Download Practice Test Solutions for Introduction to Quantum Mechanics - Fall 2004 and more Exams Physics in PDF only on Docsity! PHY4604–Introduction to Quantum Mechanics Fall 2004 Practice Test 1 solutions October 6, 2004 These problems are similar but not identical to the actual test. One or two parts will actually show up. 1. Short answer. • Explain the photoelectric effect Shine light on metal surface, only photons of sufficiently high energy (fre- quency) able to kick an electron out. Increasing intensity of low-energy light doesn’t help. A single electron can make a transition out of the solid only if a single photon carries sufficient energy h̄ω. • Explain the significance of h̄ in quantum mechanics, and give an example of a place where it shows up. Planck’s const. is a quantum (smallest possible unit) of angular momen- tum. It shows up, for example, in the orbits of the Bohr atom, where an electron may not have any value of angular momentum, but only an integer number of nh̄ • Discuss the uncertainty principle briefly If you make a measurement of an object’s position and momentum si- multaneously, you cannot measure them both arbitrarily precisely. The product of the uncertainty in knowledge of the object’s position ∆x and in momentum ∆p fulfills the Heisenberg relation ∆x∆p >∼ h̄ (1) • Explain the difference between the 2 versions of Schrödinger’s equation ih̄ ∂ψ ∂t = − h̄ 2 2m ∂2ψ ∂x2 + V ψ and − h̄ 2 2m ∂2ψ ∂x2 + V ψ = Eψ The first is the general version, the second applies for ”stationary states”, i.e. if ψ is an eigenfunction of the Hamiltonian. These special states change in time in a trivial way, ψn(x, t) = ψ(x) exp(−iEnt/h̄). 1 • What are the units of P (x, t), the probability density in 1 dimension? Justify your answer. Must be 1/volume, since ∫ dxP (x, t) = 1. • Calculate the commutator [px, x2] [px, x 2]ψ = (−ih̄∇)(x2ψ)− x2(−ih̄∇)ψ = −ih̄2xψ − ix̄2ψ′ + ih̄x2ψ′ = −ih̄2xψ. So [px, x 2] = −ih̄2x. • Calculate the expression for the Bohr levels of the hydrogen atom from the Bohr-Ehrenfest quantization condition. See notes. 2. Consider a wave packet defined by ψ(x) = ∫ ∞ −∞ dkf(k)ei(kx−ωt) (2) with ω = h̄k2/2m and f(k) given by f(k) = 0 k < −∆k/2 a −∆k/2 < k < ∆k/2 0 ∆k/2 < k (3) (a) Find the form of ψ(x) at t = 0. ψ(x) = ∫ dkf(k)eikx = a ∫ a/2 −a/2 dkeikx = = a ix (eix∆k/2 − e−ix∆k/2) = a ix 2i sin ∆kx 2 . (b) Find the value of a for which ψ(x) is properly normalized. Need ∫ dx|ψ|2 = 1, so 1 = 4a2 ∫ dx sin2 ∆kx 2 x2 = 4a2π∆k/2 2
