Physics 2D Lecture Slides Lecture 17: Feb 10 th

Transcription

1 Physics 2D Lecture Slides Lecture 17: Feb 10 th Vivek Sharma UCSD Physics

2 Just What is Waving in Matter Waves? For waves in an ocean, it s the water that waves For sound waves, it s the molecules in medium For light it s the E & B vectors that oscillate What s waving for matter waves? It s the PROBABLILITY OF FINDING THE PARTICLE that waves! Particle can be reresented by a wave acket At a certain location (x) At a certain time (t) Made by suerosition of many sinusoidal waves of different amlitudes, wavelengths λ and frequency f It s a ulse of robability in sacetime What Wave Does Not Describe a Particle y y = A cos ( kx ωt+φ) 2 k = π, w = 2 π f λ x,t What wave form can be associated with article s ilot wave? A traveling sinusoidal wave? y = A cos ( kx ωt+φ) Since de Broglie ilot wave reresents article, it must travel with same seed as article (like me and my shadow) Phase velocity (v ) of sinusoidal wave: v In Matter: h h ( a) λ = = γ mv 2 E γ mc (b) f = = h h E γ mc c v = λ f = = = γ mv v > c! = λ f Conflicts with Relativity Unhysical Single sinusoidal wave of infinite extent does not reresent article localized in sace Need wave ackets localized Satially (x) and Temorally (t)

3 Wave Grou or Wave Pulse Wave Grou/acket: Suerosition of many sinusoidal waves with different wavelengths and frequencies Localized in sace, time Size designated by x or t Wave grous travel with the seed v g = v 0 of article Constructing Wave Packets Add waves of diff λ, For each wave, ick Amlitude Phase Constructive interference over the sace-time of article Destructive interference elsewhere! Imagine Wave ulse moving along a string: its localized in time and sace (unlike a ure harmonic wave) Wave acket reresents article rob localized How To Make Wave Packets : Just Beat it! Suerosition of two sound waves of slightly different frequencies f 1 and f 2, f 1 f 2 Pattern of beats is a series of wave ackets Beat frequency f beat = f 2 f 1 = f f = range of frequencies that are suerimosed to form the wave acket

4 Addition of 2 Waves with slightly Resulting wave's "dislacement " y = y 1+ y2 : different wavelengths and y = A[ cos( k1x wt 1 ) + cos( k2x w2t) ] slightly different frequencies A+B A-B Trignometry : cosa+cos B =2cos( ) cos( ) k2 k1 w2 w1 k2 k1 w2 w1 y 2A cos( x t) cos( + x + = t) since k k k, w w w, k k, w w 2 1 ave 2 k w A' = 2A cos( x t ) = modulated amlitude 1 ave k w y = 2A cos( x t) kx w, Phase Vel wave V = kave Grou Vel w Vg = k dw Vg : Vel of enveloe= dk ' cos( t) y = A cos( kx wt) A' oscillates in x,t Wave Grou Or acket Non-reeating wave acket can be created thru suerosition Of many waves of similar (but different) frequencies and wavelengths

5 Wave Packet : Localization Finite # of diff. Monochromatic waves always roduce INFINTE sequence of reeating wave grous can t describe (localized) article To make localized wave acket, add infinite # of waves with Well chosen Aml A, Wave# k, ang. Freq. w ψ ( xt, ) = A( k) e i( kx wt) A( k) = Amlitude Fn diff waves of diff k have different amlitudes A(k) w = w(k), deends on tye of wave, media Grou Velocity V g dk dw = dk k= k0 v g t x localized Grou, Velocity, Phase Velocity and Disersion In a Wave Packet: w= w( k) Grou Velocity V k= k0 Since V = wk ( def ) w = kv g dk k = k 0 dk k= k0 g dw = dk dw dv V = = V + k usually V = V( k orλ) 1ns laser ulse diserse Material in which V varies with λ are said to be Disersive By x30 after travelling Individual harmonic waves making a wave ulse travel at 1km in otical fiber different V thus changing shae of ulse and become sread out In non-disersive media, V In disersive media V g V g = V,deends on dv dk

