UPSC Physics Optional PYQs (CSE & IFoS)PYQs

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 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?

(i) Find the lowest energy of an electron confined in a three-dimensional cubic box of each side 1\text{ \AA}. (ii) Find the temperature at which the average energy of the molecule would be equal to the lowest energy of the electron. [Given : k = 1\cdot 38 \times 10^{-23}\text{ J/K}]

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 normalized eigenfunction for the ground state of hydrogen atom is \Psi_{100} = \frac{1}{\sqrt{\pi}} \left( \frac{Z}{a_0} \right)^{3/2} e^{-Zr/a_0} Calculate the expectation value of the radius vector r of the electron in the ground state.

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.

An electron is confined to an infinite well whose width L (= 100\text{ pm}) is roughly the size of an atom. (i) What are the energies of four least energetic quantum states? (ii) What energy must be imparted to the electron to raise it from a state with n = 12 to a higher energy state with n = 25?

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?

(i) Derive the Schrodinger time-independent wave equation for matter waves. (ii) An electron is confined to move in a one-dimensional potential well of length 5\text{ \AA}. Find the quantized energy values for the three lowest energy states.

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.

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.

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.

Prove that [J^2, J_y] = 0.

Find the eigenvalues and eigenstates of the spin operator \vec{S} of an electron in the direction of unit vector \vec{n}. Assume that \vec{n} lies in the xz-plane using spin matrices.

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

Prove the following : (i) [L^2, L_x] = 0 (ii) [L_x, L_y] = i \hbar L_3

Draw a neat sketch showing potential wells and energy levels of the following : (i) Harmonic oscillator (ii) Hydrogen atom (iii) Particle in a box List the contrasting points and mention the variation of $E_n$' with n' of the above plots.

(i) Draw a neat sketch of a finite rectangular potential barrier well of height U' and width L' that contains a particle whose energy E' is less than U'. Solve the Schrödinger equations of the particles outside the well. (ii) Plot wave functions and probability densities |\psi|^2 of the particle for the first three states. Compare the results with that of a particle in a box of infinite potential (U = \infty).

Derive energies and wave functions for motion of electron in a hydrogen atom.

What is Normal Zeeman effect ? Give the classical interpretation of Normal Zeeman effect.