Find the energy of the particle of mass m moving in a potential field V(x) = \frac{2\hbar^2 b^2 x^2}{m} for which the time independent wave function is \psi(x) = \exp(-bx^2). Here b is a constant.

= \hbar L_+$ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z Prove that : (i) [L^2, L_z] = 0 (ii) [L_z, L_+] = \hbar L_+ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z where \hbar = \frac{h}{2\pi} (h is Planck's constant)

A particle limited to the x-axis has the wave function \phi(x) = bx^2 between x = 0 and x = 2; the wave function \phi(x) = 0 elsewhere. (i) Find the probability that the particle can be found between x = 1.0 and x = 1.5. (ii) Find the expectation value <x> of the particle position.

The ground state wave function of a harmonic oscillator is \psi_0(x) = \left(\frac{m\omega}{\hbar \pi}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right). (i) At which point is the probability density maximum ? (ii) What is the value of the maximum probability density ?

Show that the square of the orbital angular momentum operator (L^2) commutes with any of the components of angular momentum operator L. Is it possible to measure L^2, L_x, L_y and L_z simultaneously ? Give reasons for your answer.

Consider a stream of particles of mass m each moving in the positive x-direction with kinetic energy E towards the potential barrier V(x) = 0 \quad \text{for } x \le 0 V(x) = \frac{3E}{4} \quad \text{for } x > 0 Find the fraction of particles reflected at x = 0.

Evaluate the most probable distance of the electron of the hydrogen atom in its 2p state. What is the radial probability density at that distance ?

An operator P describing the interaction of two spin \frac{1}{2} particles is P = a + b \vec{\sigma}_1 \cdot \vec{\sigma}_2, where a and b are constants, and \vec{\sigma}_1 and \vec{\sigma}_2 are Pauli matrices of the two spins. The total spin angular momentum \vec{S} = \vec{S}_1 + \vec{S}_2 = \frac{1}{2} \hbar (\vec{\sigma}_1 + \vec{\sigma}_2). Show that P, S^2 and S_z can be measured simultaneously.

Consider the potential V(x) = \begin{cases} 0, & 0 < x < a \\ \infty, & \text{elsewhere} \end{cases} (a) Estimate the energies of the ground state as well as those of the first and the second excited states for (i) an electron enclosed in a box of size a = 10^{-10}\text{ m}. (ii) a 1\text{ g} metallic sphere which is moving in a box of size a = 10\text{ cm}. (b) Discuss the importance of the Quantum effects for both of these systems. (c) Estimate the velocities of the electron and the metallic sphere using uncertainty principle.

Consider a particle of mass m and charge q moving under the influence of a one dimensional harmonic oscillator potential. Assume it is placed in a constant electric field E. The Hamiltonian of this particle is therefore given by H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 X^2 - qEX. Obtain the energy expression and the wave function of the n\text{th} excited state of the particle.

By assuming the nucleus as a cubical box of length equal to the nuclear diameter 10^{-12}\text{ cm}, calculate the kinetic energy of the highest level occupied nucleon of iron-56 nucleus.

Calculate the zero point energy for a particle in an infinite potential well for the following cases : (i) a 100\text{ g} ball confined on a 5\text{ m} long line. (ii) an oxygen atom confined to a 2 \times 10^{-10}\text{ m} lattice. (iii) an electron confined to a 10^{-10}\text{ m} atom. Why zero point energy is not important for macroscopic objects ? Comment.

A particle constrained to move along x-axis in the domain 0 \le x \le L has a wave function \psi(x) = \sin \left( \frac{n \pi x}{L} \right), where n is an integer. Normalize the wave function and evaluate the expectation value of momentum of the particle.

A particle of mass m is in a spherically symmetric attractive potential of radius a. Find the minimum depth of the potential needed to have two bound states of zero 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})

Obtain the normalized eigenvectors of \sigma_x and \sigma_y matrices.

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

Show that E_n=\langle V\rangle in the stationary states of the hydrogen atom.

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)

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