Modern Physics Notes - St. Bonaventure University
Modern Physics Notes J Kiefer 2012
Table of ContentsTABLE OF CONTENTS .1I.RELATIVITY .2A.Frames of Reference . 2B.Special Relativity . 5C.Space-Time . 8D.Energy and Momentum . 17E.A Hint of General Relativity . 23II. QUANTUM THEORY . 25A.Black Body Radiation. 25B.Photons . 31C.Matter Waves . 34D.Atoms . 41III.QUANTUM MECHANICS & ATOMIC STRUCTURE (ABBREVIATED) . 49A.Schrödinger Wave Equation—One Dimensional . 49B.One-Dimensional Potentials . 51D.The Hydrogen Atom . 56E.Multi-electron Atoms . 63B.Subatomic . 651
I.RelativityA.Frames of ReferencePhysical systems are always observed from some point of view. That is, the displacement,velocity, and acceleration of a particle are measured relative to some selected origin andcoordinate axes. If a different origin and/or set of axes is used, then different numerical values are obtained for r , v , and a , even though the physical event is the same. An event is a physicalphenomenon which occurs at a specified point in space and time.1.Inertial Frames of Referencea.DefinitionAn inertial frame is one in which Newton’s “Laws” of Motion are valid. Moreover, any framemoving with constant velocity with respect to an inertial frame is also an inertial frame of reference. While r and v would have different numerical values as measured in the two frames, F ma in both frames.b.Newtonian relativityQuote: The Laws of Mechanics are the same in all inertial reference frames. What does “thesame” mean? It means that the equations and formulae have identical forms, while the numericalvalues of the variables may differ between two inertial frames.c.Fundamental frameIt follows that there is no preferred frame of reference—none is more fundamental than another.2.Transformations Between Inertial Framesa.Two inertial framesConsider two reference frames—one attached to a cart which rolls along the ground. Observerson the ground and on the cart observe the motion of an object of mass m. The S’-frame is moving with velocity v relative to the S-frame. As observed in the two frames:2
In S’ we’d measure t’, x’, and u x In S we’d measure t, x, and u x x . t x. tb.Galilean transformationImplicitly, we assume that t t . Also, we assume that the origins coincide at t 0. Thenx x v x t y y v y t z z v z t t t The corresponding velocity transformations aredx dx ux vx ux vxdtdtdy dy uy vy uy vydtdtdz dz uz v z u z vzdtdtFor accelerationdu dvax x ax xdtdtdu ydvyay a y dtdtdu dvaz z az zdtdt Note that for two inertial frames, the a x a x , a y a y , and a z a z .3
ExampleS-frame du dp, if m is constant.F ma m dtdtS’-frame dp du du dv , where p mu . But u u v , so F m F ma m F . That is,dtdt dt dt a a , as they must for 2 inertial reference frames.Notice the technique. Write the 2nd “Law” in the S’-frame, then transform the position andvelocity vectors to the S-frame.4
B.Special Relativity1.Michelson-Morleya.Wave speedsMidway through the 19th century, it was established that light is an electromagnetic (E-M) wave.Maxwell showed that these waves propagate through the vacuum with a speed c 3x10 8 m/sec.Now, wave motion was well understood, so it was expected that light waves would behaveexactly as sound waves do. Particularly the measured wave speed was expected to depend on theframe of reference. In the S-frame, the speed of sound is u ; in the S’-frame the speed is u . The source and the medium are at rest in the S-frame. We find (measure) that u u v , in conformity withNewtonian or Galilean relativity. We may identify a “preferred” reference frame, the frame inwhich the medium is at rest.b.Michelson-MorleyThroughout the latter portion of the 19th century, experiments were performed to identify thatpreferred reference frame for light waves. The questions were, what is the medium in whichlight waves travel and in what reference frame is that medium at rest? That hypothetical mediumwas given the name luminiferous ether (æther). As a medium for wave propagation, the ethermust be very stiff, yet offer no apparent resistance to motion of material objects through it.The classic experiment to detect the etheris the Michelson-Morley experiment. Ituses interference to show a phase shiftbetween light waves propagating thesame distance but in different directions.The whole apparatus (and the Earth) ispresumed to be traveling through the ether with velocity, v . A light beamfrom the source is split into two beamswhich reflect from the mirrors and arerecombined at the beam splitter—forming an interference pattern which isprojected on the screen. Take a look at5
