PYQ Bank

(i) Explain how the helicity of neutrino can be measured. (ii) What are 'strange particles' and what is 'strangeness number' ?

Determine from the given data whether the following reactions are exothermic or endothermic:

(i) ^{1}_{1}\mathrm{H}+{}^{3}_{1}\mathrm{H}\rightarrow{}^{2}_{1}\mathrm{H}+{}^{2}_{1}\mathrm{H}

(ii) ^{12}_{6}\mathrm{C}+{}^{12}_{6}\mathrm{C}\rightarrow{}^{20}_{10}\mathrm{Ne}+{}^{4}_{2}\mathrm{He}

Which of the following reactions are permitted or forbidden by various conservation laws?

(i) k^{\mp}p \longrightarrow \Lambda^0 + \pi^0

(ii) k^+ + n \longrightarrow \Sigma^+ + \pi^0

(iii) \pi^{\mp}p \longrightarrow k^{\mp}\Sigma^+

(iv) \pi^+ + n \longrightarrow k^+ + \Lambda^0

(v) \pi^{\mp}p \longrightarrow \Lambda^0 + k^0

(vi) \pi^{\mp}p \longrightarrow k^+ + \Sigma^-

Write down the Bethe-Weizsäcker semiempirical mass formula for a nucleus. Explain the significance of each term occurring in it. Discuss the stability of a nucleus against \beta-decay. What is the effect of pairing term on stability?

Sodium doublets are produced by transitions 3{}^2p_{1/2}\rightarrow 3{}^2s_{1/2}(D_1) and 3{}^2p_{3/2}\rightarrow 3{}^2s_{1/2}(D_2). Calculate the Landé g-factors for these levels.

(i) Explain 'term symbol' for a particular atomic state. (ii) Show that the spectral lines of alkali metals are doublets.

Calculate the two longest wavelengths of the Balmer series of triply ionized beryllium (z = 4).

What will happen to the energy level of an unpaired electron when subjected to an external magnetic field? Find the condition for electron spin resonance.

Why should rotational Raman spectrum show a separation of the first Raman line from the exciting line equal to 6B\ \mathrm{cm}^{-1}, while the separation between successive lines equals to 4B\ \mathrm{cm}^{-1}, where B is the rotational constant?

Describe the vibration-rotation spectra of a polyatomic linear molecule. What are P, Q and R branches? Draw rotational energy levels of the vibrational states v=0 and v=1 and perpendicular vibrational-rotational transitions of a linear polyatomic molecule.

Discuss the pure rotational Raman spectra of symmetric top molecules. What are R and S branches ?

Derive the rotational-vibrational energy levels of a diatomic molecule. Give the analysis of the spectral lines.

(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 ?

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

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.

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.

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

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

Given that \sigma_x,\sigma_y,\sigma_z are Pauli spin operators, prove the following relationships:

(i) \sin(\sigma_x\varphi)=\sigma_x\sin\varphi

(ii) \cos(\sigma_z\varphi)=\cos\varphi