l[Jomplex numbers are a very useful tool for calculations. but also unify several of the topics weive discussed so far in class. In discussion this week1 we*11 see how complex numbers illustrate the close relation between ordinary trig functions and hyperholic trig functions. and allow us to describe solutions to the wave equation in a convenient form. Part 1: Trig functions. ordinary and hyperbolic Recall the definitions of the hyperbolic trig functions from Discussion T: 1.1. 1.2. 1.4. c _ c :9 5.11111 '5, = i1 cosh g, = i [1} 2 2 Write the Cartesian forms for em and e49. and use them to solve for sintt and costt in terms of e'l'Jr and e'm. Your answer to problem 1.1 should look suspiciously like the hyperbolic trig formulas. hut with some extra factors of 1'. Put the 11's in the appropriate place [using 1'2 : 1 where necessary} to show the following: oosffy} = cosh y. sintiy} = isinh y. [2} 1|When y is a real number. this lets us dene the trig functions of an imaginary number. a process known as manly-tic continuation. Derive Eq. [2} in a different way. by plugging the argument 3 = iy into the Taylor series of cos 2 and sin a about 2 : fl. This is also a form of analytic continuation: in complex analysis. we can often dene functions hy their Taylor series. and simply declare that the value of the function at any point in the complex plane is the value of the Taylor series at that point. Recall that the Taylor series for the hyperholic trig functions are 3 s 2 4 lhtE+++-~. coshir1+E+E+-a- [3} Analytic continuation is used all over physics. but one particularly convenient application is the concept of \"imaginary time.\" This is really not as mysterious as it sounds. as we'll shortly see. Recall the definition of the invariant interval: as? : I? 3:2 y? 32. [4} Let T = it. Show that with this \"imaginary time"1 variable. o.-F : T2 + 2:2 + y2 + :2. [5} 3o. up to a minus sign, this analytic continuation turns Lorentz dot products into ordinary [Euclidean} dot products in 4 dimensions1 and all hyperbolic trig functions in the Lorentz transformation matrices hecome ordinary trig functions {i.e. boosts turn into rotations}. This is often used as a trick to make integrals over 4-vectors easier to compute. Part 2: The group U(1} The polar form of complex numbers makes clear that there is a very close correspondence between complex numbers and rotations in 2 dimensions. In this pent. you will show that unimodular complex numbers {the ones with |z| = 1} form a group1 called U [1}. that is identical to the rotation group 30(2} you studied in Discussion H. 2.1. Verify the formula by writing both sides in Cartesian form and using trig identities. This shows that the complex exponential function obeys the same algebraic rules as the ordinary one does. e'Hi = each. 2.2. The polar Iorm also lets us divide complex numbers: if z = a: + 3; = rem. show that = _ I? 22 if\"? [ :i by writing both sides in Cartesian form. Itemember to use the complex conjugate to ""realify"1 the denominator. 2.3. The relations {ti} and ['i'} show that if r = l [i.e. |z| = 1}. multiplication and division of complex numbers of the form ed" results in another complex number of the same form. In other words. unimodular complex numbers Irma c. group. where the group operation is [complex} multiplication. Find the complex number 2 which corresponds to the identity element1 and for a general unimodular number eta. find its inverse. 2.4. Finally, write down the 2 x 2 rotation matrix which corresponds to the group element 19 (hint: it should obey the same multiplication law as Eq. {ti}, and it should reduce to the identity element when d = fl}. This establishes the ismnorphism between the two groups I11: 1} and 501:2}: every element of Ilfl} is matched with one and only one element oI 30(2}. Part 3: Complex. wave notation and 4-vectors In PHYS 212 you learned that the solution to the 1-dimensional wave equation can be written as h[:r.f} = Acos[.lr[:r ct} + 15]. where it is the wavenumher1 1: is the wave speed. and 5 is a phase that represents the fact that the wave may be displaced from the origin by some arbitrary amount. 3.1. 1|ili'hat is the frequency f in terms of c and It? Dene the angular frequency as a: = 2111', and show that we can rewrite the wave as Mr. it} : Acosfim: alt + ti}. 3.2. Show that we can express h[:r.f} as the real part of a complex exponential: stat} = Re[Aei[*=-w=+r5}] [3} where A is a real oonstant representing the amplitude of the wave. Show also that we can write w . Mar} = sweetie-win {s} where we now use a sampler amplitude fl. 1What is .31 in terms of A and 45'? Note that this allows 11s to ahsorh the phase into the amplitude, which is often very convenient. 'With our knowledge of relativity~ the factor lor wt in the exponential looks suspiciously like a Lorentz clot product .t: - 3:, where 3:" = [tprfu [l] and k\" = {on 1:, {Lil}. 'We would he justied in combining LL? and it into a 4-vector as long as we could justify the [act that [at least for photons] they transform like a 4-vector. which we will now do. 3.4. 3.5. At a given point in spacetime+ a wave is characterized only by its amplitude and phase? it = l'te[Ae""5]. Argue that 85 is the same for all inertial observers. The easiest way to do that is to dene 5 by the value of the wave at t = :r = , and consider two different frames that have their origins synchronized. Your result from problem 3.4 shows that the phase of the complex amplitude fl is Lorentz- invariant. But since our choioe of origin was arhitrary, we oould have considered any other point [:r. t}, in which case the phase of the wave would he ks: :...It +5. Argue that this implies that the entire phase mnst he Lorentz-invariant. But since we know that 3:\" transforms like a =1-vector.~ the fact that the phase can he written as a Lorentz-invariant quantity means that It\" must transform like a vii-vector also! Quantum mechanics provides the final piece of the puzzle: the photon's 4-wayevector and 4-momentum are related by p\" = 5k\