Search, filter, and study UPSC Civil Services & IFoS Physics past year questions with rigorous derivations, formulas, and step-by-step solutions.
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.
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.
Use WKB method to estimate the energy levels of a one-dimensional harmonic oscillator.
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?
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.
Find the minimum magnetic field needed for the Zeeman effect to be observed in a spectral line of 400\,\mathrm{nm} wavelength when a spectrometer whose resolution is 0.010\,\mathrm{nm} is used. Write the answer in the nearest high integer.
Calculate the Larmor precessional frequency for a magnetic induction field of 0.5\,\mathrm{T}. Hence calculate the splitting in wave numbers of a spectral line due to normal Zeeman effect for the same field.
Describe normal and anomalous Zeeman effect. Explain how it lifts the degeneracy in hydrogen atom.
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 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 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.
Using Pauli spin matrices prove that,
(i) \sigma_x\sigma_y+\sigma_y\sigma_x=0;\ \sigma_y\sigma_z+\sigma_z\sigma_y=0;\ \sigma_x\sigma_z+\sigma_z\sigma_x=0
(ii) \sigma_+\sigma_-=2(1+\sigma_z)
(iii) \sigma_\alpha+\sigma_\beta=i\sigma_\gamma where \alpha\ne\beta\ne\gamma
How many nitrogen molecules must strike a 1\text{ cm}^2 surface each second to exert a pressure of 1\text{ atmosphere} ? (Assume the molecules are all moving at same speed, corresponding to a temperature of 300\text{ K}, and at an angle of 45^\circ to the wall). [Molecular mass of \text{N}_2 is 28\text{ u}]
A hypothetical engine, with an ideal gas as the working substance, operates in the cycle shown below. Show that the efficiency of the engine is \eta = 1 - \frac{1}{\gamma} \left( \frac{1 - \dfrac{P_3}{P_1}}{1 - \dfrac{V_1}{V_3}} \right) .
Find the pressure at which water would boil at 150^\circ\text{C} if the change in specific volume when one gm of water is converted into steam is 1676\text{ c.c.} Given J = 4\cdot 2 \times 10^7\text{ ergs/cal}, one atmosphere = 10^6\text{ dyne/cm}^2 and latent heat of vapourisation of steam = 540\text{ cal/gm}.
Write the expression for the Fermi-Dirac distribution. Plot the Fermi-Dirac distribution at T=0 and for T_1>T_2>0. Now from the plot propose two alternative definitions of the Fermi level.
Calculate the probability of an electron occupying an energy level 0.02\,\mathrm{eV} above the Fermi level at T=300\,\mathrm{K}
Write a brief note on Chandrasekhar Limit.