Practice Test 3 for PHYS 5382: Introduction to Quantum Mechanics, Exams of Quantum Mechanics

Download Practice Test 3 for PHYS 5382: Introduction to Quantum Mechanics and more Exams Quantum Mechanics in PDF only on Docsity! PHYS 5382 Introduction to Quantum Mechanics Practice Test 3 Time allowed 1 hour 15 min. Resources: Griffiths textbook, calculator, formulas from the covers of Griffiths hardback 1. Consider a particle constrained to move in a circle, whose position is described by the azimuthal angle φ in the interval [0, 2π]. Assume that wavefunctions are periodic ψ(φ) = ψ(φ + 2π), continuous, and have continuous first derivative dψ/dφ. (a) Determine whether the following operator Q is Hermitian on this interval d2 Q = dφ2 [3 points] (b) Find the normalized eigenfunctions and eigenvalues of Q and describe their degeneracy. [7 points] (c) Show that eigenfunctions with distinct eigenvalues are orthogonal to each other. [5 points] 2. A normalized wavefunction at time t = 0 is non-zero in a finite interval of the x-axis (2λ)-½ exp ( 2πi x/λ ) -λ < x < λ Ψ(x,0) = 0 x < -λ , x > λ where λ is a real parameter. (a) Determine the momentum space wavefunction Φ(p,0) at t = 0 in terms of sin(pλ/h) [7 points] (b) Sketch |Φ|2 and describe what happens to the width and height of features as λ → ∞. [5 points] (c) Which of the potentials studied in Griffiths Chapter 2 does this wavefunction belong to in the limit λ → ∞ and what happens to Δp and Δx? [3 points]