Explain the phenomenon of penetration of a particle through a barrier whose height exceeds the total energy of the particle with necessary diagram.

A bullet of mass 0\cdot 025\text{ kg} is moving with a velocity of 600\text{ m/s}. The speed is measured with an accuracy of 0\cdot 02\%. Find out the uncertainty in x. Also, comment on the result.

(i) Derive an expression for density of states for a free electron gas in one dimension. Hence, show its variation with energy for a one-dimensional metallic crystal. (ii) Where do you find the applications of free electron gas model?

The raising (J_+) and lowering (J_-) operators are defined by J_+ = J_x + i J_y and J_- = J_x - i J_y. Show that-- (i) [J_z, J_\pm] = \pm \hbar J_\pm (ii) J_+ J_- = J^2 - J_z^2 + \hbar J_z

A particle is described by the wave function \psi(x, t) = e^{i(kx-\omega t)}. (i) Is this wave function an eigenfunction corresponding to any dynamical variable or variables? If so, name the variable(s). (ii) Does this represent a ground state?

Find the energy levels of a spin S = \frac{3}{2} particle whose Hamiltonian is given by H = \frac{\alpha}{\hbar^2}(S_x^2 + S_y^2 - 2S_z^2) - \frac{\beta}{\hbar} S_z where \alpha, \beta are constants. Are these levels degenerate?

Consider a particle of mass m and charge q moving under the influence of a one-dimensional harmonic oscillator potential. Assume that it is placed in a constant electric field E. The Hamiltonian of this particle is therefore given by H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2 - qEx Derive the energy expression and wave function of the n\text{th} excited state.

Determine the size of the hydrogen atom using uncertainty principle. Given that the potential energy of electron V = \frac{-e^2}{4\pi\varepsilon_0 a}, where a is the distance of the electron from the nucleus.

Consider a particle of mass m moving freely between x = 0 and x = a inside an infinite square well potential. Calculate the expectation values \langle x \rangle_n, \langle p \rangle_n, \langle x^2 \rangle_n and \langle p^2 \rangle_n, and compare them with their classical counterparts.

Derive the degeneracy of a harmonic oscillator g_n = \frac{1}{2}(n+1)(n+2).

Consider a particle of mass m moving in the potential V(x) = \begin{cases} +\infty & ; x \le 0 \\ \frac{1}{2}m\omega^2 x^2 & ; x > 0 \end{cases} Estimate the ground state energy of this particle using the WKB method.

$, [J_z^2, J_y] and [J^2, J_y], and then show that \langle J, m | J_x^2 | J, m \rangle = \langle J, m | J_y^2 | J, m \rangle

$ and hence show that [p^2, x] = -2 i\hbar p.

Find the condition at which de Broglie wavelength equals the Compton wavelength for a particle.

(i) Derive an expression for the period of Bloch oscillation for a one-dimensional crystal having lattice period a and electric field \varepsilon. (ii) Consider an electron in a perfectly periodic lattice, wherein the energy-wavenumber relationship in the first Brillouin zone is expressed as E = \frac{\hbar^2 k^2}{5\mathrm{m}_e} where \mathrm{m}_e is the mass of an electron in free space. Find the effective mass of the electron and hence write down the time-independent Schrodinger equation for the electron. Also determine the velocity of the electron. Ignore all interactions except between the electron and the lattice.

The components of arbitrary vectors \vec{A} and \vec{B} commute with those of \vec{\sigma}. Show that (\vec{\sigma} \cdot \vec{A})(\vec{\sigma} \cdot \vec{B}) = \vec{A} \cdot \vec{B} + i\vec{\sigma} \cdot (\vec{A} \times \vec{B}).

For the n^{\text{th}} state of linear harmonic oscillator, evaluate the uncertainty product (\Delta x) \cdot (\Delta p).

The time independent wave function of a system is \psi(x) = A \exp(ikx), where k is a constant. (i) Is this wave function normalizable in the domain -\infty < x < \infty? (ii) Calculate the probability current density for this function.

The raising (J_+) and lowering (J_-) operators of total angular momentum are defined by J_+ = J_x + iJ_y and J_- = J_x - iJ_y. Find the values of the following : (i) [J_x, J_+] (ii) [J_y, J_+] (iii) J_- J_+