PYQ Bank

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?

A one-dimensional oscillator of mass m = 10^{-20}\text{ kg} oscillates under a force constant k = 10^{-4}\text{ N/m}. (i) Evaluate the zero-point energy. (ii) Calculate the classical amplitude for which the oscillator can have this energy. (iii) What is the energy for the second excited state?

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

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

The components of the angular momenta \vec{J}_1 and \vec{J}_2 satisfy commutation rules. Show that the components of the sum \vec{J} = \vec{J}_1 + \vec{J}_2 also satisfy the commutation rules. Whether the difference \vec{J}_1 - \vec{J}_2 satisfies an angular momentum?

The raising (J_+) and lowering (J_-) operators are defined by J_+=J_x+iJ_y and J_-=J_x-iJ_y respectively. Prove the following identities:

(i) [J_z,J_\pm]=\pm\hbar J_\pm

(ii) J_-J_+=J^2-J_z^2-\hbar J_z

Find the values of the following : (i) \vec{L} \times \vec{L} (ii) L_+ L_- (iii) [L_z, L_+] where \vec{L} is the angular momentum operator. L_+ and L_- are raising and lowering operators respectively.

The lifetime of a given atom in an excited state is 10^{-8}\text{ s}. It comes to the ground state by emitting a photon of wavelength 5800\text{ \AA}. Find the energy uncertainty and wavelength uncertainty of the photon.

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)

Derive an approximate expression for the transmission coefficient for a rectangular potential barrier for which \frac{a}{\hbar}\sqrt{2m(V_0 - E)} \gg 1.

A stream of electrons, each of energy E = 3\text{ eV}, is incident on a potential barrier of height V = 4\text{ eV}. The width of the barrier is 20\text{ \AA}. Calculate the percentage transmission of the beam through the barrier.

Calculate the probability of finding a simple harmonic oscillator within the classical limits if the oscillator is in its normal state. Also show that if the oscillator is in its normal state, then the probability of finding the particle outside the classical limits is approximately 16\%.

A beam of 12\,\mathrm{eV} electron is incident on a potential barrier of height 25\,\mathrm{eV} and width 0.05\,\mathrm{nm}. Calculate the transmission coefficient.

A particle is moving in a one-dimensional box of width 50\,\mathring{\mathrm{A}} and infinite height. Calculate the probability of finding the particle within an interval of 15\,\mathring{\mathrm{A}} at the centres of the box when it is in its state of least energy.

Use WKB method to estimate the energy levels of a one-dimensional harmonic oscillator.

Use the uncertainty principle to estimate (i) The ground state radius of the hydrogen atom, and (ii) The ground state energy of the hydrogen atom.

To illustrate the idea that the zero point energy gets larger by going from macroscopic to microscopic systems, calculate the zero point energy for a particle in an infinite potential well for the following three 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.

Normalised wave function of hydrogen atom for 1s state is \psi_{100}=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0},\text{ where }a_0=\frac{\hbar^2}{me^2} being the Bohr radius. Calculate the expectation value of potential energy in this state.

A particle of rest mass m_0 has a kinetic energy K, show that its de Broglie wavelength is given by \lambda=\frac{hc}{\sqrt{\left[K\left(K+2m_0c^2\right)\right]}}. Hence calculate the wavelength of an electron of kinetic energy 2\mathrm{MeV}. What will be the value of \lambda if K << m_0c^2?

Find the uncertainty in the momentum of a particle when its position is determined within 0.02 cm. Find also the uncertainty in the velocity of an electron and \alpha-particle respectively when they are located within 15 \times 10^{-8}\,cm.