Special Relativity and Quantum Mechanics: The Dirac Equation and Electromagnetic Waves - P, Study notes of Physics

Download Special Relativity and Quantum Mechanics: The Dirac Equation and Electromagnetic Waves - P and more Study notes Physics in PDF only on Docsity! SR 8 1 Special relativity, VIII The electromagnetic wave equation revisited Recall that electromagnetic waves obey ! 2 " !t 2 = c 2 ! 2 " !x 2 , where Ψ could be either E or B. Here the fields are assumed to be linearly polarized perpendicular to the direction of propagation, x. A harmonic wave traveling in the +x-direction can be expressed as ! = ! 0 sin(kx "#t) , ! = ! 0 cos(kx "#t) , or ! = ! 0 exp[i(kx "#t)] , or some combination of such expressions. In the latter form, the coefficient Ψ0 is complex. After differentiating Ψ twice in t and twice in x and substituting into the wave equation we find that in order for Ψ to be a solution to the electromagnetic wave equation c =! k . This simple relation can also be written as c = !! !k = E /P . Thus, we can think of c as being associated with the wave properties of the fields or with the energy and momentum of the quanta—i.e., the photons—of the fields. Combining the two, we can write, for example, ! = ! 0 sin( P ! x " E ! t) . In quantum terms, the wave function Ψ can be viewed as providing information both about the likelihood of detecting a photon at (x, t)—the probability (-density) is ! " 2 (which for real fields is just Ψ2)—and about the momentum and energy (P, E) it would have if detected. If someone gave you Ψ, you extract E and P from it by differentiating: ! 2 " !t 2 = # E 2 ! 2 ", ! 2 " !x 2 = # P 2 ! 2 " . Quantum field theory is predicated on the assumption that it is possible to find a wave equation for different kinds of particles whose solutions (a) determine the probability of the particles being detected at some point in space at some time, and (b) contain information about the allowed energies and momenta carried by the particles. Such wave equations involve partial derivatives, which, in this view, pull out dynamical information when they operate on the wave function. Thus, using the last two equations we can define operators, Eop 2 = !! 2 " 2 "t 2 , Pop 2 = !! 2 " 2 "x 2 . We can rewrite the electromagnetic wave equation in operator form as a kind of energy-momentum equation: E op 2 ! = c2P op 2 ! . Because a minus sign pops up in front of the derivatives when you apply them twice, the individual operators have to involve i, the square root of −1. The convention is to define Eop = i! ! !t , Pop = "i! ! !x . [Incidentally, one might expect that if Ψ contained well-defined energy information that applying Eop to it should produce Eop! = E! . But the latter equation is really the same as !" !t = E i!( )" , and the RHS of this expression contains i. Thus, the real solutions to the E op 2 ! = E2! , such as ! = ! 0 sin( P ! x " E ! t) , for example, are not solutions to E op ! = E! . Solutions to the latter are complex functions, such as ! = ! 0 exp[i( P ! x " E ! t)] (which are also SR 8 2 solutions to E op 2 ! = E2! ). This is a common occurrence: solutions to the “energy” equation are solutions to the “energy squared” equation, but not necessarily vice versa.] Relativistic wave equations for massive particles The Klein-Gordon Equation While E 2 = c 2 P 2 correctly relates energy-momentum (squared) for a massless photon, for a massive, free particle E 2 = c 2 P 2 +m 2 c 4 . So following the strategy outlined above, let’s replace E and P by derivatives and make a wave equation. In operator language, the appropriate wave equation should be E op 2 ! = (c2P op 2 + m2c4 )! (where the mass- squared “operator” just consists of multiplying Ψ by mass-squared). In terms of partial derivatives, this equation becomes ! 2 " !t 2 = c 2 ! 2 " !x 2 # m 2 c 4 ! 2 " . (1) Equation (1) was actually found by Schrödinger (in 1926) before he hit on his now famous equation [ E op ! = P op 2 2m+U( )! , or i!!"/!t = #(! 2 2m)! 2" !x2 +U" ] that underlies all of nonrelativistic quantum mechanics. For curious historical reasons (1) is called the Klein-Gordon (K-G) Equation. In any case, solutions to (1) had originally (in the 1920s) been hoped to provide probability and dynamical information. The quantity analogous to probability (-density) for K-G turns out to be !E " 2 , not ! " 2 , as for photons. The two time derivatives combined with the nonzero mass produce the extra factor of E, the particle energy. Now, formally, E = ± c 2 P 2 + m 2 c 4 . While it is tempting to discard the negative root as unphysical, both signs are necessary from a purely mathematical perspective to manufacture wave functions for all kinds of initial particle states. Choosing the negative sign renders E ! 2 nonsensical as a probability, of course. When this problem was realized early on, the K-G equation was dropped—prematurely, it turns out—like a hot potato. Schrödinger, famously, turned his attention to slowly moving quantum particles, while P.A.M. Dirac tried something nutty that—as nutty things sometimes do—turned out to have incredible consequences. The Dirac Equation We first encountered Dirac when we were discussing waves on a string. We noted at that time, that the second order equation describing transverse waves on a string can be re-expressed as a single first order equation, !" !t = v W # !" !x , but with the cost that ψ is a column vector and α is a matrix. In 1928, Dirac recognized that the problem with the K-G equation was a combination of mass plus the two time derivatives. To solve the problem of negative probabilities, he proposed obtaining a wave equation for a freely moving massive particle with just one time derivative by writing E op ! = c2P op 2 + m2c4( )! = c !" i ! P op +#mc2( )! . A particle’s momentum in 3+1 space-time has three components—thus, ! P op = P op ,x , P op ,y , P op ,z( )— and each can have its own α matrix, i.e., ! ! = ! x ,! y ,! z( ) . Because ψ is a column vector η is also a matrix.