6 Grou Velocity of Wave Packets: V g Consider An Electron: mass = m velocity = v, momentum = 2 2π Energy E = hf = γ mc ; ω = 2 π f = γmc h h 2π 2π Wavelength λ = ; k = k = γ mv λ h dw dw / dv Grou Velocity : Vg = = dk dk / dv v g t 2π 2 dw d mc 2 mv 2 m = h π dk d 2π π & dv dv v = = v v 2 1/ 2 v [1-( ) ] h[1-( ) ] dv dv mv = h[1-( ) ] h[1-( ) ] c c c c dw dw / dv Vg = = = v dk dk / dv Grou velocity of electron Wave acket "ilot wave" is same as electron's hysical velociy t 2 2 1/ 3/ 3/2 But velocity of individual waves making u the wave acket V 2 = w c c! (not hysical) k = v > x Wave Packets & Uncertainty Princiles We added two Sinusoidal waves k w y = 2A cos( x t) cos( kx wt) Amlitude Modulation x 1 x 2 Distance X between adjacent minima = (X 2 ) node -(X 1 ) node Define X 1 =0 then hase diff from X 1 X 2 = π (similarly for t 1 t 2 ) What can we learn from this simle model? w k Node at y = 0 = 2A cos ( t x), Examine x or t behavior in x: k. x= π Need to combine many waves of diff. k to make small x ulse π x=, for small x 0 k & Vice Verca k ad n In t : w. t = π Need to combine many waves of diff ω to make small t ulse π t=, for small t 0 ω & Vice Verca ω

7 Signal Transmission and Bandwidth Theory Short duration ulses are used to transmit digital info Over hone line as brief tone ulses Over satellite link as brief radio ulses Over otical fiber as brief laser light ulses Ragardless of tye of wave or medium, any wave ulse must obey the fundamental relation» ω t π Range of frequencies that can be transmitted are called bandwidth of the medium Shortest ossible ulse that can be transmitted thru a medium is t min π/ ω Higher bandwidths transmits shorter ulses & allows high data rate Wave Packets & Uncertainty Princiles of Subatomic Physics in sace x: k. x= π 2π h since k =, = λ λ. x= h/2 usually one writes. x /2 aroximate relation In time t : w. t = π since ω=2 π f, E = hf E. t = h/2 usually one writes E. t /2 aroximate relation What do these inequalities mean hysically?

8 Know the Error of Thy Ways: Measurement Error Measurements are made by observing something : length, time, momentum, energy All measurements have some (limited) recision` no matter the instrument used Examles: How long is a desk? L = (5 ± 0.1) m = L ± L (deends on ruler used) How long was this lecture? T = (50 ± 1)minutes = T ± T (deends on the accuracy of your watch) How much does Prof. Sharma weigh? M = (1000 ± 700) kg = m ± m Is this a correct measure of my weight? Correct (because of large error reorted) but imrecise My correct weight is covered by the (large) error in observation Length Measure Voltage (or time) Measure r Measurement Error : x ± x Measurement errors are unavoidable since the measurement rocedure is an exerimental one True value of an measurable quantity is an abstract concet In a set of reeated measurements with random errors, the distribution of measurements resembles a Gaussian distribution characterized by the arameter σ or characterizing the width of the distribution Measurement error large Measurement error smaller

9 Interreting Measurements with random Error : True value Where in the World is Carmen San Diego? Carmen San Diego hidden inside a big box of length L Suose you can t see thru the (blue) box, what is you best estimate of her location inside box (she could be anywhere inside the box) X=0 X=L x Your best unbiased measure would be x = L/2 ± L/2 There is no erfect measurement, there are always measurement error

