Download Practice Assignment 6 - Introduction to Quantum Mechanics I | PHY 4604 and more Assignments Physics in PDF only on Docsity! PHY4604 Fall 2007 Problem Set 6 Department of Physics Page 1 of 2 PHY 4604 Problem Set #6 Due Wednesday November 14, 2007 (in class) (Total Points = 130, Late homework = 50%) Reading: Griffiths Chapter 4 Sections 4.4. Problem 1 (20 points): The Pauli spin matrices are given by โโ โ โ โโ โ โ = 01 10 xฯ โโ โ โ โโ โ โ โ = 0 0 i i yฯ โโ โ โ โโ โ โ โ = 10 01 zฯ (a) (10 points) Show that ฯโi = ฯi , det(ฯi) = -1, Tr(ฯi) = 0, [ฯi,ฯj] = 2iฮตijkฯk, and {ฯi,ฯj} = 2ฮดij. Note that [A, B] = AB โ BA and {A, B} = AB +BA. (b) (10 points) Show that โ+= l lijlijji i ฯฮตฮดฯฯ . (Hint: see Griffiths section 4.3 and problem 4.26) Problem 2 (20 points): An electron is in the spin state โโ โ โ โโ โ โ >= 4 3 | i Aฯ . (a) (4 points) Determine the normalization constant A. (b) (8 points) Find the expectation value of Sx, Sy, and Sz for the state |ฯ>, where ฯ rr h 2=S . (c) (8 points) Find the โuncertaintiesโ ฮSx, ฮSy, and ฮSz for this state. (Hint: see Griffiths section 4.3 and problem 4.27) Problem 3 (25 points): The most generalized normalized spinor |ฯ> can be written >+>=โโ โ โ โโ โ โ >= โ+ ฯฯฯ ||| bab a where โโ โ โ โโ โ โ >=+ 0 1 | ฯ and โโ โ โ โโ โ โ >=โ 1 0 | ฯ with |a|2 +|b|2 = 1. (a) (5 points) Find the expectation value of Sx, Sy, and Sz for the state |ฯ>, where ฯ rr h 2=S . (b) (10 points) Find the expectation values of Sx2, Sy2, and Sz2, for this state and check that <Sx2> + <Sy2> + <Sz2> = <S2>. (c) (10 points) Find the eigenvalues and the eigenspinors of Sy. If you measure Sy on a particle in the general state |ฯ>, what values might you get, and what is the probability of each? Check that the probabilities add up to one. (Hint: see Griffiths problem 4.28 and 4.29) Problem 4 (20 points): Construct the spin matrices Sx, Sy, and Sz for a particle of spin 1 and verify that โ โ โ โ โ โ โ โ โ โ =++= 100 010 001 2 22222 hzyx SSSS . (Hint: see Griffiths problem 4.31)
