PYQ Bank

Determine the discrete energy levels and the corresponding eigenfunctions for a particle in an infinitely deep potential well inside a cube of dimension L, i.e., assume \begin{gathered} V(x,y,z) = 0 \quad \text{for } 0 < x < L;\\ 0 < y < L;\\ 0 < z < L = \infty \quad \text{elsewhere} \end{gathered}

(i) Define angular momentum, Express it in operator form and show that \vec{L} \times \vec{L} = i\hbar \vec{L} Explain the physical significance of this relation. (10) (ii) Let Y_{lm} be an eigenstate of L^2 and L_z with eigenvalues l(l + 1)\hbar^2 and m\hbar, respectively. Show that \phi = (L_x + i L_y) Y_{lm} is likewise an eigenstate of L^2 and L_z, and determine the eigenvalues. (10)

(i) State and explain Heisenberg's uncertainty principle. Experimental data reveal that no electron in an atom has energy greater than 4 MeV. Assuming that the radius of a nucleus is 10^{-14}\text{ m}, show using Heisenberg's uncertainty principle that an electron cannot exist inside the nucleus. (10) (ii) Calculate the de Broglie wavelength of thermal neutrons at 300 K. (10)

Distinguish between potential well and potential barrier. Given their illustrations. Considering one-dimensional potential step, prove that sum of the reflection and transmission coefficients is unity.

(i) Using Pauli matrices \sigma_x, \sigma_y and \sigma_z, show that \left(\vec{\sigma} \cdot \vec{r}\right) \left(\vec{\sigma} \cdot \vec{p}\right) = \vec{r} \cdot \vec{p} + i \vec{\sigma} \cdot \vec{L} \hfill 10 (ii) For the radiation of wavelength 6000~\text{\AA}, determine the wavelength separation between its two component lines which are observed in the normal Zeeman effect. The magnetic field used is \frac{\pi}{4}~\text{weber}/\text{m}^2 and the specific charge of the electron = 1.76 \times 10^{11}~\text{C}~\text{kg}^{-1}. \hfill 10

Explain how the problem of the hydrogen atom could be solved using Schrodinger equation. Also derive an expression for its energy eigenvalue and discuss the associated bound states of this case.

Prove that \frac{d}{dt}(x) = \frac{1}{m}(p_x) Define all the terms of this relation and give its physical interpretation.

(i) Calculate the de Broglie wavelength of a thermal neutron at 27~^\circ\text{C} temperature. \hfill 10 (ii) Considering one-dimensional case for free particle, show that the plane wave function is the eigenfunction of a linear momentum operator and kinetic energy operator. \hfill 10

Consider a particle of mass m in an infinite one dimensional potential well of width a. The particle is found in the state given by \psi (x) = c \left[ \sin \frac{\pi x}{a} + \frac{1}{2} \sin \frac{2 \pi x}{a} \right] (i) Calculate c. (ii) If a measurement of energy is made, what are the possible results and what are the probabilities for each one of them ?

Discuss WKB approximation and apply the same to determine the transition probability for leakage through a potential barrier.

(i) The ground state wavefunction of a linear simple harmonic oscillator is \psi = A \exp \left( - \frac{\alpha^2 x^2}{2} \right) Calculate the constant A and the average values of x^2 and x. Given that \int_0^\infty e^{-x^2} = \frac{\pi^{1/2}}{2} (ii) How can the pure rotation spectrum of \text{H}_2 molecule be observed ? If the bond length of \text{H}_2 molecule is 0 \cdot 07417 \text{ nm}, what would be the spacing of lines in its spectrum ?

(i) Write the commutation relations for the position variable x and the momentum components p_x, p_y and p_z. Explain the physical significance of these relations. (ii) Calculate the de Broglie wavelength of an electron moving with a kinetic energy of 1 MeV.

Set up the time-independent Schrodinger equation for an electron moving in Coulomb field, V(r) = \frac{Ze^2}{4 \pi \varepsilon_0 r}, in polar coordinates. Solve the radial equation to get the energy eigen values.

A hydrogen atom is in the following state : \Psi_{nlm} (r, 0) = (\sqrt{1/14}) [2\psi_{100}(r) - 3\psi_{200}(r) + \psi_{322}(r)] (i) What is the probability of finding the system in the state (200) ? (ii) What are \langle H \rangle and \langle L_z \rangle ?

Set up the time-independent Schrodinger equation for an electron moving in Coulomb field, V(r) = \frac{Ze^2}{4\pi\varepsilon_0 r} in polar coordinates. Solve the radial equation to get the energy eigen values.

A particle of mass m, with energy E such that -V_0 < E < 0, is trapped in a potential wall as shown below :

Write time independent Schrodinger equation in regions (i) 0 < x < a and (ii) x > a.

(i) Show that the radial probability density of the ground state of the hydrogen atom has a maximum at r = a. The ground state wave function of the hydrogen atom is given by \psi (r) = \frac{1}{\sqrt{\pi a^{3/2}}} e^{-r/a} where a is the Bohr radius. (ii) Calculate the Larmor frequency of a spin 1/2 particle in a magnetic field B.

= 0$. What is the significance of this commutation relation ? (ii) Show that the Pauli Matrices anti-commute.

(i) Explain what do you understand by Heisenberg uncertainty principle. Using this principle, determine the energy of the ground state of a one dimensional simple harmonic oscillator. (ii) An electron having an energy 2 eV is travelling in the region where V(x) varies as shown below:

Calculate the de Broglie wavelength of the electron in regions I, II and III.

Number at non-interacting electrons are confined in a cube of volume L^3. Obtain an expression for the Fermi energy.