Lecture notes in modern Physics

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Published Date:25-07-2017
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Class Notes for Modern Physics, Part 1 J. Gunion U.C. Davis 9D, Spring Quarter J. GunionWhat is Modern Physics? The study of Modern Physics is the study of the enormous revolution in our view of the physical universe that began just prior to 1900. At that time, most physicists believed that everything in physics was completely understood. Normal intuition and all experiments fit into the context of two basic theories: 1. Newtonian Mechanics for massive bodies; 2. Maxwell’s Theory for light (electromagnetic radiation). Consistency of the two required that there be a propagating medium (and, therefore, a preferred reference frame) for light. However, even a little thought made it clear that there was trouble on the horizon. And then came many new experimental results that made it clear that the then-existing theoretical framework was woefully inadequate to describe nature. In a relatively short period of time, physicists were compelled to adopt: J. Gunion 9D, Spring Quarter 11. the theory of special relativity based on the idea that there was no propagating medium for light (so that light traveled with the same speed regardless of the “frame” from which the light was viewed); 2. the theory of quantum mechanics, according to which the precise position and precise momentum of a particle cannot both be determined simultaneously. In fact, one must think of particles not as particles, but as waves, much like light. 3. At the same time, experiments made it clear that light comes in little quantum particle-like packets called photons. 4. In short, both particles and light have both a particle-like and wave-like nature. It is useful to focus first on the inconsistencies of the “ether” picture and of the above-outlined naive picture of space and time. This will lead us to the theory of special relativity. ThelatterinconsistenciesarerevealedbythinkingcarefullyaboutGalilean transformations between coordinate systems that underpinned the pre- relativity view of space and time. J. Gunion 9D, Spring Quarter 2Before proceeding, let me just emphasize that in this course we will be embarking on an exploration that has been repeated in a certain sense several times now. Indeed, the business of looking for inconsistencies in existing theories now has a long history of success, beginning with the development of special relativity, general relativity, and quantum mechanics. We have learned not to be arrogant, but rather to expect that the best theories of a given moment are imperfect and to look for difficulties (perhaps subtle ones) or extensionsthataresuggestedbythoughtexperimentsthatpushthetheories into a new domain. As an example, the development of the Standard Model of fundamental interactions (that you may have heard of) began with the realization that thetheorythatwasdevelopedtoexplaintheweakinteractionswouldviolate the laws of probability conservation when extended to high energies. In fact, nowadays, we have many arguments that suggest that the Standard Model is itself little more than an “effective” theory valid at the energy scales that we have so far been able to probe. It has undesirable features when we try to extend it to higher energies (e.g. from the scale of the masses of the new W and Z bosons to the Planck mass scale that is some 16 orders of magnitude larger). J. Gunion 9D, Spring Quarter 3The ether picture for light propagation • At the end of the 19th century, light waves were an accepted fact, but all physicists were “certain” that there had to be a medium in which the light propagated (analogous to water waves, waves on a string, etc.). • However, the “ether” in which light propagated had to be quite unusual. The speed of light was known to be very large (the precise value we 8 now know is c = 3.00× 10 m/s). A medium that supported this high speed had to be essentially incompressible (i.e. something vastly more incompressible than water, and even more vastly incompressible than air). • And yet, it was clear that light traveled over great distances from the stars, implying that this ether extended throughout a large section of the universe. This means that the planets, stars, galaxies, . . . , were traveling through this ether according to Newton’s laws without feeling any frictional, viscosity, . . . , type of effects. J. Gunion 9D, Spring Quarter 4• Well, foranyonethinkingaboutthisnowadays, thisisobviouslyridiculous. But, at the end of the 1900’s it was impossible for physicists to accept the fact that there was no ether medium in which light traveled and it was bizarre to imagine that light could travel through “vacuum”, despite the fact that Maxwell’s equations were most easily understood in this context. • WewillshortlyturntotheMichelson-Morley(MM)experimentperformed in 1887 in which MM set out to demonstrate the existence of the ether. We will learn that they failed. To show how entrenched thinking can become, it should be noted that Michelson (who was quite a brilliant guy) never believed the result of his experiment and spent the next 20 years trying to prove hisoriginal result was wrong. He failed, but provided ever-increasing accuracy for