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^-
On the basis of binding energy curve explain why energy is released both in Fission and Fusion.
What is Q value of a nuclear reaction ? Obtain an expression for the Q-value in terms of mass and energy of the particles.
Discuss the four basic types of fundamental interactions in nature and compare them.
Discuss the pure rotational Raman spectra of symmetric top molecules. What are R and S branches ?
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.
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?
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.
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.
Derive the rotational-vibrational energy levels of a diatomic molecule. Give the analysis of the spectral lines.
Calculate the two longest wavelengths of the Balmer series of triply ionized beryllium (z = 4).
(i) Explain 'term symbol' for a particular atomic state. (ii) Show that the spectral lines of alkali metals are doublets.
Consider the one-dimensional wavefunction \psi(x)=Axe^{-kx}, (0\leq x\leq\infty;\,k>0)
(i) Calculate A so that \psi(x) is normalized.
(ii) Using Schrödinger's equation find the potential V(x) and energy E for which \psi(x) is an eigenfunction. (Assume that as x \to \infty, V(x) \to 0).
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.
(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 ?
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.
What do you understand by expectation value ? Prove that \frac{d}{dt}\langle x\rangle = \frac{1}{m}\langle p_x\rangle
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
(i) Solve the radial part of the time-independent Schrödinger equation for a hydrogen atom. Obtain an expression for the energy eigenvalues.
(ii) What is the degree of degeneracy of the energy eigenvalues? What happens if the spin of the electron is taken into account?
