Download Quantum Mechanics I: Angular Momentum Algebra and Eigenfunctions - Prof. Paul M. Goldbart and more Assignments Quantum Mechanics in PDF only on Docsity! Physics 580 Handout 13 28 October 2008 Quantum Mechanics I http://w3.physics.uiuc.edu/∼goldbart Homework 9 Prof. P. M. Goldbart 2115 & 3135 ESB University of Illinois 1) The algebra of angular momentum – optional: a) By using the definition L = R×P, and the canonical commutation relations [Ri, Pj] = ih̄δij, establish the following results: a.i) [Li, Lj] = ih̄²ijkLk; a.ii) [L · L, Li] = 0; a.iii) [R, a · L] = ih̄a×R for c-numbers a; a.iv) [Ri, Lj] = ih̄²ijkRk; a.v) [Pi, Lj] = ih̄²ijkPk; a.vi) [R ·R, Li] = 0; a.vii) [R ·RPi, Lj] = ih̄²ijkR ·RPk. b) The raising and lowering operators, L±, are defined via L± = Lx ± iLy. Use them to establish the following results: b.i) [L±, L2] = 0, where L2 ≡ L · L; b.ii) [Lz, L±] = ±h̄L±; b.iii) [L+, L−] = 2h̄Lz; b.iv) L2 = L+L− + L2z − h̄Lz. c) For the Hilbert space of functions on a sphere, discuss why the set {L2, Lz} forms a complete set of commuting observables (CSCO), i.e., argue that inclusion of Lx or Ly violates the commuting property. Would {L2, Lx} also form a CSCO? d) Show that the eigenvalues of L2 are positive or zero. Can they always be written in the form h̄2l(l + 1) with l dimensionless and greater than or equal to zero? e) Denote the set of simultaneous eigenstates of L2 and Lz by |l, m〉, where L2|l, m〉 = h̄2l(l + 1)|l, m〉; Lz|l, m〉 = h̄m|l, m〉. As yet, there is no restriction on l and m, except that l ≥ 0. We shall now derive constraints on l and m by using only the algebraic properties of the angular momentum operators, as specified by the commutation relations. e.i) By using the hermitean property, (L−)† = L+, show that 〈`,m|L+L−|l, m〉 ≥ 0. e.ii) By using part (b-iv), show that l(l + 1)−m(m− 1) ≥ 0. e.iii) Similarly, by considering 〈`,m|L−L+|l, m〉, show that l(l + 1)−m(m + 1) ≥ 0. e.iv) Hence, show that −l ≤ m ≤ l. f) Show that the ket L−|l, m〉 is f.i) an eigenket of L2 with eigenvalue h̄2l(l + 1); and 1 f.ii) an eigenket of Lz with eigenvalue h̄(m− 1). Thus, L− lowers the z-component of angular momentum by h̄. g) Show, by a suitable choice of phase, that g.i) L−|l, m〉 = h̄ √ l(l + 1)−m(m− 1)|l, m− 1〉; and g.ii) L+|l, m〉 = h̄ √ l(l + 1)−m(m + 1)|l,m + 1〉. h) Prove that the inequality −l ≤ m ≤ l will not be violated if h.i) 2l is an integer greater than or equal to zero; and h.ii) m = −l,−l + 1, . . . l − 1, l. j) State the orthonormality condition on {|l, m〉}. k) Consider a wave function ψ(θ, φ) that describes the quantum mechanics of a particle moving on the surface of a unit sphere. A suitable basis set is χlm(θ, φ) ≡ 〈θ, φ|l,m〉. Show that if ψ(θ, φ) is single-valued, then |ψ〉 = ∑ l ∑ m ψlm|l,m〉 can only include terms with integral value of m. What does this imply about allowed values of the angular momentum quantum number l? l) State the orthonormality condition for {|θ, φ〉}. 2) Unit angular momentum: a) Use the angular momentum raising and lowering operators L± to construct the two 3× 3 matrices that represent L± in the l = 1 sector and in the Lz basis. b) Use the matrices found in part (a) to construct the matrices that represent Lx and Ly in the same sector. Write down the matrix that represents Lz, and compute the matrix that represents L2. c) Show, by calculating commutators, that your three matrices representing Lx, Ly and Lz obey the angular momentum commutation relations. 3) Particle confined to a disk: Solve the energy eigenproblem (i.e., find the energy eigenvalues and eigenfunctions) for a particle confined to a disk of radius d. 4) Orbital angular momentum (after Shankar, 12.3.3): A particle moving in two dimensions is described by the wave function ψ(ρ, φ) = A e−ρ 2/2∆2 cos2 φ, where ρ and φ are plane polar coordinates and ∆ is a real parameter. Show that the probabilities of observing the z component of angular momentum and finding the results 0h̄, 2h̄ and −2h̄ are, respectively, 2/3, 1/6 and 1/6. Note that it is unnecessary to compute any radial integrals. 2
