quantum physics

Problem 4.32 (a) Find the eigenvalues and eigenspinors of Sy. (b) If you measured Sy on a particle in the general state x (Equation 4.139), what values might you get, and what is the probability of each? Check that the probabilities add up to 1. Note: a and b need not be real! (c) If you measured S, what values might you get, and with what probabilities?4.4.1 Spin 1/2 By far the most important case is s = 1/2, for this is the spin of the particles that make up ordinary matter (protons, neutrons, and electrons), as well as all quarks and all leptons. Moreover, once you understand spin 1/2. it is a simple matter to work out the formalism for any higher spin. There are just two eigenstates: (3 2 ). which we call spin up (informally. (), and 1 (-) ), spin down (4). Using these as basis vectors, the general state of a spin-1/2 particle can be represented by a two-element column matrix (or spinor): x = (") = ax+ +bx-. (4.139) with *+ = (0) (4.140) representing spin up, and x- =(") (4.141) for spin down. With respect to this basis the spin operators become 2 x 2 matrices,"which we can work out by noting their effect on x+ and x-. Equation 4.135 says S. Townsend, A Modern Approach to Quantum Mechanics, 2nd edn, University Books, Sausalito, CA. 2012.) But the price of mathematical simplicity is conceptual abstraction, and I prefer not to do it that way. 39 If it comforts you to picture the electron as a tiny spinning sphere. go ahead: I do. and I don't think it hurts, as long as you don't take it literally, 40 I'm only talking about the spin state. for the moment. If the particle is moving around, we will also need to deal with its position state (1), but for the moment let's put that aside. I hate to be fussy about notation, but perhaps I should reiterate that a ket (such as (s m)) is a vector in Hilbert space (in this case a (2s + 1)-dimensional vector space), whereas a spinor y is a set of components of a vector, With respect to a particular basis ( } }) and } - 1). in the case of spin + ). displayed as a column. Physicists sometimes write, for instance. 2 3) = x+. but technically this confuses a vector (which lives "out there" in Hilbert space) with its components (a string of numbers). Similarly. S (for example) is an operator that acts on kets; it is represented (with respect to the chosen basis) by a matrix S, (sans serif). which multiplies spinors-but again, S. = 52, though perfectly intelligible, is sloppy language