De Broglie s Pilot Waves

Transcription

1 De Broglie s Pilot Waves Bohr s Moel of the Hyrogen tom: One way to arrive at Bohr s hypothesis is to think of the electron not as a particle but as a staning wave at raius r aroun the proton. Thus, nλ πr an nλ r with n,, 3, π The orbital angular momentum is nλp h L rp n nh, π π which is Bohr s hypothesis provie that p h/λ! Bohr s Postulate! r λ πr/4 Pilot Waves: In his octoral issertation (94 Louis De Broglie suggeste that if waves can act like particles (i.e. the photon, why not particles acting like waves? He calle these waves pilot waves (wave of what? an assigne them the following wavelength an frequency: h E λ f or E h ω p hk p h Plane Wave where k π/λ an ω πf. Phase Velocity of a Plane Wave: For a traveling plane (i.e. monochromatic wave given by ( x, sin( kx ω the position of the n th noe (i.e. zeros is given by kx n ωt nπ an the spee of the n th noe along the x-axis is xn ω v phase λf t k Phase Velocity of a Plane Pilot Wave: For a plane De Broglie pilot wave we get E E ( cp + ( mc v phase ω c c, k p cp cp which is greater than c for particles with non-zero rest mass! - v phase x Particles cannot be plane pilot waves! Department of Physics Chapter5_.oc University of Floria

2 Wave Superposition an Wave Packets Suppose we a two plane waves together one with wave number k an frequency ω an the other with wave number k+ k an frequency ω+ ω as follows ( x, ( x, + ( x, ( x, sin( kx ω ( x, sin(( k + k x ( ω + ω - - Wave Superposition x Then k ω k + k ω + ω ( x, cos x t sin x t where I use sin + sin B cos( ( Bsin( ( + B Now suppose that k << k an ω << ω so that k ω ( x, cos x t sin( kx ω P( x, sin( kx ω. The term P(x, escribes the wave-packet. The position of the n th noe (i.e. zeros of the wave-packet is given by kx n ω t ( n + π an the spee of the wave-packet along the x-axis is xn ω ω vgroup t k k The pilot wave-packet travels at the spee of the particle! Wave Packet! Group Velocity of a Pilot Wave-Packet: For a De Broglie pilot wave-packet we get E cp v group ω c, k p E where I use ω E / h, k p / h, an c ( ( cp ( mc E p +. p Note that for pilot waves v phase v group c Wave Packet x v group Department of Physics Chapter5_.oc University of Floria

3 Representing Waves as Complex Numbers Im( kx-ω Im( sin(kx-ω Crest Phase φ kx- ωt Re( Trough t Wave Function We can use complex numbers to represent traveling waves. If we let kx ω then Im( sin( kx ω is a traveling plane wave with wave number k π/λ, angular frequency ω πf, an amplitue. The intensity, I, is proportional to. mplitue (, iωt I Phase-Shift Due to a Path Length Difference (r, kr-ω Intensity x φ - ωt Distance r x r φ kr- ωt Consier two traveling wave that are in phase at their source, but wave travels a istance r an wave travels a istance r to the point P. The phase ifference between the two waves at the point P is given by φ φ φ k( r r π r λ The conition for maximal constructive interference is φ πm r mλ m, ±, ±,K (max constructive The conition for maximal estructive interference is π + πm r m + λ m, ±, ±,K φ (max estructive Department of Physics Chapter5_3.oc University of Floria

4 The Complex Plane Magnitue Im(z y z φ ± ± iφ e φ e i cosφ ± isinφ x Phase z x+iy Re(z z z Re( z + i Im( z Re( z i Im( z z e z e iφ iφ x Re( z z cosφ ( z + z y Im( z z sinφ ( z z i φ arctan( y / x z x + y zz i ± i e π ± i Im(z Using Complex Numbers to Represent SHM z iωt φ ωt Re(z Im(z sinωt t We can use complex numbers to represent simple harmonic motion. If we let iωt then z Re(z cosωt SHM with amplitue an angular frequency ω Re( z Im( z cosωt sinωt z t Department of Physics Chapter5_4.oc University of Floria

