PYQ Bank

A beam 4.0 keV electrons from a source is incident on a target 50.0 cm away. Find the radius of the electron beam spot due to Heisenberg’s uncertainty principle.

Calculate the probability of transmission of an electron of 1.0\,\mathrm{eV} energy through a potential barrier of 4.0\,\mathrm{eV} and 0.1\,\mathrm{nm} width.

Show that

(i) \hat{L} \times \hat{L} = i \hbar \hat{L}

(ii) [\hat{L}_+, \hat{L}_-] = 2\hbar \hat{L}_z

(iii) \hat{\sigma}_x \hat{\sigma}_y \hat{\sigma}_z = -i

Describe the differences in the behaviour of a quantum harmonic oscillator with the corresponding classical oscillator.

Using Schrodinger equation, obtain the eigenfunctions and eigenvalues of energy for a 1- dimensional harmonic oscillator. Sketch the profiles of eigenfunctions for first three energy states.

A particle of mass m is confined in a one dimensional box on the x-axis between x = 0 and x = L. Obtain the allowed energy states of the particle and the corresponding eigenfunctions. What will be the energy pattern in the limit of L \to \infty ?

Explain why the square of the angular momentum (L^2) and only one of the components (L_x,L_y,L_z) of L are regarded as constants of motion.

Estimate the de Broglie wavelength of the electron orbiting in the first excited state of the hydrogen atom.

Draw a schematic diagram of the single particle energy levels in a shell model including the effect of spin-orbit coupling. Show how it explains magic numbers in nuclei. Give two examples to show how this scheme predicts the spins and parities of odd A nuclei.

Use the uncertainty principle to estimate the binding energy of the hydrogen atom in the ground state in terms of m_\text{e}, e, \hbar and c. Estimate the answer in eV.

An electron is described by the following wave function : \begin{aligned} \psi(x) &= 0 && \text{for } x < 0 \\ &= C e^{-x}(1 - e^{-x}) && \text{for } x \ge 0 \end{aligned} where C is a constant. Find out the average position of the electron.

Estimate the probability of finding the particle between x = 0 and x = \frac{L}{3} in the lowest energy state.

Evaluate the most probable distance of the electron from nucleus of a hydrogen atom in its 2p state. What is the probability of finding the electron at this distance?

Calculate the lowest energy of an electron confined to move in a 1-dimensional potential well of width 10\,\mathrm{nm}.

Generalize the results obtained in part (a) to obtain the allowed energy states and the corresponding eigenfunctions of a particle in a three dimensional isotropic box of dimension L. What is the degeneracy of the first two excited states of the particle ?

\questiondiagram{
} A beam of particles of mass m and energy E is incident on a step potential of height V_0 from the left as shown in the figure. Discuss the behaviour of the particle for (i) E < V_0 and (ii) E > V_0. Obtain expressions for the reflection and the transmission coefficients and sketch them as a function of the incident energy of the particle.

Explain Bose-Einstein distribution and obtain the same from the grand canonical ensemble.

Derive the mathematical expression for the total energy of a degenerate Fermi gas at a temperature T and calculate the specific heat of the Fermi gas at this temperature.

Derive Clausius-Clapeyron equation. How does it explain the effect of pressure on melting point of solids and boiling point of liquids?

Prove the thermodynamic relation : \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V and hence show that \frac{\mathrm{d}P}{\mathrm{d}T} = \frac{L}{T (V_2 - V_1)} ; all the terms have their usual meanings.