UPSC Physics Optional PYQs (CSE & IFoS)PYQs

Prove the following : (i) [L^2, L_x] = 0 (ii) [L_x, L_y] = i \hbar L_3

Derive energies and wave functions for motion of electron in a hydrogen atom.

(i) What do you mean by expectation value of a physical quantity ? How does it help to extract information from a wave function ? (ii) A particle limited to move along x-axis has the wave function \psi = ax between x = 0 and x = 1, \psi = 0 elsewhere. Find the probability that the particle can be found between x = 0\cdot 45 and x = 0\cdot 55. Find the expectation value <x> of the particles position x = 0 to x = 1.

(i) Discuss about the validity criterion for the WKB approximation. (ii) Qualitatively explain alpha (\alpha) particle emission by WKB approximation.

Draw a neat sketch showing potential wells and energy levels of the following : (i) Harmonic oscillator (ii) Hydrogen atom (iii) Particle in a box List the contrasting points and mention the variation of $E_n$' with n' of the above plots.

(i) Draw a neat sketch of a finite rectangular potential barrier well of height U' and width L' that contains a particle whose energy E' is less than U'. Solve the Schrödinger equations of the particles outside the well. (ii) Plot wave functions and probability densities |\psi|^2 of the particle for the first three states. Compare the results with that of a particle in a box of infinite potential (U = \infty).

In the |jm\rangle basis formed by the eigenkets of J^2 and J_z, show that \langle jm| J_- J_+ |jm\rangle = (j-m)(j+m+1)\hbar^2 where J_+ = J_x + i J_y and J_- = J_x - i J_y.

Show that the phase velocity v_p for a particle with rest mass m_0 is always greater than the velocity of light and that v_p is a function of wavelength.

An electron is confined in the ground state of a one-dimensional harmonic oscillator such that \Delta x = 10^{-10}~\text{m}. Assuming \langle T \rangle = \langle V \rangle, find the energy in eV required to excite it to the first excited state.

If the Hamiltonian of a system be H = (p_x^2 / 2m) + V(x), obtain the value of the commutator [x, H]. Hence find the uncertainty product (\Delta x)(\Delta H).

Consider a particle whose wave function is given by \psi(x) = A e^{-\alpha x^2}.

(i) What is the value of A if this wave function is normalized?

(ii) Calculate the expectation value of x for this particle.

For Pauli's matrices, prove that (i) [\sigma_x, \sigma_y] = 2i\sigma_z and (ii) \sigma_x \sigma_y \sigma_z = i.

An electron and a photon each has a wavelength of 2~\text{\AA}. Calculate (i) their momenta and (ii) the ratio of their kinetic energies.

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.

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

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.

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).

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?

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.

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