Problem Set 9 - Introduction to Quantum Mechanics I | PHY 4604, Assignments of Physics

Download Problem Set 9 - Introduction to Quantum Mechanics I | PHY 4604 and more Assignments Physics in PDF only on Docsity! PHY4604–Introduction to Quantum Mechanics Fall 2004 Problem Set 9 Nov. 9, 2004 Due: Nov. 17, 2004 Reading: Griffiths Chapter 4 1. Ladder Operators. A simultaneous eigenfunction of the anuglar momentum operators L2 and Lz can be labelled as ψ` m. In the standard sign convention, the raising angular momentum ladder operator satisfies the sign convention L+ψ` m = h̄(`(` + 1)−m(m + 1))1/2ψ` m+1. (1) The square root factor on the right hand side preserves the normalization. (a) Use the expression for L−L+ in notes and the known results of operating on ψ` m with L 2 and Lz to derive the square root in equation (1). (b) Use L− operatoing on (1) to find the normalizing constant c in the analo- gous equation L−ψ` m = c ψ` m−1. 2. Angular momentum states of 2 particles. Read Griffiths 4.4.3 before doing this problem. Consider a two-particle system, with angular momentum observables L(a) and L(b). The complete set of observables for this system is L(a)2, L(b)2, Lz(a), and Lz(b), (2) with eigenfunctions φ(`a,ma, `b,mb), (3) or L2, Lz, L(a) 2, and L(b)2, (4) with eigenfunctions ψ(`,m, `a, `b). (5) In this problem, L2 is the square of total angular momentum operator associated with the two particles, L = L(a) + L(b), and Lz is its z component. (a) By using the equations Lz = Lz(a)+Lz(b), and L± = L±(a)+L±(b), show that the state with `a = ma = 2 and `b = mb = 1 is an eigenfunction of the set of observables in Eq. (4) with ` = m = 3, i.e. show that φ(2, 2, 1, 1) = ψ(3, 3, 2, 1) (6) 1