(i) State and explain position momentum uncertainty principle. Justify that this principle is not just a negative statement rather a useful tool, with one example. (ii) The lifetime of an excited state of an atom is about 10^{-8}\text{ sec.} Calculate the minimum uncertainty in the energy of the excited state.

A particle is described by the wave function \Psi(x)=\left(\frac{\pi}{2}\right)^{-1/4}e^{-ax^2/2}. Calculate \Delta x and \Delta p for the particle, and verify the uncertainty relation \Delta x\Delta p=\frac{\hbar}{2}.

Prove that Bohr hydrogen atom approaches classical conditions, when n becomes very large and small quantum jumps are involved.

(i) What do you mean by expectation value of a physical quantity ? How does it help to extract information from a wave function ? (ii) A particle limited to move along x-axis has the wave function \psi = ax between x = 0 and x = 1, \psi = 0 elsewhere. Find the probability that the particle can be found between x = 0\cdot 45 and x = 0\cdot 55. Find the expectation value <x> of the particles position x = 0 to x = 1.

Find the probability current density for the wave function \Psi(x,t)=\left[Ae^{ipx/\hbar}+Be^{-ipx/\hbar}\right]e^{-ip^2t/2m\hbar} Interpret the result physically.

Derive energies and wave functions for motion of electron in a hydrogen atom.

Prove the commutation relation for the angular momentum: [L^2,L_z]=0 Also show that (\vec{L}\times\vec{L})=i\hbar\vec{L}.

Consider a Hermitian operator A with property A^3=1. Show that A=1.

Using the uncertainty principle \Delta x\Delta p \geq \hbar/2, estimate the ground state energy of a harmonic oscillator.

If the z-component of an electron spin is +\frac{\hbar}{2}, what is the probability that its component along a direction z' (forming an angle \theta with z-axis) is \frac{\hbar}{2} or -\frac{\hbar}{2}? What is the average value of spin along z'?

Prove the following : (i) [L^2, L_x] = 0 (ii) [L_x, L_y] = i \hbar L_3

A blue lamp emits light of mean wavelength of 4500\ \mathring{\mathrm{A}}. The rating of the lamp is 150 W and its 8% of the energy appears as light. How many photons are emitted per second by the lamp?

Calculate (i) the internal energy of the electron gas per unit volume and (ii) the molar specific heat at constant volume for sodium at 100\text{ K} containing one free electron per atom. Given that the density of sodium = 0{\cdot}97 \times 10^3\text{ kg m}^{-3} and the atomic weight of sodium = 23.

10\text{ g} of water at 60\text{ }^\circ\text{C} is mixed with 30\text{ g} of water at 20\text{ }^\circ\text{C}. Will the entropy of the system increase or decrease? Calculate the change.

Deduce Clausius-Clapeyron equations based on reversible cycle. Show that the specific heat of steam is negative. What is the significance of negative specific heat?

State the first law of thermodynamics for a diffusively interacting system. The temperature of 10\ \mathrm{g} of air is raised by 2^\circ\mathrm{C} at constant volume. Calculate the increase in its internal energy. Given: C_v=0.172\ \mathrm{cal\ g^{-1}\ ^\circ C^{-1}}.

(i) The energy level of a quantum harmonic oscillator with frequency \nu is given by E_n=\left(n+\frac{1}{2}\right)h\nu,\quad \text{where } n=0,1,2,\ldots Calculate its partition function.

(ii) Calculate the partition function of a two level system.

There are g cells of energy \varepsilon. Show that the number n of bosons of energy \varepsilon distributed among these cells is given by n = \frac{g}{e^{(\varepsilon - \mu)/kT} - 1} What is \mu and how will you find it?

Obtain the Clausius-Clapeyron equation. Using this equation, show that for the phase boundary of the liquid and vapour phases, p--T relation can be written as p=p_0e^{-L/kT}. Here it has been assumed that the latent heat L is independent of temperature, that vapour is treated as an ideal gas and that V_{\mathrm{vapour}}=V\gg V_{\mathrm{liquid}} and that p\to p_0 as T\to\infty.

Discuss the principle of adiabatic demagnetization process to achieve low temperatures. Determine the fall in temperature produced by adiabatic demagnetization of a paramagnetic material at initial temperature of 3\ \mathrm{K} when the magnetic field is switched off from 10{,}000 oersted to zero. Given: heat capacity at constant magnetic field =0.2\ \mathrm{J\ g^{-1}\ K^{-1}} and Curie constant per gram mole per \mathrm{cm^3} =0.042\ \mathrm{erg\ K^{-1}\ g^{-1}\ Oe^{-2}}.