Determine the ground state energy of an electron in an infinite potential well of width of 2\text{ \AA}.

For a potential with the boundary conditions V(x) = \begin{cases} 0, & x < -a \\\\ V, & -a < x < a \\\\ 0, & x > a \end{cases} solve the Schrödinger's equation in one dimension and find out the conditions for tunnelling.

By applying the Schrödinger's equation to the ground state of hydrogen atom, determine the zero-point energy.

A particle is described by the wave function \psi(x, t) = e^{i(kx-\omega t)}. (i) Is this wave function an eigenfunction corresponding to any dynamical variable or variables? If so, name the variable(s). (ii) Does this represent a ground state?

Show that the energy of the triplet state (S = 1) is not equal to the energy of the singlet state (S = 0).

The raising (J_+) and lowering (J_-) operators are defined by J_+ = J_x + i J_y and J_- = J_x - i J_y. Show that-- (i) [J_z, J_\pm] = \pm \hbar J_\pm (ii) J_+ J_- = J^2 - J_z^2 + \hbar J_z

State how for spin-half particles, the spin (\sigma) can be expressed by its three components \sigma_x, \sigma_y and \sigma_z.

A bullet of mass 0\cdot 025\text{ kg} is moving with a velocity of 600\text{ m/s}. The speed is measured with an accuracy of 0\cdot 02\%. Find out the uncertainty in x. Also, comment on the result.

Explain how the uncertainty in position is different from the uncertainty or inaccuracy of the measuring instruments.

Explain the phenomenon of penetration of a particle through a barrier whose height exceeds the total energy of the particle with necessary diagram.

Is it possible for a photon to transfer all its energy to a free electron? Give reasons.

Consider \text{N} non-interacting ideal spinless particles (Bose gas) are occupying a volume \text{V}. Find out the temperature '\text{T}' below which B-E condensation takes place.

Consider a mixture of N_A molecules of a monatomic gas A and N_B molecules of a monatomic gas B. For this mixture, obtain the Helmholtz free energy and pressure. (The particle partition function for a monatomic gas is q = \left(\frac{2\pi m k T}{h^2}\right)^{\frac{3}{2}} V).

Consider that an ideal non-interacting Fermi gas with internal energy '\text{U}' at temperature \text{T} is kept in a cubical box of volume \text{V}. Find the pressure for the gas in terms of \text{U} and \text{V}.

A reversible heat engine operates with three reservoirs at 300\text{ K}, 400\text{ K} and 1200\text{ K}. It absorbs 1200\text{ kJ} energy as heat from the reservoir at 1200\text{ K} and delivers 400\text{ kJ} work. Determine the heat interactions with the other two reservoirs.

One gram of water (1\text{ cm}^3) becomes 1671\text{ cm}^3 of steam when boiled at constant pressure of 1\text{ atm} (1\cdot 013 \times 10^5\text{ Pa}). The heat of vapourisation at this pressure is \text{L} = 2\cdot 256 \times 10^6\text{ J/kg}. Calculate :

(i) The work done by the water when it vapourizes, and

(ii) Increase in its internal energy.

Assuming the Maxwell's velocity distribution formula, find out the value of :

(A) Mean velocity (\bar{\text{v}}),

(B) The most probable velocity (\text{v}_{\text{mp}}), and

(C) Root mean square speed (\text{v}_{\text{rms}}) in terms of the Boltzmann constant (\text{k}_{\text{B}}) and show that : \text{v}_{\text{rms}} > \bar{\text{v}} > \text{v}_{\text{mp}}. If nitrogen molecules are kept at 27^{\circ}\text{C}, find out the value of \text{v}_{\text{rms}}, \bar{\text{v}} and \text{v}_{\text{mp}}. Given : Molecular mass of nitrogen \text{M} = 28 \times 10^{-3}\text{ kg/mol} and gas constant \text{R} = 8\cdot 314\text{ J.mol}^{-1}\text{K}^{-1}.

Explain why, at equilibrium, the chemical potential of a component must be the same in all coexisting phases. Derive the equilibrium condition for a binary liquid-vapour system in terms of chemical potential.

A ternary system consists of three components (A, B and C) in equilibrium with two phases. Determine the number of degrees of freedom using the Gibb's phase rule and discuss the effect of pressure and temperature variations on the phase equilibrium.