5 Double Slit Interference The simplest way to prouce a phase shift a ifference in the path length between the two wave sources, S an S is with a ouble slit. The point P is locate on a screen that is a istance L away from the slits an the slits are separate by a istance. Double Slit Double Slit r S r S L P y S S r r If L >> then to a goo approximation the path length ifference is, r r r sin. sin Maximal Constructive Interference: The conition for maximal constructive interference is sin m λ m, ±, ±,K (Bright Fringes - max constructive Orer of the Bright Fringe Maximal Destructive Interference: The conition for maximal estructive interference is λ sin m + m, ±, ±,K (Dark Fringes - max estructive S S Double Slit r L y L tan P y Department of Physics Chapter5_5.oc University of Floria

6 Double Slit Intensity Pattern S S Double Slit r r r sin We form the superposition of the two waves at the point P on the screen as follows. an thus tot + tot + ik r kr ωt ik r / ik r / ik r / ( + e e ( e + e e ik r / cos( + k r where k π/λ. The path ifference (for L >> is given by r r r sin The istance from the center of the two slits an the point P on the screen is given by r r + r/ an hence kr ωt tot( r, cos( π sin / λ. The intensity is proportional to the amplitue square an hence I ( 4I cos ( π sin / λ where I is the intensity of a single wave (wave or wave. Double Slit r P 5/ / S S L / -/ sin/λ - Central Bright Spot -3/ - Intensity Department of Physics Chapter5_6.oc University of Floria

7 The Davison-Germer Experiment Bragg X-Ray Scattering (photons: Consier the scattering (or reflection of X- rays from two ajacent planes of atoms (calle Bragg planes. To a goo approximation the path length ifference between ray an ray "grazing angle φ is, r r r sinφ, where is the atomic spacing an φ is the grazing angle. Thus maximal constructive interference will occur at sin φ m λ m, ±, ±,K with the st orer maximum at Bragg X-Ray Scattering Incient ray Scattere ray Incient ray Scattere ray λ sinφ. Davison-Germer (electrons instea of photons: The first evience of De Broglie s hypothesis came in 95 when Davison an Germer scattere electrons off a Ni crystal. The spacing for Ni was known from X-ray scattering to be.9 nm. Davison an Germer use with a kinetic energy of 54 ev. The De Broglie wavelength of an electron with a given kinetic energy is h hc hc hc λ p cp ( KE + me c ( mec KE + ( mec KE For non-relativistic electrons KE << m e c an hc hc λ KE m c << e. KE + ( mec KE ( mec KE For 54 ev electrons λ.67 nm an λ.67nm sinφ.98 an φ (.9nm 65 o. 5 o The etector angle is relate to the grazing angle by o π φ 5. The value of 5 o agree with the experimental ata! Note that the interference pattern occurs even with only one electron in transit at a time! sinφ "etector angle φ φ φ φ sinφ φ Detector ngle Number of Scattere 54 ev Department of Physics Chapter5_7.oc University of Floria

8 Inescapable Consequence of Wave Packets Wave of what? 3 De Broglie postulate that the pilot wavepacket governs the motion of the electron. x However, the electron is a point like elementary particle (i.e. raius zero!. So - where exactly is the electron within the pilot - wave-packet? The electron moves with spee -3 v group somewhere within the pilot wave packet x given by k ω ( x, cos x t sin( kx ω P( x, sin( kx ω. The term P(x, escribes the wave-packet. s a crue guess lets say that at fixe time t the electron must be somewhere in the region x x n+ - x n, where the position of the n th noe (i.e. zeros of the wave-packet is given by kx n ω t ( n + π. Thus at fixe time t, (( n + + π ωt (n + π + ωt π x xn+ xn k k an hence x k π. Here x represents an uncertainty in the electrons position at time t an k represents an uncertainty in the electrons precise wave number. Due to the nature of wave superposition asking for a small uncertainty in the position x of the electron requires a larger k an hence a bigger uncertainty in the electrons wave number. Similarly at fixe position x, 3 Wave Packet Wave Packet electron t v group v group t fixe time t t ω π. electron Here t represents an uncertainty in the time an - -3 ω represents an uncertainty in the electrons t angular frequency. gain ue to the nature of wave superposition asking for a small uncertainty in the time t requires a larger ω an hence a bigger uncertainty in the electrons angular frequency. - t fixe x Department of Physics Chapter5_8.oc University of Floria

