Quantum Physics Problem Set 2: Understanding Quantum Mechanics & Uncertainty - Prof. Paulo, Assignments of Quantum Physics

Download Quantum Physics Problem Set 2: Understanding Quantum Mechanics & Uncertainty - Prof. Paulo and more Assignments Quantum Physics in PDF only on Docsity! QUANTUM PHYSICS I PROBLEM SET 2 due September 29 A. (Griffths 1.18) When is quantum mechanics necessary ? In general, quantum mechanics is relevant when the de Broglie wavelength of the particle in question (h/p) is comparable to the characteristic size of the system (d). In thermal equilibrium a temperature T , the average kinetic energy of a particle is p2 2m = 3 2 kT, (1) where k is the Boltzmann constant, so the typical wavelength is λ = h√ 3mkT . (2) The purpose of this problem is to anticipate which systems will have to be treated quantum mechanically, and which can safely be described classically. a) Solids: The lattice spacing in a typical solid is around d ≈ 0.3nm (not very different from the size of an atom). Find the temperature below which the free electrons in a solid are quantum mechanical. Below which temperature are the nuclei in a solid quantum mechanical (use sodium as a typical example) ? b) Gases: For what temperatures are the atoms in an ideal gas at pressure P quantum mechanical ? Hint: use the ideal gas law PV = NkT to deduce the interatomic spacing (the answer is T < (1/k)(h2/3m)3/5P 2/5). Obviously we want m to be as small as possible and P as large as possible for the gas to show quantum behavior. Put in numbers for helium at atmospheric pressure. Is hydrogen in outer space (interatomic distance ≈ 1cm and temperature ≈ 3K) quantum mechanical ? B. (Griffiths, 2.4) Uncertainty on the square well states Calculate 〈x〉, 〈x2〉, 〈p〉 and 〈p2〉 for the nth stationary state of the infinite square well. Check that the uncertainty principle is satisfied. Which state comes closer to the uncertainty limit ? C. (Griffiths, 2.5 and 2.6, sort of ...) When is the wave function phase relevant ? A particle on an infinite square well has an inital wave function that is an equal superposition of the two first states: Ψ(x, 0) = A(ψ1(x) + ψ2(x)). (3) i) Normalize Ψ(x, 0). ii) Find Ψ(x, t) and |Ψ(x, t)|2. To simplify the result, let ω = π2~/(2ma2). iii) Compute 〈x〉 and notice it is oscillatory. What is the amplitude and angular frequency of this oscillation ? Although the overall phase of the wave function is of no physical significance (it cancels out whenever you calculate a measurable quantity), the relative phase of the coefficients in eq. (3) does matter. For example, suppose we take Ψ(x, 0) = A(ψ1(x) + eiψψ2(x)) (4) instead of eq. (3). Find iv) Ψ(x, t), v) |Ψ(x, t)|2 and vi) 〈x〉 and compare with the φ = 0 case.