the two light rays as observed in the ether rest frame.The sideward ray:The time required for the light ray to travelfrom the splitter to the mirror is obtainedfrom v2 (ct ) (vt ) t 1 2 c c 222 12.Now c v, so use the binomial theorem to simplify 1 x n 1 v2 .t 1 c 2 c 2 n(n 1) 2 1 nx x 2!The total time to return to the splitter is twice this: t1 2t 2 1 v 2 1 .c 2 c 2 For the forward light ray, the elapsed time from splitter to mirror to splitter is 2 v 2t2 1 2c v c v c c 1 2 v 2 1 c c 2 . The two light rays recombine at the beam splitter with a phase difference [let .]:c t c2 1 v 2 c v 2. t 2 t1 c 2 c2 c2 t 0 , the two light rays are out of phase even though they have traveled the same distance. By measuring t one could evaluate v .Since However, no such phase difference was/is observed! So, there is no ether, no v with respect tosuch an ether. This null result is obtained no matter which way the apparatus is turned. Theconclusion must be that either the “Laws” of electromagnetism do not obey a Newtonianrelativity principle or that there is no universal, preferred, rest frame for the propagation of lightwaves.c.Expedients to explain the null resultlength contraction—movement through the ether causes the lengths of objects to be shortened inthe direction of motion.ether-drag theory—ether is dragged along with the Earth, so that near the Earth’s surface theether is at rest relative to the Earth.6
Ultimately, the expedients were rejected as being too ad hoc; it’s simpler to say there is no ether.This still implies that the “Laws” of electromagnetism behave differently under a transformationfrom one reference frame to another than do the “Laws” of mechanics.2.Postulates of Special Relativitya.Principle of Special RelativityIt doesn’t seem sensible that one “part” of Physics should be different from another “part” ofPhysics. Let’s assume that they are not different, and work out the consequences. This is whatEinstein did. He postulated that ‘All the “Laws” of Physics are the same in all inertial referenceframes.’b.Second PostulateThe second postulate follows from the first. ‘The speed of light in a vacuum is (measured to be)the same in all inertial reference frames.’When the speed of light is measured in the two reference frames, it is found that c c v ,rather c c . Evidently, the Galilean Transformation is not correct, or anyway not exact. In anycase, we assume the postulates are true, and work out the consequences.An event may be regarded as a single observation made at a specific location and time. Onemight say that an event is a point in space-time (x,y,z,t). Two events may be separated byintervals in either space or in time or in both.c.Time intervalsConsider a kind of clock:We observe two events: i) the emission of a flash at O’ and ii) thereception of the flash at O’. In this case, x y z 0 . The2d time interval between the two events is t .cNow let’s view the same two events from the point of view ofanother frame, S. As shown below, the S’-frame is moving to theright with speed v relative to the S-frame. In the S-frame, x 0 .2dThe elapsed time is t , where d 2 d 2 2 . Substitute forc d , d , and in terms of t , t , c, and v.7
