(i) What do you mean by expectation value of a physical quantity ? How does it help to extract information from a wave function ? (ii) A particle limited to move along x-axis has the wave function \psi = ax between x = 0 and x = 1, \psi = 0 elsewhere. Find the probability that the particle can be found between x = 0\cdot 45 and x = 0\cdot 55. Find the expectation value <x> of the particles position x = 0 to x = 1.

(i) Discuss about the validity criterion for the WKB approximation. (ii) Qualitatively explain alpha (\alpha) particle emission by WKB approximation.

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.

$. 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.

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.

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.

= 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.

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.

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).

Use uncertainty principle to estimate the ground state energy of a linear harmonic oscillator.

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.

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.

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.

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

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 ?

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

(i) \hat{L} \times \hat{L} = i \hbar \hat{L}

(ii) [\hat{L}_+, \hat{L}_-] = 2\hbar \hat{L}_z

(iii) \hat{\sigma}_x \hat{\sigma}_y \hat{\sigma}_z = -i

Use the uncertainty principle to estimate the binding energy of the hydrogen atom in the ground state in terms of m_\text{e}, e, \hbar and c. Estimate the answer in eV.