UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Prove the commutation relation for the angular momentum: [L^2,L_z]=0 Also show that (\vec{L}\times\vec{L})=i\hbar\vec{L}.

What is Zeeman effect? Explain Zeeman effect on the basis of classical electron theory.

Using the uncertainty principle \Delta x\Delta p \geq \hbar/2, estimate the ground state energy of a harmonic oscillator.

Define angular momentum of a particle and find out the three components of the angular momentum operator \hat{L} in Cartesian coordinates. Show that \hat{L}^2=-\hbar^2\left[r^2\nabla^2-\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)\right] Prove that the operator \hat{L}^2 can also be expressed as \hat{L}^2=-\hbar^2\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}\right] in spherical polar coordinates (r,\theta,\phi).

Describe normal and anomalous Zeeman effect. Explain how it lifts the degeneracy in hydrogen atom.

Define mathematically the Bohr radius of a hydrogen atom and show that the binding energy at state n of this atom can be given by E_n=-\frac{1}{2}\frac{Ze^2}{(a/Z)4n^2\pi\epsilon_0} where Z is the atomic number of H atom. Calculate the numerical values of a and E_1 of H atom.

State and express mathematically the three uncertainty principles of Heisenberg. Highlight the physical significance of these principles in the development of Quantum Mechanics.

For a free quantum mechanical particle under the influence of a one-dimensional potential, show that the energy is quantized in discrete fashion. How do these energy values differ from those of a linear harmonic oscillator?

Show that the mass and linear momentum of a quantum mechanical particle can be given by m=h/(\lambda v) and p=h/\lambda, respectively, where h, \lambda and v are Planck’s constant, wavelength, and velocity of the particle, respectively. Comment on the wave-particle duality from these relations.

How do you define density of states? Show that the density of states with wave vector less than \vec{k} in a three-dimensional cubic box of volume V can be given by D(\omega)=\frac{V}{2\pi^2}k^2\left(\frac{dk}{d\omega}\right) in the frequency spectrum between \omega and \omega+d\omega. Here, assume that the number of modes per unit range of k is L/(2\pi), L being the length of each side of the cubic box.

Estimate the size of hydrogen atom and the ground state energy from the uncertainty principle.

Define Pauli spin matrices \sigma_x, \sigma_y and \sigma_z. Using these definitions, prove the following:

(i) \sigma_x^2=\sigma_y^2=\sigma_z^2=1

(ii) \sigma_x\sigma_y=i\sigma_z;\ \sigma_z\sigma_x=i\sigma_y;\ \sigma_y\sigma_z=i\sigma_x

Write down the Hamiltonian operator for a linear harmonic oscillator. Show that the energy eigenvalue of the same can be given by E_n=\left(n+\frac{1}{2}\right)\hbar\omega_0 at energy state n with \omega_0 being the natural frequency of vibration of the linear oscillator. Prove that n=0 energy state has a wave function of typical Gaussian form.

The general wave function of harmonic oscillator (one-dimensional) are of the form u_n(x)=\sum_{k=0}^{\infty}a_k y^k e^{-y^2/2} With y=\sqrt{\frac{m\omega}{\hbar}}x, and coefficients a_k are determined by recurrence relations a_{k+2}=\frac{2(k-n)}{(k+1)(k+2)}a_k Corresponding energy levels are E_n=\left(n+\frac{1}{2}\right)\hbar\omega. Discuss the parity of these wave functions. What happens, if the potential for x\leq 0 is infinite (half harmonic oscillator)?

Calculate the zero-point energy of a system consisting of a mass of 10^{-3}\,\mathrm{kg} connected to a fixed point by a spring which is stretched by 10^{-2}\,\mathrm{m} by a force of 10^{-1}\,\mathrm{N}. The system is constrained to move only in one direction.

Prove the following identities:

(i) [\hat{p}_x,\hat{L}_y]=i\hbar\hat{p}_z

(ii) e^{i(\hat{\sigma}\cdot\hat{n})\theta}=\cos\theta+i(\hat{\sigma}\cdot\hat{n})\sin\theta

Show that for free electron gas, the density of states in three dimensions (3D) varies as E^{1/2}, and this dependence changes to E^0 for 2D (quantum well), E^{-1/2} for 1D (quantum wire) and \delta function for 0D (quantum dot).

The wave function of a particle is given as \psi(x)=\frac{1}{\sqrt{a}}e^{-\lvert x\rvert/a}. Find the probability of locating the particle in the range -a\leq x\leq a.

Which of the following functions is/are acceptable solution(s) of the Schrodinger equation?

(i) \psi(x)=Ae^{-ikx}+Be^{ikx}

(ii) \psi(x)=Ae^{-kx}+Be^{kx}

(iii) \psi(x)=A\sin 3kx+B\cos 5kx

(iv) \psi(x)=A\sin 3kx+B\sin 5kx

(v) \psi(x)=A\tan kx Explain your answer.

A beam of particles of energy 9\,\mathrm{eV} is incident on a potential step 8\,\mathrm{eV} high from the left. What percentage of particles will reflect back?