UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Explain the origin of the anomalous Zeeman effect.

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.

Derive an expression for transmission coefficient for a particle through a rectangular potential barrier.

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.

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

Write down Schrödinger 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.

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.

(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.

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.

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

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?

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.

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.

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

(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 ?

(i) What angles do the \vec{L} \text{(vector)} make with the z-axis when l = 2 for an electron ? (ii) Determine the values of the total angular momentum for a 3d electron.

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.

Solve the Schrodinger equation for a potential step function given by \begin{aligned} v(x) &= 0 \text{ for } x < 0 \\ &= v_0 \text{ for } x > 0 \end{aligned} and calculate the reflection and transmission coefficients. Show that for E < v_0 there is a finite probability of finding the particle in a classically forbidden region.