Show that the Pauli Spin Matrices obey the following relations : (i) \text{Tr} (\sigma_x) = \text{Tr} (\sigma_y) (ii) \det (\sigma_y) = \det (\sigma_z) (iii) The eigenvalues of \sigma_z and \sigma_x are the same. (iv) Write down the y-component of the spin angular momentum matrix corresponding to an antineutrino.
Show that for the one dimensional wave function \psi(x) = \begin{cases} \frac{1}{\sqrt{2a}} & , \quad |x| < a \\ 0 & , \quad |x| > a \end{cases} where a is a real constant, the rms uncertainty in momentum is infinite.
An electron is moving freely in a one-dimensional infinite potential box with walls at x = 0 and x = a. If the electron is initially in the ground state of the box and if suddenly the wall at x = a is moved x = 4a, calculate the probability of finding the particle in the ground state of the new box.
Show that the probability of transmission across the step barrier represented by the potential V(x) = \begin{cases} 0 & \text{for } x < 0 \\ V_0 & \text{for } x > 0 \end{cases} is T = \frac{4 k_1 k_2}{(k_1 + k_2)^2}, where k_1 and k_2 are wave numbers in regions x < 0 and x > 0, respectively.
The Hamiltonian of a particle moving along the x-axis is given by \hat{H} = -\alpha \frac{d^2}{dx^2} + 16\alpha \hat{x}^2, where \alpha is a real and positive constant having dimensions of energy. (i) If \psi(x) = A e^{-2x^2}, find the normalization constant A. Check whether \psi is an eigen function of \hat{H}. If yes, find the corresponding eigen value. (ii) Calculate the probability of finding the particle anywhere along the negative x-axis. (iii) Find the eigen value of \hat{H} corresponding to the eigen function \phi(x) = x\psi(x), where \psi(x) is the same as in part (i). (iv) Are the wave functions \psi(x) and \phi(x) orthogonal ?
Write (do not derive) the formula for the energy levels of a particle in a three dimensional cubical box of side L. How many electrons can occupy the level having energy 66h^2/8mL^2 ?
$ and [\hat{L}_-, \hat{L}_z]. Show that \hat{L}_+ |l, m\rangle = \sqrt{l(l+1) - m(m+1)} |l, m+1\rangle, where |l, m\rangle is the state with definite values for L^2 and L_z.
Explain normal and anomalous Zeeman effect. Obtain expression for Zeeman splitting of an alkali metal spectral line, and illustrate with an example.
An electron is in the spin state \chi = A \begin{pmatrix} 3i \\ 4 \end{pmatrix}. Determine the normalization constant A. Find the expectation value of the spin operator \hat{S}_x and also the uncertainty in the value of S_x in this state.
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}
A one-dimensional potential barrier is represented by the function V(x) = 0 \quad \text{for } x < 0 = V_0 \quad \text{for } x > 0 Where V_0 is positive. Find the transmission coefficient for particles of mass m incident from the left on the barrier.
(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)
(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)
In the free electron theory of metals, a conductor is regarded as consisting of free electrons in a three-dimensional box. Using the results of (a), obtain an expression for the density of states,
Solve the Schrodinger equation for a linear harmonic oscillator. Obtain the eigenvalues and the corresponding eigenfunctions.
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.
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.
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
(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