10 Wave Packets & Matter Waves What is the Wave Length of this wave acket? λ λ < λ < λ+ λ De Broglie wavelength λ = h/ Momentum Uncertainty: - < < + Similarly for frequency ω or f ω ω < ω < ω+ ω Planck s condition E= hf = hω/2 E- E < E < E + E Back to Heisenberg s Uncertainty Princile & x. h/4π If the measurement of the osition of a article is made with a recision x and a SIMULTANEOUS measurement of its momentum x in the X direction, then the roduct of the two uncertainties (measurement errors) can never be smaller than h/4π irresective of how recise the measurement tools E. t h/4π If the measurement of the energy E of a article is made with a recision E and it took time t to make that measurement, then the roduct of the two uncertainties (measurement errors) can never be smaller than h/4π irresective of how recise the measurement tools These rules arise from the way we constructed the Wave ackets describing Matter ilot waves Perhas these rules Are bogus, can we verify this with some hysical icture??

11 The Act of Observation (Comton Scattering) Act of observation disturbs the observed system Comton Scattering: Shining light to observe electron Light (hoton) λ=h/= hc/e scattering = c/f off an electron I watch the hoton as it enters my eye hgg g The act of Observation DISTURBS the object being watched, here the electron moves away from where it was originally

12 Act of Watching: A Thought Exeriment Observed Diffraction attern Photons that go thru are restricted to this region of lens Eye Diffraction By a Circular Aerture (Lens) See Resnick, Halliday Walker 6 th Ed (on S.Reserve), Ch 37, ages Diffracted image of a oint source of light thru a lens ( circular aerture of size d ) First minimum of diffraction attern is located by sinθ = 1.22 d λ See revious icture for definitions of ϑ, λ, d

13 Resolving Power of Light Thru a Lens Image of 2 searate oint sources formed by a converging lens of diameter d, ability to resolve them deends on λ & d because of the Inherent diffraction in image formation d X Not resolved barely resolved resolved Resolving ower x λ 2sinθ ϑ Deends on d Putting it all together: act of Observing an electron Photons that go thru are restricted to this region of lens Observed Diffraction attern Eye Incident light (,λ) scatters off electron To be collected by lens γ must scatter thru angle α -ϑ α ϑ Due to Comton scatter, electron icks u momentum P X, P Y h h sinθ Px sinθ λ λ electron momentum uncertainty is ~2h sinθ λ After assing thru lens, hoton diffracts, lands somewhere on screen, image (of electron) is fuzzy How fuzzy? Otics says shortest distance between two resolvable oints is : λ x = 2sinθ Larger the lens radius, larger the ϑ better resolution 2hsinθ λ. x = h λ 2sinθ. x / 2

14 Pseudo-Philosohical Aftermath of Uncertainty Princile Newtonian Physics & Deterministic hysics toles over Newton s laws told you all you needed to know about trajectory of a article Aly a force, watch the article go! Know every thing! X, v,, F, a Can redict exact trajectory of article if you had erfect device No so in the subatomic world! Of small momenta, forces, energies Cant redict anything exactly Can only redict robabilities There is so much chance that the article landed here or there Cant be sure!...cognizant of the errors of thy observations Philosohers went nuts!...what has haened to nature Philosohers just talk, don t do real life exeriments! All Measurements Have Associated Errors If your measuring aaratus has an intrinsic inaccuracy (error) of amount Then results of measurement of momentum of an object at rest can easily yield a range of values accommodated by the measurement imrecision : - Similarly for all measurable quantities like x, t, Energy!

15 Matter Diffraction & Uncertainty Princile x Y Incident Electron beam In Y direction Momentum measurement beyond slit show article not moving exactly in Y direction, develos a X comonent Of motion P X =h/(2π a) Probability slit size: a 0 P X X comonent P X of momentum Particle at Rest Between Two Walls Object of mass M at rest between two walls originally at infinity What haens to our ercetion of George as the walls are brought in? L m 0 George s Momentum On average, measure <> = 0 but there are quite large fluctuations! Width of Distribution = P ( P ) ave ( Pave) ; P = P L