Quantum Mechanics: and its applications

EPL202 Date: 15.02.10 Problem Set 4

  1. Show that in co-ordinate the representation of the angular momentum operators are the following

  1. (a)Show that the eigenstates of the operator are given by spherical harmonics .

(b) Show that the following equation is satisfied by

© Show that the spherical harmonics are also eigenstates of the parity operator.

  1. The wavefunction of a particle subjected to a spherically symmetric potential is given by

(a)Is an eigenfunction of ? If so what is its corresponding eigenvalue. If not what are the possible values we shall obtain when we shall measure .

(b)What are the probabilities of finding out the particle in various states?

  1. (a) A particle is in a spherically symmetric potential is known to be an eigenstate of with eigenvalues and . Prove that the expectation values between states satisfy
  1. Consider a system made up of two spin particles. Observer specializes in measuring the spin components of one of the particles ()while the observer B measures the spin component of the other particle. Suppose the system is known to be in the spin-singlet state, that is . (a) What is the probability of for observer A to obtain when the observer B makes no measurement. ?Same problem for .

(b) Observer B determines the spin of particle 2 to be in the state with certainty. What can be then said about the observer A’s measurement if (i) A measures and (ii) if A measures ? Justify your answer.

  1. A beam of spin atoms goes through a series of Stern Gerlach type of measurements as follows

(a)The first measurement accepts atoms and rejects type of atoms.

(b)The second measurement accepts atoms and rejects type of atoms, where is the eigenvalue of the operator with is an unit vector making an angle in the x-z plane with respect to the axis.

(c)The third measurement accepts atoms and rejects type of atoms

What is the intensity of the final beam when the beam surviving the first measurement is normalized to unity? How must we orient the second measuring apparatus if we are to maximize the intensity of the final beam ?

  1. Find out the bound state solutions for the particle in the following spherically symmetric square well potential

8 A system of two angular momentum of respective magnitude , is described by the basis. The system is in a state , where J is the total angular momentum and M is the -component of J. Consider in particular , the states (a) . For each state calculate the probability of measuring each pair of possible values find the expectation values of (c) Calculate the expectation values of in the state