The normalized eigenfunction for the ground state of hydrogen atom is \Psi_{100} = \frac{1}{\sqrt{\pi}} \left( \frac{Z}{a_0} \right)^{3/2} e^{-Zr/a_0} Calculate the expectation value of the radius vector r of the electron in the ground state.

Show that for a given principal quantum number n, there are n^2 possible states of the atom.

What is the spin wave function (for s=\frac{1}{2}) if the spin component in the direction of unit vector \eta has a value of \frac{1}{2}\hbar?

The components of the angular momenta \vec{J}_1 and \vec{J}_2 satisfy commutation rules. Show that the components of the sum \vec{J} = \vec{J}_1 + \vec{J}_2 also satisfy the commutation rules. Whether the difference \vec{J}_1 - \vec{J}_2 satisfies an angular momentum?

An electron in a one-dimensional infinite potential well, defined by V(x)=0 for -a\leq x\leq a and V(x)=\infty otherwise, goes from n=4 to n=2 level and emits photon of frequency 3.43\times10^{14}\,\mathrm{Hz}. Calculate the width of the well. (Assume Plank's constant h=6.626\times10^{-34}\,\mathrm{J.S.} and mass of electron m=9.11\times10^{-31}\,\mathrm{kg})

Set up the Schrodinger's wave equation for one dimensional potential barrier and obtain the probability of tunnelling.

What is de Broglie concept of matter wave? Evaluate de Broglie wavelength of Helium that is accelerated through 300\mathrm{V}. (Given mass of proton = mass of neutron = 1.67\times10^{-27}\,kg)

Find the values of the following : (i) \vec{L} \times \vec{L} (ii) L_+ L_- (iii) [L_z, L_+] where \vec{L} is the angular momentum operator. L_+ and L_- are raising and lowering operators respectively.

The lifetime of a given atom in an excited state is 10^{-8}\text{ s}. It comes to the ground state by emitting a photon of wavelength 5800\text{ \AA}. Find the energy uncertainty and wavelength uncertainty of the photon.

Assume that the Earth's atmosphere is pure nitrogen in thermodynamic equilibrium at a temperature of 300\ \mathrm{K}. Calculate the height above sea level at which the density of the atmosphere is one-half its sea level value. (Molecular weight of \mathrm{N}_2 is 28\mathrm{gm/mole})

Liquid ^4\text{He} has normal boiling point at 4\cdot 2\text{ K}. It is observed that it boils at 1\cdot 2\text{ K} at a pressure of 1\text{ mm of Hg}. Calculate the average latent heat of vaporization of ^4\text{He} in this temperature range. (\text{Given : Density of Hg} = 13\cdot 6 \times 10^3\text{ kg m}^{-3}).

A thermally insulated cylinder, closed at both ends, is fitted with a frictionless heat-conducting piston which divides the cylinder in two parts. Initially, the piston is clamped in the centre, with one litre of air at 200\ \mathrm{K} and 2\ \mathrm{atm} pressure on one side and one litre of air at 300\ \mathrm{K} and 1\ \mathrm{atm} pressure on the other side. The piston is released and the system reaches equilibrium in pressure and temperature, with the piston at a new position. Compute the final pressure and temperature.

Calculate the Fermi energy of aluminium at absolute zero. The density of aluminium is 2\cdot 7 \times 10^3\text{ kg m}^{-3} and its atomic weight is 26\cdot 98\text{ kg (k mol)}^{-1}. Show that the electron gas in aluminium is strongly degenerate.

Two Carnot engines \text{C}_1 and \text{C}_2 operate in series, Engine \text{C}_1 absorbs heat at T and rejects heat to a sink at temperature 300\text{ K}. Engine \text{C}_2 absorbs \frac{1}{4}\text{th} of the heat rejected by engine \text{C}_1 and rejects heat to the sink at 200\text{ K}. If the work done in both the cases is the same, find the temperature T.

What do you understand by negative temperature? Write and explain various restrictions on a system for the concept of negative temperature to be meaningful.

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A gas of interacting atoms has an equation of state and heat capacity at constant volume given by the expressions p(T,V)=aT^{1/2}+bT^3+cV^{-2} C_v(T,V)=dT^{1/2}+eT^2V+fT^{1/2} where a through f are constants which are independent of T and V. Find the differential of the internal energy dU(T,V) in terms of dT and dV.

In the case of a gas obeying the equation of state \frac{\text{Pv}}{\text{RT}} = 1 + \frac{\beta}{\text{v}}, where \beta is a function of T only, find the expression of the heat capacity at constant volume.

One mole of gas obeys van der Waals equation of state. If its molar internal energy is given by u=cT-a/V (in which V is the molar volume, a is one of the constants in the equation of state and c is a constant), calculate the molar heat capacities C_v and C_p.

Estimate the temperature at which the root mean square speed of nitrogen molecules exceeds their most probable speed by 100\text{ ms}^{-1}.