Quantum Mechanics: Solutions to Problems in PC2130 AY2008/2009 Semester 2, Exams of Quantum Mechanics

Download Quantum Mechanics: Solutions to Problems in PC2130 AY2008/2009 Semester 2 and more Exams Quantum Mechanics in PDF only on Docsity! PC2130 QUANTUM MECHANICS AY2008/2009 Semester 2 Suggested Solution Question 1 (i) Infinitely many pure states that are different from |ψ〉. (ii) Two outcomes. Another outcomes is orthogonal to the |ψ〉 |ψ〉 = √ 1 3 |+z〉 − i √ 2 3 |−z〉 (iii) Pψ = |ψ〉〈ψ| = 23 |+z〉〈+z| − i √ 2 3 |+z〉〈−z| +i √ 2 3 |−z〉〈+z|+ 1 3 |−z〉〈−z| (iv) ρ = 2 3 |+z〉〈+z|+ 1 3 |−z〉〈−z| Expand to matrix (v) 〈A〉ρ = tr{Aρ}, 〈A〉ψ = tr{APψ} and compare. Question 2 (i) H =  2E 0 00 E 0 0 0 −E  Sx = 1√ 2  0 1 01 0 1 0 1 0  (ii) |u0〉 = 1√ 2  10 −1  |u−1〉 = 1 2  1−√2 1  |u1〉 = 1 2  1√2 1  (iii) |ψ(0)〉 = 1 2  1−√2 1  (iv) |ψ(t)〉 = 1 2  e −2iEt h̄ − √ 2e −iEt h̄ e iEt h̄  1 (v) 〈ψ(t)|Sx|ψ(t)〉 = − 1 2 ( cos ( Et h̄ ) + cos ( 2Et h̄ )) = −1 ⇒ cos ( Et h̄ ) = 1 and cos ( 2Et h̄ ) = 1 ⇒ t = 2πh̄ E Question 3 (i) For bound states to exist, we demand E < 0. (ii) Using Schrödinger equation, −h̄2 2m ∂2φ1 ∂x2 − V0φ1 = Eφ1 0 < x < a −h̄2 2m ∂2φ2 ∂x2 = Eφ2 x > a =  ∂2φ1 ∂x2 = 2m(E + V0) −h̄2 φ1 0 < x < a ∂2φ2 ∂x2 = 2mE −h̄2 φ2 x > a k1 = √ 2m(E + V0) h̄2 k2 = √ 2mE h̄2 (iii) Continuity conditions at x = 0 and x = a, φ1(0) = 0 φ1(a) = φ2(a) dφ1(x) dx ∣∣∣∣ x=a = dφ2(x) dx ∣∣∣∣ x=a (iv) −ik1 k2 cot(k1a) = A2e 2ika −B2 A2e2ika +B2 2