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# Intermediate Modern Physics PHYS 3710

Utah State University

GPA 3.6

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This 4 page Class Notes was uploaded by Talon Thompson on Wednesday October 28, 2015. The Class Notes belongs to PHYS 3710 at Utah State University taught by David Peak in Fall. Since its upload, it has received 18 views. For similar materials see /class/230477/phys-3710-utah-state-university in Physics 2 at Utah State University.

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Date Created: 10/28/15

Special relativity VIII The electromagnetic wave equation revisited 32 BZ I 72 C2 72 where I could be either 5 or B 3t 3x Here the elds are assumed to be linearly polarized perpendicular to the direction of propagation x A harmonic wave traveling in the xdirection can be expressed as I 11 sinkx cot Recall that electromagnetic waves obey I P0 COSkx cot or I P0 expikx wt or some combination of such expressions In the latter form the coef cient PO is complex After differentiating I twice in t and twice in x and substituting into the wave equation we nd that in order for I to be a solution to the electromagnetic wave equation 6 60 k This simple relation can also be written as c hathie EP Thus we can think ofc as being associated with the wave properties ofthe elds or with the energy and momentum of the quantaiie the photonsiof the elds P E Combining the two we can write for example I P0 51mg 96 gt In quantum terms the wave function I can be viewed as providing information both about the likelihood of detecting a photon at x tithe probability density is M W 2 which for real elds is just I 27and about the momentum and energy P E it would have if detected If someone gave you I you extract E and P from it by differentiating 32111 E 32111 11 3 if 3x2 if Quantum eld theory is predicated on the assumption that it is possible to nd 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 canied by the particles Such wave equations involve partial derivatives which in this view pull out dynamical information when they operate on the wave function 32 P2 h2 i Thus using the last two equations we can de ne operators E p h2 y 0p 3x2 We can rewrite the electromagnetic wave equation in operator form as a kind of energymomentum equation 2 2 2 EUV I c ng l 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 l The convention is to de ne 0 0 Eap lhi P0p lhi 9t 9 Incidentally one might expect that if 1 contained wellde ned energy information that applying Eap to it should produce EW I E I But the latter equation is really the same as JPBt Eih I and the RHS of this expression contains i Thus the real solutions to the Ejp l EZ I such as I 11 Sln x gt for example are not solutions to 15ng E I P E Solutions to the latter are complex functions such as I P0 exp1 x 0 which are also solutions to Ejp I EZ I 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 4sz correctly relates energymomentum squared for 2 a massless photon for a massivefree particle E2 4sz m 44 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 Ejp I c213 m2c4 I where the mass squared operator just consists of multiplying I by masssquared In terms of partial derivatives this equation becomes 7 c 7 7 I 1 Biz 3x2 hz Equation 1 was actually found by Schrodinger in 1926 before he hit on his now famous equation Egp I 2m U lI or ihQ I 3t h22m BAPsz U I that underlies all of nonrelativistic quantum mechanics For curious historical reasons 1 is called the KleinGordon KG 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 2not Ml I nonzero mass produce the eXtra factor of E the particle energy 2 turns out to be M E l I as for photons The two time derivatives combined with the Now formally E ix CZPZ mic4 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 l P Z nonsensical as a probability of course When this problem was realized early on the KG equation was dropped7prematurely it turns out7like a hot potato Schrodinger famously turned his attention to slowly moving quantum particles while PAM Dirac tried something nutty that7as nutty things sometimes do7tumed out to have incredible consequences The Dirac Equation We rst 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 reeXpressed as a single rst order equation vwag but with the cost that yis a column x vector and 06 is a matrix In 1928 Dirac recognized that the problem with the KG 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 Egpy 1lc2P0 m2c4I cat15 nmc2y A particle s momentum in 31 spacetime has three components7thus 130 lip1137plyPWZ7 and each can have its