the precise speed of light. • It would be natural to presume that Einstein’s theory of special relativity was a response to this experiment. But, in fact, he stated that when he developed his theory he was completely unaware of the MM result. He simply was thinking of J. Gunion 9D, Spring Quarter 5Maxwell’s theory of light as a medium independent theory and asking about its consequences. This is not totally implausible given the fact that the MM experiment was performed in the “back-woods”, frontier town of Cleveland Ohio (some would say that the MM experiment is still the most important thing, other than some baseball greats, to come out of Cleveland). And communications were not so hot back in those days. To understand the ideas behind the MM experiment and to set the stage for how we discuss space and time in an “inertial” frame, we must consider how to relate one frame to another one moving with constant velocity with respect to the first frame. The1900’sviewofthisrelationshipisencodedin“Galileantransformations”. J. Gunion 9D, Spring Quarter 6Galilean transformations A coordinate system is to be thought of as: 1. a system of “meter” sticks laid down in the x,y,z directions throughout all of space. 2. a single universal clock time that applies throughout all of space (i.e. is the same no matter where in space you are). An event is thus specified by its location in the (t,x,y,z) space. But, now suppose that there is another person moving with constant velocity in the original coordinate system in the positive x direction. His coordinates will be related to (t,x,y,z) by: 0 x = x− vt 0 y = y 0 z = z 0 t = t, (1) J. Gunion 9D, Spring Quarter 7where the last equation, according to which both observers can use the time, or set of clocks, is particularly crucial. This relation between frames is illustrated below. C lick to add title Fig. 1-2, p. 4 Fig. 1-1, p. 4 Figure 1: The Galilean Transformation. Of course, if we are tracking an object moving in the x-direction in the two coordinate systems, we may compare its velocity and acceleration as viewed in the two coordinate systems by taking derivatives of the first J. Gunion 9D, Spring Quarter 80 equation above to obtain (using dt =dt from above) 0 dx dx 0 u ≡ = − v≡ u − v, and x x dt dt 0 du du x 0 x a ≡ = ≡ a ,. (2) x x dt dt Note that the accelerations are the same, which is consistent with the idea that the force causing the acceleration should be the same as viewed by the two different observers (given that forces depend on separations between objects which will be the same in the two different frames): 0 0 0 0 F =ma =ma =ma =F , (3) x x x x x 0 where we assumed m = m is frame independent. The fact that 0 0 F = ma and F = ma have the same form in the two different x x x x reference systems is called covariance of Newton’s 2nd law. But, there is already a problem with covariance in the case of light. Light cannot be subject to the same covariance rules without conflicting with the idea of an ether in which it propagates. J. Gunion 9D, Spring Quarter 9In particular, imagine you are at rest in the ether and look at your reflection in the mirror. No problem – it just takes a very short time for the light to travel to the mirror and back. However, now suppose you are on a rocket ship moving with velocity v c with respect to the ether. Then the light traveling with speed c in the ether never makes it to the mirror • Thus, 1. either we are forced to give up the general concept that motion with constant velocity is indistinguishable from being at rest (i.e. there must be a preferred rest frame), or 2. the Galilean transform equations eq. (1) are wrong. It is the latter that is true. These thoughts led to the Michelson-Morley experiment. They showed that either there is no ether or that the earth is not moving through the ether (a very geo-centric point of view by then, since it would make much more sense if the earth was moving with its orbital velocity through an ether J. Gunion 9D, Spring Quarter 10that was a rest with respect to the galaxy or universe as a whole). The Michelson Morley Experiment The experimental arrangement for the MM experiment appears in the diagram below. Fig. 1-4, p. 8 Figure 2: The Michelson Morley experimental set-up. In the pictured arrangement, the light (wave) is split by a half-silvered mirror into two components, one traveling parallel to the earth’s motion, J. Gunion 9D, Spring Quarter 11the other traveling perpendicular to the earth’s motion through the ether. The time of travel for the horizontal light to and back from the mirror will (Galilean assumed) be L L t = + . (4) horizontal c+v c− v The time of travel for the vertical light (which must actually be aimed “up-stream” in order to return to the splitting mirror) is given by 2L t =√ . (5) vertical 2 2 c − v 4 8 From this, we find (for v =v ∼ 3× 10 m/s andc' 3× 10 m/s) earth 2 v Δ t = t − t 'L for vc or h v 3 c −7 2 −8 Δ d≡ cΔ t ∼ 10 m for L = 10 m and (v/c) ∼ 10 . (6) Although this is a very small number, an interferometer which measures the interference between the vertical and horizontal light waves can be sensitive to it. J. Gunion 9D, Spring Quarter 12◦ We now rotate the apparatus by 90 so that the roles of the two light paths are interchanged. Since the waves are sensitive to the