9 Opposite to Classical Mechanics! Heisenberg Uncertainty Principle (97 Due to the nature of wave superposition an wave packets we see that x k π an t ω π. The energy an momentum of a particle are given by E h ω p x hk x an hence we see that x p x h an t E h where p x is the uncertainty in the x-component of the momentum an E is an uncertainty in the energy. Thus, there is a funamental limit on the ultimate precision with which we can know both the position (xcoorinate of a particle an its momentum (x-componen. In aition, a measurement of a particles energy performe uring a time interval t must be uncertain by an amount E. Exact Uncertainty Relations: We mae a crue approximation in eriving the above relations. Later we will erive the precise form of the uncertainty relations: h h h h x px y p y z pz t E, where correspons to the root-mean-square eviation from the mean (i.e. stanar eviation. Note that this correspons to a lower limit. It is always possible to know things less well. Uncertainty as a Function of Time: If at t we have localize the particle to within x, then at t p x h/ x an v x h/(m x. t a later time t, x v x t ht/(m x an hence,. The better we know the particle s position at t (i.e. smaller x, the worse we know it s position at a later time t (i.e. larger x.. The uncertainty in the particle s position x increases with time. In classical physics you can know precisely the position an momentum of a particle (i.e. no limit on the precision. Furthermore, if you know the position an momentum of an isolate particle at t, then the exact position of the particle can be preicte for all future times! Department of Physics Chapter5_9.oc University of Floria

10 Center-of-Mass Frame (before ecay m e v Virtual Particles Center-of-Mass Frame (after ecay electron p e photon γ p γ Energy an momentum conservation forbi an electron from ecaying into another electron plus a massless photon. This is easiest to see in the center-of-mass frame. Energy conservation gives me c Ee + Eγ ( cp + ( mec + cp, where I have use momentum conservation to set p e p γ p an E γ cp. We see that m ( me c cp ( cp + ( mec Thus, ( me c + ( cp ( mec ( cp ( cp + ( mec an ( m e c ( cp, which implies that either cp E γ or m e. Hence, the process cannot occur. Similarly, an electron with mass m e cannot absorb a photon an remain an electron with mass m e. e c cp ( cp + ( mec an However ue to the uncertainty principle an electron can emit a photon an remain an electron for a short time t h/ E, where the E is the amount that energy conservation has been violate. These virtual photons form a clou aroun the electron. Electrons are constantly emitting an absorbing virtual photons. This clou is the classical electric fiel. Time virtual particle γ t ~ h/ E Space The virtual photon can be absorbe by another electron (within the time allowe by the uncertainty principle in such a way that the overall energy an momentum is then conserve. These virtual photons carry the information about the force between the two electrons. Time Q em γ Q em t ~ h/ E Virtual particles are the basic quantum (i.e. carriers of the forces between the real particles. Space Department of Physics Chapter5_.oc University of Floria

11 Feynman Diagram + + QED an the Quantum Vacuum QED Quantum Electroynamics Momentum-Space p 3 p 4 Q em γ Q em time Vertex (QED ~ p q ~ Q em γ p ~ ~ p Space-like virtual photon ~ p p ~ ~ p +q ~ 4-Momentum Space: t a vertex aitive quantum numbers are conserve an the 4-momentum is conserve, but particles may or may not be on their mass shell. r r ~ m p E p p ~ r r m p E p p ( on shell real particle ( on shell virtual particle The invariant mass of a virtual particle can by positive (i.e. time-like or negative (i.e. space-like. The Quantum Vacuum: The uncertainty principle makes the vacuum a very complicate object! The vacuum is an object with zero energy. But there is an uncertainty in the zero an everything that is not forbien by other conservation laws happens for short times t h/ E, where the E is the energy of the state. The energy E is borrowe from the uncertainty principle, but it can only be borrowe for the time allowe by the uncertainty principle. The vacuum we live in is full there virtual states being born an then ying. The classical vacuum containing nothing oes not exist! Real particles moving through the quantum vacuum interact with the virtual states an are affecte by the vacuum. If you a energy to the vacuum a virtual state can absorb it an thereby become real, since energy an momentum can now be conserve. For example, it is possible to collie two photons an prouce an electron-positron pair, γ + γ e + +, an you can prouce a protonantiproton pair by annihilating an electron-positron pair, Time e + e p + p +. Virtual particleantiparticle pair! e + t ~ h/(m e c Space Department of Physics Chapter5_.oc University of Floria