c 2 t 2 c 2 t 2 v 2 t 2 444 c2 Solve for t t 22 c vC.Space-Time1.Definitions 12 t v21 2c.Time intervals are not absolute, after all, as has been assumed in classical physics.a.Inertial reference frame or “observer”An inertial observer is a coordinate system for space-time; it records the position (x) and time (t)of any event. [We’ll restrict our attention to one spatial dimension, at least for the moment.] Thespace-time has the following properties:i) the distance between x1 and x2 is independent of t.ii) at every point in space there is a clock; the clocks are synchronized and all run at thesame rate.iii) the geometry of space at any fixed time, t, is Euclidian. This is the assumption thatmakes special relativity special.b.ObservationAn observation is the act of assigning a coordinate, x, to an event, and the time, t, on the clock atthat point.c.UnitsIt is convenient to introduce a set of units in which time is measured in meters. One meter oftime is the elapsed time for a photon to travel one meter of distance. In this system of units, thespeed of light is c 1 (dimensionless) and speeds are effectively multiples of c, alsodimensionless. All other familiar physical units, like Newton and Joule are converted by theconversion factor 3 10 8 m 1 sec .8
d.Space-time diagramWe set up the time axis for the rest frame of anobserver at the origin. In our relativistic units, thedivisions on both the space and time axes are meters.The space axis is defined as those points for which thetime coordinate is t 0. Photons that are emitted fromthe point (-a,0) and reflect back to the point (a,0) forma right angle at the point (0,x). This will be true in anyinertial reference frame, because the slope of aphoton’s trajectory is observed to be 45 degrees in allinertial reference frames. An event is a (t,x) point inspace-time.e.World lineA world line is a trajectory through space-time. At any point, the slope of the world-line is 1/v.The world line of a photon is a straight line with a slope of 1 in our relativistic units. The worldline of a massive object has a slope on the t vs. x graph 1, since v c. Notice that the worldline of the observer is the time axis itself, because the observer is at rest.f.Light coneImagine a photon emitted at the origin of coordinates.Imagine its world-line rotated about the t-axis,sweeping out a cone—that’s called a light cone.The light cone divides the space-time into fourregions: past, present, future, and elsewhere. Wecannot receive information from any event that iselsewhere. The present is the origin.The world lines of photons arriving at or emitted fromthe event A define the light cone for the event A.g.IntervalThe interval is the separation between two events inspace-time. s 2 t 2 x 2Why is it t 2 ? Consider the distance of event Afrom the origin, O. A photon emitted from the originreaches the point A at a time t. It will have travelleda spatial distance x 2 y 2 z 2 c 2 t 2 from theorigin. Observed in another frame, whose origin O’ coincided with O when the photon wasemitted, x 2 y 2 z 2 c 2 t 2 . For the sake of argument, subtract these two equations.9
x 2 y 2 z 2 c 2 t 2 x 2 y 2 z 2 c 2 t 2 . With one-dimensional space, this reduces tox 2 c 2 t 2 x 2 c 2 t 2 . The quantity called the interval is invariant, the same in all inertialreference frames. With c 1, x 2 t 2 s 2 . We use this definition of interval because it isinvariant.spacelike s 2 0timelike s 2 0lightlike s 2 0The equation t 2 x 2 a 2 defines a hyperbola inspace-time. All points on that hyperbola are the samedistance in space-time from the origin. Likewise,t 2 x2 a2 .2.Consequences of the Postulates of Special Relativitya.Two inertial observersLet’s say that the origin, O’, of a moving inertial framecoincides with O at t 0. The t’-axis as drawn in theS-frame is just the world line of O’ in the S-frame.The slope of the t’ axis is 1/v, where v is the relativevelocity of the S’-frame.The x’-axis is a little trickier. A photon emitted at(t’ -a,x’ 0), reflected at (t’ 0,x’), and received at(t’ a,x’ 0) travels on lines with slopes of /- 1 in theS-frame. The x’-axis is the locus of points such thatthe world lines of the emitted and reflected photons areperpendicular.[Recall that photon world lines have a slope of /- 1 in all inertial reference frames.Transform coordinates:t t x.x t xEquations of the t’ axis and the x’ axis in the S-frame:10vt x 0vx t 0.
