Homework 1 Questions - Quantum Mechanics I - Fall 2007 | PHYS 580, Assignments of Quantum Mechanics

Download Homework 1 Questions - Quantum Mechanics I - Fall 2007 | PHYS 580 and more Assignments Quantum Mechanics in PDF only on Docsity! PHYSICS 580 – FALL 2007 PROBLEM SET 1 – DUE: Thursday, Sept. 6 1. Consider a two-slit diffraction experiment, with a slit separation b and the screen a distance L from the slits. See Fig. 1. Assume that a plane wave eikz impinges on the slits from the left. a) Show that the angular width δθ of the central peak on the screen is ∼ λ/b, where λ is the wavelength. b) Suppose that the experiment is done with electrons impinging on the screen in a state ∼ eikz on the left. i) Before the electrons reach the slits what is their component of momentum in the x-direction? ii) What is the uncertainty of their x position before they reach the slits? iii) For the electrons passing through the slits what is the uncertainty ∆x in their x coordinate just after they pass through the slits? iv) Using the results of a) show that ∆px∆x>∼h̄, where ∆px is the uncertainty in their x component of momentum just after they pass through the slits. L b dq x z FIG. 1: 2. A beam of monochromatic light of wavelength λ falls on electrons initially at rest. Calculate the wavelength of scattered radiation as a function of the angle θ through which it is scattered. (Treat the electrons relativistically.) 3. A beam of radiation of wavelength 200Å falls on a gas of hydrogen atoms which are initially at rest and in their ground states. Calculate all the frequencies of scattered radiation. The energy levels of the hydrogen atom are given by En = −me4/2h̄2n2, where n = 1, 2, 3, . . . and m and e are the mass and charge of the electron. Discuss the dependence of the frequencies of scattered radiation on the angle of scattering. LQM 1-1. A man sends a beam of red light along the z axis through a polaroid filter that passes only x polarized light. The beam is initially polarized at 30o to the x axis, and the total energy content of the beam is quite accurately 10 joules. Estimate the fluctuations in the energy of the beam, i.e., the range of likely energy values, after it passes through the polaroid. How do the fluctuations depend on h̄? LQM 1-2. a) Write down a basis corresponding to 45o, 135o polarizations. b) Write down a basis that is neither plane nor circularly polarized. LQM 1-3. a) Calculate the transformation matrix from the x, y basis to the R, L basis. b) Calculate the transformation matrix from the R, L basis to the basis devised in Problem LQM 1-2(b) above. c) Calculate the transformation matrix from the x,y basis to the basis in Problem 1-2(b), and show that it is the product of the matrix calculated in Problem 1-3(b) and the matrix calculated in Problem 3(a). In which order must you multiply these matrices? d) Show in general that the product of i) the transformation matrix from a basis, 1, to a basis 2, with ii) the transformation matrix from basis 2 to a third basis, 3, is the transformation matrix from basis 1 to basis 3. Hint: make extensive use of the completeness relation, e.g., (LQM Eqs. 1.69, 1.70), for a basis. LQM 1-4 a) Show that 〈Φ|MΨ〉∗; b) Show that if M |Ψ〉 = λ|Ψ〉 then 〈Ψ|M † = λ∗〈Ψ|. LQM 1-5 Show that the transformation matrix from one basis to another is unitary. LQM 1-6.. a)Show that the matrix |Φ〉〈Φ| is Hermitian. b) Show that the photon spin operator S is Hermitian. Generally all physical quantities are represented by Hermitian matrices. LQM 1-7. Let |x′(θ)〉, |y′(θ)〉 denote the basis tilted at an angle θ to the x, y basis. Show that the components of a vector |Ψ〉 in this basis 〈x′(θ)|Ψ〉, 〈y′|(θ)|Ψ〉 obey the differential equations −i ∂ ∂θ 〈x′(θ)|Ψ〉 = 〈x′(θ)|S|Ψ〉 −i ∂ ∂θ 〈y(θ)|Ψ〉 = 〈y′(θ)|S|Ψ〉. Solve these equations explicitly for |Ψ〉 = R〉 and |Ψ〉 = L〉. LQM 1-8. The probability that a photon in state |Ψ〉 passes through an x-polaroid is the average value of a physical observable which might be called the “x-polarizedness.” Write down the operator, Px, corresponding to this observable. Show that it is Hermitian. What are its eigenvalues and eigenstates? Write down its representation in terms of its eigenvalues and eigenstates [as in (LQM Eq. 1-71)]. Verify that the probability that a photon in state |Ψ〉 passes through the x-polaroid is 〈Ψ|Px|Ψ〉. LQM 1-9. Photons polarized at 30o to the x axis are sent through a y-polaroid. An attempt is made to determine how frequently the photons that pass through the polaroid, pass through “as right circularly polarized photons,” and how frequently they pass through “as left circularly polarized photons”; this attempt is made as follows: First, a prism that passes only the right circular polarized light is placed between the source of the 30o polarized photons and the y-polaroid, and it is determined how frequently the 30o photons pass through the y-polaroid. Then this experiment is repeated with a prism that passes only left circular polarized light instead of the one that passes only right. Show by explicit calculation that the sum of the probabilities for passing through the y-polaroid measured in these two experiments