbar}}\left|2\right > \\ \end{align*}$$ Hence, $$\left < \psi(t)\right|\left.\psi(t)\right > =\sin^2\theta+\cos^2\theta=1$$

Hence, answer is (A)

5. The specific heat per molecule of a gas of diatomic molecules at high temperatures is
1. $8k_B$
2. $3.5k_B$
3. $4.5k_B$
4. $3k_B$

At high temperatures, the specific heat at constant volume $C_v$ has three degrees of freedom from rotation, two from translation, and two from vibration. Hence, $f=7$. $$E=\frac{7}{2}RT$$ $$C_v=\frac{dE}{dT}=\frac{7}{2}R$$

Hence, answer is (B)