Using WKB approximation find out the lifetime of \alpha-emitter.

Normalize the ground state wave function \psi_0(x) = A e^{(-\alpha x^2/2)} for the simple harmonic oscillator and find expectation values \langle x \rangle and \langle x^2 \rangle.

Show that

(i) no two of the three components of angular momentum operator commute and

(ii) the third component of the angular momentum operator commutes with the square of angular momentum operator.

Explain the origin of the anomalous Zeeman effect.

What is the de Broglie wavelength of an electron whose kinetic energy is 100\text{ eV} ?

(i) Enumerate the possible values of quantum numbers j and m_j for state in which l = 2 and s = 1/2.

(ii) Draw the corresponding vector model diagram.

The particle in a box has a ground state wave function given as \psi(x) = \frac{1}{\sqrt{l}} \cos \frac{\pi x}{2l} The box width is 2l and the particle is confined within (-l, +l). Calculate the expectation value of x^2.

Write down Schr"{o}dinger equations for a particle of energy E < V_0, incident on a step potential height V_0. Solve them to find out the transmission and reflection coefficients in terms of k and k', where k = \sqrt{\frac{2mE}{\hbar^2}} and k' = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}}. Show that in this case, there is a finite probability of finding the particle in a classically forbidden region.

A particle is bound in a potential well given by V(x) = \begin{cases} \infty & \text{for } x \le 0 \\ cx & \text{for } x > 0 \end{cases} Estimate the ground state energy of the system from uncertainty principle.

Show that the spherical harmonics Y_{lm}(\theta, \phi) are simultaneous eigenfunctions of L^2, L_z and L_z^2. What are their corresponding eigenvalues?

Develop and write down the expressions of L^2, L_z and L_z^2 in angular momentum operator algebra.

(i) Show that the orientation of the spin angular momentum vector of an electron with respect to the z-axis is less than one radian for spin up electron. (ii) When the orbital angular momentum vector \vec{L} has the magnitude \sqrt{6}\hbar, calculate the L_z components. What angles does the \vec{L} vector make with the z-axis ?

A lead ball of mass 0\cdot 1\text{ g} is thrown with a velocity of 10^3\text{ cm/sec} through a hole 1\text{ cm} in radius. Calculate the uncertainty in the angle of emergence.

Solve the Schrödinger equation for a gas of non interacting electrons enclosed in a cube of volume L^3. What do you mean by density of states ? Calculate the density of states and Fermi energy for the above system.

Solve the Schrödinger equation to obtain the energy levels and eigenfunctions of a particle in a one dimensional infinitely deep potential well given by \begin{aligned} V(x) &= 0 \text{ for } 0 < x < a \\ &= \infty \text{ for } x < 0 \text{ and for } x > a \end{aligned} Show that they form an orthonormal set of functions. What do you understand by completeness condition ? Show that they form a complete set of functions.

Starting from Schrödinger equation obtain an expression for the probability current density. Hence give an interpretation to the wave function.

What do you understand by expectation value ? Prove that \frac{d}{dt}\langle x\rangle = \frac{1}{m}\langle p_x\rangle

Solve the eigen value equation L^2 Y(\theta, \phi) = \lambda \hbar^2 Y(\theta, \phi) and obtain the eigen values and eigen functions of L^2.

For a quantum mechanical system prove that all energy eigen-values E_n are real and if E_n \neq E_k, then the corresponding eigen functions are orthogonal.

What are Pauli spin matrices ? 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}) where \vec{\sigma} are the Pauli spin matrices and \vec{A} and \vec{B} are vector operators which commute with \vec{\sigma}, but do not necessarily commute with each other.