own amatriX ie o a1xyxz Because yis a column vector 17 is also a matrix Like the situation described above solutions to the energy equation should also be solutions to the energy squared equation Thus it must be that E0p2142P0p2 m2041 namely that you get the K G equation back when you square the operator With this requirement we nd that 06106 06061 0 where i andj can be x y or 2 but with i 1 0617117061 0 for any i 0612 1 for any i and 172 1 In these expressions the bold 0 signi es a matrix with 0 everywhere while 1 signi es a matrix with 1 along the diagonal and 0 everywhere else Example Suppose M 11 i J N i 11 Then OHM11N11M12N217 O12 MHN12 MUN22 and so on The result show it is MN 02 3 J On the other hand NM 0 2 0 2 So MN NM 0 0 0 Similarly the 2 0 2 0 0 0 matrices Q MR Nhavethepropertythat Q2R21 11 Dirac found that the smallest matrices that would do the trick had to have 4 rows and 4 columns hence not the matrices in the example he also found that yhad to be a column vector consisting of 4 rows and 1 column The collection of matrices 07 is a generalization of the 06 matrix for waves on a string 06 1 1 J expressed as o 0 1 where 0 is a 2x2 matrix of all zeroes and each component of 639 is also a 2x2 matrix whose square is the 2x2 1 0 1 The generalization of 06 contains a surprise It turns matrix 1 the matrix 17 is 17 out that the orbita angular momentum L of a freely translating Dirac mass relative to some a h a h s pomt1s not conserved however the quantity L 70 is conserved Thus 7039 acts like angular momentum but not one associated with orbita motion Moreover for a freely translating particle 813 has two possible values gt 0 when 639 has a component parallel to P and lt 0 when it has a component antiparallel to l3 These properties suggest that a Dirac mass has spin 12 with it s z component either along the particle momentum or opposite itistates of right handed and left handed helicity respectively Solutions to the Dirac Equation yield as Dirac had hoped a happy unambiguous probability density 1y negative energies are still required for a complete theory Dirac s interpretation of the two 2 which is not proportional to Eidespite the fact that both positive and different energy shows that the Dirac Equation contains something even more monumental than spin Consider the two possibilities E ix CZPZ mic4 When the particle is at restiie when P 0715 imcz In other words negative energy seems to imply negative rest mass In fact there are two ways of writing Egplil for a free particle namely icol30p nmCZV That is there are really two Dirac Equations one for positive energy the second for negative Now let s add to E a potential energy U Suppose U qV the potential energy associated with a charge 1 moving through an electric potential V The two equations Dirac are now EWV teat130 i nmc2 EV1 where the sign goes with positive energy and the 7 sign with negative It is straightforward to convert the negative energy equation into a positive energy equation just multiply both sides by 71 The negative energy equation then becomes E py 665130 nmc2 qV 11 Egpll gives a positive energy value This equation describes a 0 positive energy positive mass particle but with opposite sign charge If the original positive energy equation describes an electron with charge 7e the second describes a particle with the same spin and mass as the electron but with charge e The latter are called positrons positrons are said to be antielectrons The Dirac Equation therefore automatically has in it not only spin but also antimatter Carl Anderson con rmed the eXistence of the positron in 1932 using a cloud chamber to investigate particles produced by highenergy cosmic rays Anderson wrote later that he vaguely knew of Dirac s work but didn t consciously think that the apparently positively charged electrons he was occasionally observing had anything to do with it Incidentally armed with all of this about the Dirac Equation it is possible to usefully reinterpret the anomalous probability density of the KleinGordon Equation The alleged 2 and the interpretation should then be electric charge density The reason for this is that the positive energy solutions of the original Dirac Equation are electrons with negative charge while the negative energy solutions are really positrons with positive charge Solutions to the Dirac Equation are also solutions to KG Whether KG is good for anything elseieg for describing massive bosonsiis a matter we will return to later probability density really should have a minus sign ie should be ec E CI

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