PYQ Bank

= 0$, where i = x, y, and z and comment on the measurability of J^2 and its components as operators.

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.

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.

\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.

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

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

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.

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.

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

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

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 ?

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 ?

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

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?

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.

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.

In a nuclear experiment, beams of electrons and protons are moving with velocities c / 10 and c / 20 respectively. Calculate the de Broglie wavelengths of both the particles in metre (c is the speed of light = 3 \times 10^8\text{ m / s}).

A typical atomic radius is about 5 \times 10^{-15}\,m and the energy of \beta-particle emitted from a nucleus is at most of the order of 1\,\mathrm{MeV}. Prove on the basis of uncertainty principle that the electrons are not present in nuclei.

What does the uncertainty principle signify? Consider an electron in a box of size 10^{-10}\text{ m}. What will be its maximum uncertainty in momentum?