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.

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

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 ?

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 ?

{ifos-q-5-049-fig-1} 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.

Recall the Pauli matrix representation of two-state system of electron, \sigma_x, \sigma_y, \sigma_z, and show that they do not commute.

Obtain the solution of one-dimensional free particle Schrödinger equation. Show that it corresponds to plane monochromatic (constant angular frequency) wave.

Consider a two-electron system in ground state with orbital quantum number zero (l = 0). Use Pauli's exclusion principle to obtain orientations of the two electrons.

Explain the phenomenon of Zeeman effect of atomic physics.

Consider a one-dimensional harmonic oscillator potential V(x) = \frac{1}{2} m \omega_0^2 x^2 where m represents the mass of a particle, \omega_0 is the classical frequency of the oscillator and x is the displacement from equilibrium position. Obtain the solution of the corresponding Schrödinger equation. Using the normalization condition on the solution, show that the ground-state wave function has the form \psi_0(x) = \left( \frac{\alpha}{\pi} \right)^{1/2} e^{-\frac{1}{2} \alpha^2 x^2} \alpha = \left( \frac{m \omega_0}{\hbar} \right)^{1/2}

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}).

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?

Calculate the most probable value of 'r' for an electron in the ground state of the hydrogen atom.

Solve the Schr"odinger equation for an electron of mass m confined in a one-dimensional potential well of the form \begin{align*} V &= 0 \text{ when } 0 \le x \le L \ &= \infty \text{ when } x < 0; \ x > L \end{align*} Obtain the discrete energy levels and the normalized eigen functions.

Find the de Broglie wavelength of a neutron moving with a kinetic energy of 500\text{ eV}. (1\text{ eV} = 1\cdot 602 \times 10^{-19}\text{ J})

Deduce the commutation relations between the components of angular momentum operator \mathbf{L} [L_x, L_y] = i \hbar L_z [L_y, L_z] = i \hbar L_x [L_z, L_x] = i \hbar L_y using the commutation relations [x, p_x] = [y, p_y] = [z, p_z] = i \hbar

Give an account of Heisenberg's Uncertainty principle. Outline an idealised experiment to bring out its significance.

Determine the values of the total angular momentum for a 3d electron.

Derive an expression for transmission coefficient for a particle through a rectangular potential barrier.