UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Show that the energy of the triplet state (S = 1) is not equal to the energy of the singlet state (S = 0).

State how for spin-half particles, the spin (\sigma) can be expressed by its three components \sigma_x, \sigma_y and \sigma_z.

Obtain the expressions for reflection coefficient (R) and transmission coefficient (T) for reflected waves and transmitted waves from an infinite thin barrier.

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

An alpha particle is trapped in a nucleus of radius 1\cdot 4\text{ fm}. What is the probability that it will escape from the nucleus if its energy is 2\text{ MeV}? The potential barrier at the surface of the nucleus is 4\text{ MeV}. [Given : The mass of the alpha particle is 6\cdot 64 \times 10^{-27}\text{ kg}]

Show that the square of the orbital angular momentum operator (L^2) commutes with any of the components of angular momentum operator L. Is it possible to measure L^2, L_x, L_y and L_z simultaneously ? Give reasons for your answer.

The ground state wave function of a harmonic oscillator is \psi_0(x) = \left(\frac{m\omega}{\hbar \pi}\right)^{1/4} \exp\left(-\frac{m\omega x^2}{2\hbar}\right). (i) At which point is the probability density maximum ? (ii) What is the value of the maximum probability density ?

(i) Assuming the potential seen by a neutron in a nucleus to be schematically represented by a one-dimensional, infinite rigid wall potential of length 10^{-15}\text{ m}, estimate the minimum kinetic energy of the electron. (ii) Estimate the minimum kinetic energy of neutron bound within the nucleus as described above. Can an electron be confined in a nucleus ? Explain.

Calculate [J_x^2, J_y], [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

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

A particle limited to the x-axis has the wave function \phi(x) = bx^2 between x = 0 and x = 2; the wave function \phi(x) = 0 elsewhere. (i) Find the probability that the particle can be found between x = 1.0 and x = 1.5. (ii) Find the expectation value <x> of the particle position.

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.

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.

(ii) [L_z, L_+] = \hbar L_+ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z Prove that : (i) [L^2, L_z] = 0 (ii) [L_z, L_+] = \hbar L_+ (iii) [L_+, L_-] = 2\hbar L_z (iv) L_+ L_y = L^2 - L_z^2 + \hbar L_z where \hbar = \frac{h}{2\pi} (h is Planck's constant)

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.

Find the energy of the particle of mass m moving in a potential field V(x) = \frac{2\hbar^2 b^2 x^2}{m} for which the time independent wave function is \psi(x) = \exp(-bx^2). Here b is a constant.

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.

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?

A particle constrained to move along x-axis in the domain 0 \le x \le L has a wave function \psi(x) = \sin \left( \frac{n \pi x}{L} \right), where n is an integer. Normalize the wave function and evaluate the expectation value of momentum of the particle.

By assuming the nucleus as a cubical box of length equal to the nuclear diameter 10^{-12}\text{ cm}, calculate the kinetic energy of the highest level occupied nucleon of iron-56 nucleus.