PHYS 5382 Introduction to Quantum Mechanics Practice Test 1, Exercises of Quantum Mechanics

Download PHYS 5382 Introduction to Quantum Mechanics Practice Test 1 and more Exercises Quantum Mechanics in PDF only on Docsity! PHYS 5382 Introduction to Quantum Mechanics Practice Test 1 Time allowed 1 hour 15 min. Resources allowed: Griffiths textbook, calculator, formulas from the covers of Griffiths hardback 1. Given the spatial wavefunction ψ(x) = Ax exp(-kx) (0 < x < ∞, k > 0), ψ(x) = 0 (0 > x), (a) what value must the constant A take in terms of k in order that ψ is normalized? [5 points] (b) If k = 0.5 x 1010 m-1, what is the probability of finding the particle described by ψ between x = 0 m and x = 2.0 x 10-10 m? [5 points] 2. Consider a stationary state having a wave function periodic in time, namely, that there exists a time T for which ψ(x,t) = ψ(x,t + T). Find the allowed values of the energy in terms of T. [5 points] 3. Consider a particle of mass m in the one-dimensional infinite square well potential V(x) = +∞ {x < − L and x > L} V(x) = 0 {− L < x < L} (a) Write down all the normalized stationary state wavefunctions ψ(x) in terms of L and also the corresponding energies (you may either rewrite those already shown in Griffiths, or derive them directly by solving the Schrodinger equation). [5 points] (b) For the odd wavefunctions, ψ(x) = - ψ(-x) , calculate the spatial uncertainty Δx = σx and the momentum uncertainty Δp = σp and then verify that the uncertainty principle is satisfied. [10 points]