PYQ Bank

Define angular momentum of a particle and find out the three components of the angular momentum operator \hat{L} in Cartesian coordinates. Show that \hat{L}^2=-\hbar^2\left[r^2\nabla^2-\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)\right] Prove that the operator \hat{L}^2 can also be expressed as \hat{L}^2=-\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] in spherical polar coordinates (r,\theta,\phi).

= 2i\sigma_z$ and (ii) \sigma_x \sigma_y \sigma_z = i.

In the |jm\rangle basis formed by the eigenkets of J^2 and J_z, show that \langle jm| J_- J_+ |jm\rangle = (j-m)(j+m+1)\hbar^2 where J_+ = J_x + i J_y and J_- = J_x - i J_y.

= 0$, where i = x, y, and z and comment on the measurability of J^2 and its components as operators.

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 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 ?

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.

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.

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.

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.

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)?

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?

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.

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).

Calculate the radius of electron orbit for \mathrm{Li}^{++} in ground state.

What is Zeeman effect? Discuss the factors on which Larmor frequency is dependent.

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.

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.