vt t v x Match these up when x’ 0 & t’ 0:, whence. x x vt v x vt 1 vx 1 .Now, we must have c c 1 , thereforet t v x 1 vFinally, what is ?t t v x t t v x ; at the same time the inverse transformation must besince the twox x vt x x vt frames are equivalent. So take the equation for x and substitute for x’ and t’.x x vt x vt v t vx 2 x 2 vt 2 vt 2 v 2 x x 2 1 v 2 2 11 v211
b.Time dilationConsider an event, B, on the t’-axis. The coordinates ofB in the S-frame are (t,x), and in the S’ frame (t’,0).s 2 s 2 t 2 x 2 t 2 0 2 t 2 ( vt ) 2 t 2t t 1 v2The elapsed time between O and B is measureddifferently in the two frames.example (prob. 1-10 in the text)The lifetime of a pion in its own rest frame is t 2.6 x10 8 sec. Consider a pion moving withspeed v 0.95c in a lab—what will be measured as its lifetime in the lab? t t 1 v2 2.6 x10 8 sec1 0.95 2 2.6 x10 8 sec 8.33 x10 8 sec .0.312c2The lifetime of a fast-moving particle is measured by noting how far it travels before decaying.In this example v t 0.95c 8.33 x10 8 sec 23.7 m. In practice, we measure and compute t .Proper timeThe proper time is the time interval measured by an observer for whom the two events occur atthe same place, so that x y z 0 .c.Length contractionConsider a rigid rod at rest in the S’ frame, along the x’axis. It’s ends are at events A (t’ 0,x’ 0) & C(t’ 0,x’ ). The world line of the right end passesthrough the point B. So, the length of the rod in the Sframe is x . Relative to the (t,x) axes,Bs C2 xC2 t C2Where xC is the x-coordinate of the point C in the Sframe.12
t C vxC xC 1 v2x xBv CtC t BNote: the slope of the x’ axis is v,or the inverse of the slope of the t’axis.Let tB 0 and solve for xB.x B xC vt CxB 2v 2 2 1 v 21 v1 vThe contraction takes place in the direction of the relative motion. Lengths perpendicular to vare not affected. So for instance in the situation discussed above the width and thickness of themeter stick are still measured the same in both reference frames.Proper length is the length measured in the rest frame of the rod, in this case .example (prob. 1-8 in the text)A meter stick moving in a direction parallel to its length appears to be 75 cm long to an observer.What is the speed of the meter stick relative to the observer? 1.0m 0.75m 1 v 22 v 1 1 0.5625 0.4375 v 0.661That is, in SI units, v 0.661c 1.98 10 8 m / s .2d.non-simultaneityIn the S-frame, all events on a line parallel to the x-axis have the same t coordinate, and areobserved to be simultaneous in S. A line parallel to the x-axis is not parallel to the x’-axis sothose events are not simultaneous in the S’-frame. Simultaneity is not absolute, or invariant.e.synchronizationWe would like all clocks in a reference frame to display exactly the same readingsimultaneously, but can this be arranged? Only by the exchange of signals, which is another wayof saying only in terms of intervals. However, as we have seen, intervals are not the same forobservers in different inertial reference frames.13
4.Lorentz TransformationNow we wish to derive the transformation equations for the displacement and velocity of anobject—the relativistic version of the Galilean transformation equations. In what follows, we’ll1be setting .v21 2ca.Two framesConsider two inertial reference frames, S & S’ and assume that O O’ at t’ 0.What is the x-distance from O to the point P, as measured in the S’ frame? In effect, then, we’llhave t t and t t . x vt xIn the S frame, x , so also. Set ‘em equal. x x vt x x vt x On the other hand, as measured in the S frame, x vt . Set them equal. x x vt vt Solve for t.x vt x vt v t t 2 x c b.Transformation equationsWe have, then, for relative motion along the x-axis:v x x vt ; y y ; z z ; t t 2 x c 14
Notes: i) the inverse transformation is obtained by replacing v with –v.ii) for v c, these reduce to the Galilean transformation.c.4-vectorsSuppose that when O O’, a flash of light is emitted from the origin O. In the S frame, thedistance the light wave front travels in time t is r 2 x 2 y 2 z 2 c 2 t 2 . Measured in the S’frame, it’s r 2 x 2 y 2 z 2 c 2 t 2 . Subtract the second expression from the first andcollect the S frame on one side of the equal sign, the S’ frame on the other side.r 2 r 2 c 2t 2 c 2t 2r 2 c 2t 2 r 2 c 2t 2There is this quantity, a generalized displacement (call it s) which is the same in the two inertialreference frames.s 2 s 2We see that the quantity (ict) “acts like” a component of displacement along a fourth axis. Theinterval between any two events in space-time is s 2 x 2 y 2 z 2 c 2 t 2 . The intervalis invariant under the Lorentz Transformation.
5 B. Special Relativity 1. Michelson-Morley a. Wave speeds Midway through the 19th century, it was established that light is an electromagnetic (E-M) wave. Maxwell showed that these waves propagate through the vacuum with a speed c 3x108 m/sec. Now, wave motion was well und
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