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 ?
(i) Assuming the potential seen by a neutron in a nucleus to be schematically represented by a one-dimensional, infinite rigid wall potential of length 10^{-15}\text{ m}, estimate the minimum kinetic energy of the electron. (ii) Estimate the minimum kinetic energy of neutron bound within the nucleus as described above. Can an electron be confined in a nucleus ? Explain.
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.
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.
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.
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 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.
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.
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.
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.
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.
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.
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.
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 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})
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 the spin wave function (for s=\frac{1}{2}) if the spin component in the direction of unit vector \eta has a value of \frac{1}{2}\hbar?
Show that for a given principal quantum number n, there are n^2 possible states of the atom.
Obtain the normalized eigenvectors of \sigma_x and \sigma_y matrices.
