Download Quantum Mechanics Lecture: Spin and Rotations - PHY662, Spring 2004 and more Assignments Quantum Mechanics in PDF only on Docsity! PHY662, Spring 2004 Outline for Tues. Jan. 13, 2004 13th January 2004 Introduction • Course syllabus - review in detail. Pay special attention to homework procedures and practice with collaboration and research. • Summarize calendar. • Reading assignment for Thursday: Feynman, Leighton, and Sands, Ch. 5 (skim latter part of Ch. 5), Secs. 6-1 through 6-3. Shankar, pp. 373-385. Note that they use very dierent approaches. (Feynman is also building the ideas of representations and transformations, while Shankar already has that technology available for discussion.) • It has been a long time since I have had to use a lot of quantum mechanics. So I will be grateful for corrections and input. • Physics is based on observation - must not forget this. • Mathematics is a surprisingly successful method for organizing our ob- servations. The tools developed to study QM are of great use in many dierent areas. Concepts guide mathematics and mathematics suggests concepts. • Thought experiments can be of high importance: explore possible results, draw conclusions about the logical structure of a theory. • Quantum mechanics - introductory remarks. Is, like all physical theories, incorrect. But it is amazing that a linear theory (in the sense of superposition of states) works so well and is the basis for so much else (quantum eld theory, etc.) Spin is a realm which has no (single-particle) classical limit, though there are parallels. 1 One can deduce a lot from symmetries, continuity, and lin- earity. We will be using this for spin, especially. Spin-1/2 is most important case (photon close second). Very general, also, e.g., two-level systems. One measures probabilities, not amplitudes. But (relative) ampli- tudes can be deduced from probabilities. That is amazing, but also good - otherwise why would we need amplitudes to describe experi- ments? Starting Spin Conceptual background for describing spin, experiments indicating its existenc, and start of building a consistent mathematical description. 1. Rotations: transformations on our space and the objects in it. (a) One type of spatial transformation: translations. Do they commute? (b) What is P? (reminder) Does P commute with translations? (c) What is SO(3)? Do rotations commute? (d) What is the global structure of SO(3)? What is its dimensionality? [What is the dimensionality of SO(4)?] Rotation by 2π about a given axis does what to a vector? (e) Experiment in physical space on a sequence of rotations. Map a sequence of rotations R(t) to a twisted object. Gives information on connectedness of the set of rotations. (f) Also use example of 4π rotation as mentioned in Wald's text on general relativity: i. Rotate by 4π about the z-axis: this is a sequence R(t), with R(t) = R(4πtẑ). R(0) = R(1/2) = R(1) = I. ii. Hold rst half of sequence xed, modify second half while main- taining R(1/2) = R(1) = I, by slowly changing the rotation axis from ẑ to −ẑ. iii. Cancel out rst half with second half, to get identity sequence. (g) Continuity gives constraints on the physical eect of a sequence of rotations that is related to the trivial sequence. What about the phase, for example? Spinors will provide a representation of SO(3) where a 2π rotation will not be equal to the identity in phase. 2. Representations of the rotation group. (a) Mathematical description of the eect of rotations. (b) Orbital angular momentum ~L = ~R × ~P . This is derived from the operations on ψ(x, y, z) that generate rotations. 2