wave pattern oscillation   2π sin (x− ct) (7) λ with t = t or t = t , one sees a shift (relative to that if t = t ) in h v h v the constructive interference fringe corresponding to an angular amount given by 2π 2π (c2Δ t) = 2Δ d (8) λ λ (factor of 2 from fact that net time shift is twice the time difference between wave arrival times in any one set-up). This would be noticeable −7 for λ ∼ few× 10 m in a typical laboratory sized set-up using visible light. What did they see? No shift in the interference pattern. The time was ripe for a new idea. Enter Einstein in 1905. J. Gunion 9D, Spring Quarter 13Special Relativity Postulates of special relativity: 1. The Principle of Relativity: All the laws of physics have the same form in all inertial reference frames. Inotherwords, covarianceappliestoelectromagnetism(thereisnoether) as well as to mechanics. 2. The Constancy of the Speed of Light: The speed of light in vacuum has 8 the same value, c = 3.00× 10 m/s, in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light. This postulate is in fact more or less required by the first postulate. If the speed of light was different in different frames, the Maxwell equations governing the propagation of light would have to be frame-dependent. In fact, Einstein said he was completely unaware of the MM experiment at the time he proposed his postulates. He was just thinking about the theory of light as being absolute and frame independent. J. Gunion 9D, Spring Quarter 14These two apparently simple postulates imply dramatic changes in how we must visualize length, time and simultaneity. 1. Thedistancebetweentwopointsandthetimeintervalbetweentwoevents both depend on the frame of reference in which they are measured. 2. Events at different locations that occur simultaneously in one frame are not simultaneous in another frame moving uniformly with respect to the first. To see what exactly is true, we need to first think about how an inertial reference frame is defined. We use a coordinate grid and a set of synchronized clocks throughout all space. As an aside, we should note that we are already questioning that such a picture actually exists when looking at very tiny distance scales where effects of quantum gravity are expected to enter. An inertial reference frame is probably only an effective description that is only valid up to the Planck mass scale. J. Gunion 9D, Spring Quarter 15C lick to add title Fig. 1-8, p. 13 Figure 3: Picture of an inertial reference frame. Let’s return to the concept of time. Example A Suppose time were uniquely definable and the same in all frames. Consider a (small) plane moving at speed v (and very close to ground) in the +x direction relative to someone on the ground. J. Gunion 9D, Spring Quarter 16x’=−D x’=0 v The whole (P) plane picture above is moving with velocity v relative to the ground (G). Figure 4: The frame for a plane moving relative to the earth. 0 0 When the plane is at x = 0 someone at x =− D flashes a light (these are the plane’s coordinates). If time is universal then bothP (plane) and 0 G (ground) agree that the light flashes at a certain time, say t =t = 0. 0 The time at which P thinks the light arrives at x = 0 (the plane never 0 0 moves from x = 0 – he is at rest in his coordinate system) is t =D/c (assuming light travels with velocity c). The time at which G thinks the light arrives at the plane would also be 0 t = t = D/c if time is universal. However, since the plane has moved J. Gunion 9D, Spring Quarter 17by an amount D extra distance =v (9) c according to G while the light has been traveling, the G observer concludes that the velocity of light is D D +v distance c = =v +c. (10) D time c Well, this contradicts Einstein’s postulates of relativity. It has to be that the clocks in theG andP frames are not synchronized in the manner we assumed or that distance scales are not the same in the two frames. In fact, both apply. Example B • Consider2observersAandB thatpassoneanother, with, say,B moving with velocity v in the x direction relative to A who we envision is at rest in “our” frame. A burst of light is emitted as they pass one another. Each claims that the light travels outward in spherical waves with velocity c, with the spheres centered on themselves. J. Gunion 9D, Spring Quarter 18A modern day application is that a terrorist dropping a bomb (that immediately detonates) from a fast moving car might hope to quickly leave behind the destruction and explosion. But, to the extent that electromagnetic radiation was the only consideration, he would always be at the center of the explosion no matter how fast he was moving in some other frame. • Thisiscompletelydifferentfromwhatonewouldconcludeiflighttraveled in a medium like water. Consider two boats, one (A) at rest in a pond, the other (B) moving rapidly (but without creating any wake) relative to the first boat. B drops a rock in the pond as he passes A. Because A is at rest in the pond, the ripples spread out in concentric circles from his position and B, looking back, agrees. Indeed, he could even go faster than the ripples, in which case they would never catch up to him. Putting Einstein’s visualization into mathematical language, we would say J. Gunion 9D, Spring Quarter 19

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