Search, filter, and study UPSC Civil Services & IFoS Physics past year questions with rigorous derivations, formulas, and step-by-step solutions.
The most intense vibrational bands of CO and HCl molecules have the wave numbers 2\cdot 143 \times 10^5\text{ m}^{-1} and 2\cdot 886 \times 10^5\text{ m}^{-1} respectively. Calculate the force constants of these molecules.
Discuss the fine structure of hydrogen spectrum. How is it of importance in the astronomical observations?
Write down the Schrödinger wave equation for a one-dimensional harmonic oscillator in which a particle of mass m and frequency \omega is subject to a parabolic potential V(x) = m\omega^2 x^2/2. Let one of the possible energy eigen states be given by \psi(x) = Ax e^{-x^2/x_0^2}. Find the energy E corresponding to the eigen state given by \psi(x). Is it a ground state energy or one of the excited state energies ? Comment.
= 0$, where i = x, y, and z and comment on the measurability of J^2 and its components as operators.
Which of the following functions is/are acceptable solution(s) of the Schrodinger equation?
(i) \psi(x)=Ae^{-ikx}+Be^{ikx}
(ii) \psi(x)=Ae^{-kx}+Be^{kx}
(iii) \psi(x)=A\sin 3kx+B\cos 5kx
(iv) \psi(x)=A\sin 3kx+B\sin 5kx
(v) \psi(x)=A\tan kx Explain your answer.
Electrons with energies of 1\cdot 0\text{ eV} and 2\cdot 0\text{ eV} are incident on a barrier 10\cdot 00\text{ eV} high and 0\cdot 50\text{ nm} (nanometer) wide. Calculate the ratio of their respective transmission probabilities across the barrier.
The radial part of Schrödinger wave function for hydrogen atom for spherical symmetric potential V(r) = -\frac{e^2}{4\pi \varepsilon_0 r} is given as : \frac{1}{r^2}\frac{d}{dr}\left(r^2 \frac{dR}{dr}\right) + \frac{2\mu}{\hbar^2}\left[ E - V(r) - \frac{\hbar^2}{2\mu}\frac{l(l+1)}{r^2}\right] R = 0, where \mu = \frac{mM}{m+M} is reduced mass and m and M are mass of electron and proton respectively. (i) Obtain the form of the above equation for ground state electron of hydrogen atom. (ii) Also starting with a trial wave function for the radial equation, R = A e^{-r/a_0}, for the l = 0 state, find the expressions of energy E and orbital radius for the ground state of hydrogen atom.
Use uncertainty principle to estimate the ground state energy of a linear harmonic oscillator.
The wave function of a particle is given as \psi(x)=\frac{1}{\sqrt{a}}e^{-\lvert x\rvert/a}. Find the probability of locating the particle in the range -a\leq x\leq a.
Calculate the zero-point energy of a system consisting of a mass of 10^{-3}\,\mathrm{kg} connected to a fixed point by a spring which is stretched by 10^{-2}\,\mathrm{m} by a force of 10^{-1}\,\mathrm{N}. The system is constrained to move only in one direction.
The general wave function of harmonic oscillator (one-dimensional) are of the form u_n(x)=\sum_{k=0}^{\infty}a_k y^k e^{-y^2/2} With y=\sqrt{\frac{m\omega}{\hbar}}x, and coefficients a_k are determined by recurrence relations a_{k+2}=\frac{2(k-n)}{(k+1)(k+2)}a_k Corresponding energy levels are E_n=\left(n+\frac{1}{2}\right)\hbar\omega. Discuss the parity of these wave functions. What happens, if the potential for x\leq 0 is infinite (half harmonic oscillator)?
Calculate the radius of electron orbit for \mathrm{Li}^{++} in ground state.
A beam of particles of energy 9\,\mathrm{eV} is incident on a potential step 8\,\mathrm{eV} high from the left. What percentage of particles will reflect back?
What is Zeeman effect? Discuss the factors on which Larmor frequency is dependent.
Consider a particle trapped in a box of width L and its \text{n}^{\text{th}} state wave function is given by \psi_n (n) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right). Calculate the expectation value of the position <x> of the particle. How is this result different from classical consideration of finding the particle inside the box ?
Given the time dependent one-dimensional Schrödinger wave equation as H\psi(t) = i\hbar \frac{\partial \psi(t)}{\partial t}, \text{ with } H = \frac{\vec{p}\cdot \vec{p}}{2m} + V(\vec{x}) as Hermitian operator for a particle of momentum \vec{p} under the influence of a potential V(\vec{x}). Find the value of \frac{d}{dt}\left(\int \psi^*(t) \psi(t) \, dx\right).
Consider an experiment in which a beam of electrons is directed at a plate containing two slits, labelled as A and B. Beyond the plate is a screen, where electrons hit the screen and are detected. For each of the following cases sketch the variation of the relative number of incident electrons as a function of position along the screen and also provide brief explanation about each observation : (i) Slit A open, slit B closed, (ii) Both A and B are open.
Show that for free electron gas, the density of states in three dimensions (3D) varies as E^{1/2}, and this dependence changes to E^0 for 2D (quantum well), E^{-1/2} for 1D (quantum wire) and \delta function for 0D (quantum dot).
The ground state wave function for hydrogen atom is \psi(r)=\frac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0} where a_0 is the Bohr radius. Sketch the wave function and the probability density as a function of the separation distance r. Calculate the probability that the electron in the ground state is found beyond the Bohr radius.
Prove the following identities:
(i) [\hat{p}_x,\hat{L}_y]=i\hbar\hat{p}_z
(ii) e^{i(\hat{\sigma}\cdot\hat{n})\theta}=\cos\theta+i(\hat{\sigma}\cdot\hat{n})\sin\theta