Solutions to Quantum Mechanics Homework Set 1 for PHY 389K, Assignments of Health sciences

Download Solutions to Quantum Mechanics Homework Set 1 for PHY 389K and more Assignments Health sciences in PDF only on Docsity! PHY 389K Quantum Mechanics, Homework Set 1 Solutions Matthias Ihl 02/01/2008 Note: I will post updated versions of the homework solutions on my home- page: http://zippy.ph.utexas.edu/~msihl/teaching.html 1 Problem 1 We know that A|ai〉 = ai|ai〉. Therefore, f(A)|ai〉 = (A− a1)...(A− aN )(A− ai)|ai〉 = 0, (1) since one can commute the corresponding factor to the right. For an arbitrary state |ψ〉 = ∑N j=1 cj|aj〉, it follows that f(A)|ψ〉 = ∑ j cjf(A)|aj〉 = 0. (2) Since this is true for all |ψ〉, f(A) = 0 on the space spanned by {|ai〉}i. 2 Problem 2 (a) Hermiticity: (A†A)† = A†(A†)† = A†A. (3) (b) trace of A†A: Let U be a unitary transformation such that |bi〉 = Uij|aj〉, ∀i, j = 1, . . . , N. (4) 1 which rotates an arbitrary basis {|bi〉}i to the eigenbasis {|ai〉}i of A. Then, trA†A = trU †A†AU = ∑ i 〈ai|U †A†AU |ai〉 = ∑ i,j 〈ai|U †A†|bj〉〈bj|AU |ai〉 = ∑ i,j ∣ ∣ ∣ 〈bj |AU |ai〉 ∣ ∣ ∣ 2 = ∑ i,j ∣ ∣ ∣ 〈bj |A|bj〉 ∣ ∣ ∣ 2 . (c) Similarly, for 〈A†A〉, we get 〈ψ|A†A|ψ〉 = ∑ i,j 〈ai|c ∗ iA †Acj|aj〉 = ∑ i |ci| 2|ai| 2 ≥ 0. 3 Problem 3 We consider the operator ~S · ~n (see Sakurai, page 62). The fact that the unit vector ~n lies in the yz-plane means α = ±π. Moreover, β = γ in this problem. We will define |γ,+〉 := |~S · ~n,+;α = π, β = γ〉 = cos γ 2 |+〉 + i sin γ 2 |−〉. (5) The probability that anmSz = − ~ 2 atom is accepted by the second experiment is given by: P2 = |〈γ,+|−〉| 2 = sin2 γ 2 . (6) Likewise, for the third experiment, we get P3 = |〈−|γ,+〉| 2 = sin2 γ 2 . (7) This is maximized for γ = π.