Problem set 94

1. The energy density for photons in a cavity is proportional to
1. $T^3$
2. $T$
3. $T^4$
4. $T^{4/3}$

According to Stefan Boltzmann law energy density is proportional to fourth power of temperature.

Hence, answer is (C)

2. Let $\rho$ be the density matrix for a system. Then
1. $Tr(\rho)=0$
2. $Tr(\rho) < 0$
3. $0\leq Tr(\rho) < 1$
4. $Tr(\rho)=1$

The Density matrix for the pure state $|\psi>$ is given by $$\rho=|\psi><\psi|$$ This density matrix has the following properties:

1. $\hat\rho^2=\hat\rho$ projector
2. $\hat\rho^\dagger=\hat\rho$ hermiticity
3. $T_r\hat\rho=1$ normalization
4. $\hat\rho\geq 0$ positivity

Hence, answer is (D)

3. A system has only two energy levels $E_1$ and $E_2$. In equilibrium at temperature $T$, the number of particles occupying level $E_1$ is double of those occupying level $E_2$. The value of $E_2-E_1$ must be ($k$ is Boltzmann constant):
1. $kT\ln2$
2. $kT\ln3$
3. $3kT$
4. $2kT$

$$n_1\propto e^{-E_1/kT}$$ $$n_2\propto e^{-E_2/kT}$$ $$\frac{n_1}{n_2}=e{(E_2-E_1)/kT}=2$$ $$E_2-E_1=kT\ln2$$

Hence, answer is (A)

4. The quantities (i) isothermal compressibility (ii) volume coefficient of expansion are :
1. Extensive and intensive respectively
2. Intensive and extensive respectively
3. Both extensive
4. Both intensive

The properties that increase proportionally with the size of the system, such as $n$ and $V$, called extensive properties, and those defined for each small region in the system, such as $P$ and $T$, called intensive properties. Terms that are added together or are on opposite sides of an equal sign must contain the same number of extensive variables. The quotient of two extensive variables is an intensive variable.

Isothermal compressibility is given by $K=-\frac{1}{V}\left(\frac{\partial V}{\partial P}\right)_T$ and volume coefficient of expansion is given by $\alpha=-\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P$.

Both of these are intensive properties.

Hence, answer is (D)

5. The chemical potential in classical limit is:
1. Zero
2. Negative
3. Positive
4. Complex quantity

Negative

Hence, answer is (B)