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Quantum Mechanics: and its applications EPL202 Date: 15.02.10 Problem Set 4 1. Show that in r , , co-ordinate the representation of the angular momentum operators are the following L1 L x i (sin cos cot ) L2 L y i ( cos cos cot ) L3 L z i 1 1 2 L (i ) (sin ) sin 2 2 sin 2. (a)Show that the eigenstates of the operator L2 are given by spherical (2l 1)(l m)! m Pl (cos )e im . harmonics Yl m ( , ) (1) m 4 (l m)! 2 2 (b) Show that the following equation is satisfied by Pl m (x) (1 x 2 ) d 2 Pl m dPl m m2 2 x ( l ( l 1 ) ) Pl m 0 dx dx 2 (1 x 2 ) © Show that the spherical harmonics are also eigenstates of the parity operator. 3. The wavefunction of a particle subjected to a spherically symmetric potential V (r ) is given by (r ) ( x y 3z ) f (r ) (a) Is an eigenfunction of L2 ? If so what is its corresponding eigenvalue. If not what are the possible values we shall obtain when we shall measure L2 . (b) What are the probabilities of finding out the particle in various m states? 4. (a) A particle is in a spherically symmetric potential is known to be an eigenstate of L2 & Lz with eigenvalues l (l 1) 2 and m . Prove that the expectation values between states | l, m satisfy l (l 1) 2 m 2 2 2 2 (a) Lx L y 0, Lx L y 2 1 5. Consider a system made up of two spin particles. Observer 2 A specializes in measuring the spin components of one of the particles ( s1x , s1z etc. )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 stotal 0 . (a) What is the probability of for observer A to obtain s1 z when the observer B makes no measurement. ?Same 2 problem for s1 x . 2 (b) Observer B determines the spin of particle 2 to be in the state s 2 z with certainty. What can be then said about the observer A’s 2 measurement if (i) A measures s1z and (ii) if A measures s1x ? Justify your answer. 1 atoms goes through a series of Stern Gerlach type 2 of measurements as follows (a) The first measurement accepts s z atoms and rejects 2 s z type of atoms. 2 (b) The second measurement accepts s n atoms and rejects 2 s n type of atoms, where s n is the eigenvalue of the 2 operator S nˆ with n̂ is an unit vector making an angle in 6. A beam of spin the x-z plane with respect to the z axis. (c) The third measurement accepts s z atoms and rejects 2 s z type of atoms 2 What is the intensity of the final s z beam when the s z beam 2 2 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 s z beam ? 2 7. Find out the bound state solutions for the particle in the following spherically symmetric square well potential 0, r a V0 ,r a V 8 A system of two angular momentum of respective magnitude 1 1 j1 1 & j 2 , is described by the basis | j1 1, m1 | j 2 , m2 . The system 2 2 is in a state | JM , where J is the total angular momentum and M is the z component of J. Consider in particular , the states (a) 3 3 1 | J , M & (b) | J 12 , M . For each state calculate the probability of 2 2 2 measuring each pair of possible values (m1 , m2 ) & find the expectation values of J 1z & J 2 z (c) Calculate the expectation values of J y in the state |J 1 1 ,M 2 2