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.
What is Normal Zeeman effect ? Give the classical interpretation of Normal Zeeman effect.
(i) State and explain position momentum uncertainty principle. Justify that this principle is not just a negative statement rather a useful tool, with one example. (ii) The lifetime of an excited state of an atom is about 10^{-8}\text{ sec.} Calculate the minimum uncertainty in the energy of the excited state.
Show that the phase velocity v_p for a particle with rest mass m_0 is always greater than the velocity of light and that v_p is a function of wavelength.
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.
For Pauli's matrices, prove that (i) [\sigma_x, \sigma_y] = 2i\sigma_z and (ii) \sigma_x \sigma_y \sigma_z = i.
Consider a particle whose wave function is given by \psi(x) = A e^{-\alpha x^2}.
(i) What is the value of A if this wave function is normalized?
(ii) Calculate the expectation value of x for this particle.
If the Hamiltonian of a system be H = (p_x^2 / 2m) + V(x), obtain the value of the commutator [x, H]. Hence find the uncertainty product (\Delta x)(\Delta H).
An electron is confined in the ground state of a one-dimensional harmonic oscillator such that \Delta x = 10^{-10}~\text{m}. Assuming \langle T \rangle = \langle V \rangle, find the energy in eV required to excite it to the first excited state.
An electron and a photon each has a wavelength of 2~\text{\AA}. Calculate (i) their momenta and (ii) the ratio of their kinetic energies.
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.
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.
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).
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.
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.
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 ?
The quantum theory describes the angular momentum (\vec{J}) as a linear operator with three components J_x, J_y and J_z in three dimensions. For the angular momentum operator prove that [J^2, J_i] = 0, where i = x, y, and z and comment on the measurability of J^2 and its components as operators.
Estimate the probability of finding the particle between x = 0 and x = \frac{L}{3} in the lowest energy state.
Generalize the results obtained in part (a) to obtain the allowed energy states and the corresponding eigenfunctions of a particle in a three dimensional isotropic box of dimension L. What is the degeneracy of the first two excited states of the particle ?
