debroglie and his matter waves, and its consequences for physics and our concept of reality Everything is a wave?

Transcription

1 debrglie and his matter waves, and its cnsequences fr physics and ur cncept f reality Everything is a wave?

2 de Brglie s strange idea f matter wave in 94 ( λ, ) r f ( p, E) r ( p, E) wave : radiatin particle : Nature particle : matter Wave :? (λ, f) 93 thesis P h, E hf λ mv p r mv, r E Extend ntatin f wave-particle duality frm light t matter. de Drglie Fr Phtn, P E c Suggests fr matter, hf c h λ λ f h P E h λ h P, f E h de Brglie wavelength de Brglie frequency

3 Why quantizatin f angular mmentum in the Bhr mdel? A hydrgen-lie atm : r The stable rbiting electrn L nh h rmv nh n π n,, 3, psitive integer An integer number f wavelengths fits int the circular rbit n h λ π r where λ de Brglie wavelength P

4 w/. the cnsideratin f special relativity E P c + A particle f charge q and rest mass m is accelerated frm rest by a ptential difference V. ( ) ( m c + KE m c qv) 4 E P c + mc + P λ h P qvm c + q qvm h qvm hc c + V + q c V qv m c / hc qvm c + q V qvm c m c / 4 in a nn-relativistic limit v<<c KE qv << m c P / qv + m c γm v mke and λ fr nn-relativistic mtin h qvm h m KE

5 What is the scale f wavelength fr a baseball? m0.45g, v43m/s h λ P Jule sec 0.45g 43m/sec N way t detect! m ~ 0-5 nm >> the finest aperture ~ 0.nm Hw abut electrns? KE0eV h λ m E e Jule sec 30 8 ( g)(.6 0 Jule) ~0.39nm Typical wavelength ~ interatmic spacings Suggesting we l fr interference in scattering frm crystals, similar t scattering f X rays. 0 m

6 The Davissn-Germer Experiment in 97 clear-cut prf f wave nature f electrns 95 Elsasser : electrn diffractin wuld explain the slw electrn scattering experiments 97 Davissn Davissn and Germer at the Bell lab. In USA Thmsn in England. perfrmed the experiments 937 The Davissn and Germer apparatus Thmsn Three parameters : Electrn energy V Ni target rientatin Scattering angle φ Scatter X-rays frm Ni Crystal lattice spacing d0.09nm α

7 Scatter electrn KE54eV in nn-relativistic regin λ h m E e 30 9 ( g)( Jule) 0.67nm Jule sec Scattering intensity fr 54eV electrns A cnstructive pea 50 Atmic planes Incident electrn φ/ Bragg cnditin φ cs λ d d cs(φ/) Reflective electrn φ d cs nλ 0.67 cs 0.8 d 47

8 Kept detectr at a fixed angle φ and varied the accelerating vltage V φ d cs nλ cnst. h h λ me m ev λ V Cnstructive interference Current pea cnst. ( ) n cnstant n Experiment f Davissn and Germer cnfirmed that lw speed particles with mass (v<<c) d have wave-lie prperties. Thmsn : bserved diffractin pattern f high speed electrns thrugh thin crystalline films. Independent cnfirmatin f de Brglie s idea, matter wave. taen accunt f the special thery f relativity 0~40eV

9 Wave-lie prperties f neutral atms and mlecules? (ther than electrns) 930 Estermann, Stern, and Frisch Prjectile : He r H Target : LiF crystal Observed diffractin pattern Kinetic energy? 3 Thermal energy h h λ P mke B T ( 7 )( g Jule) He 34 3 KE ( meV/K)( 300K) 38.8meV Jule sec Å S lng the Bragg relatin is hld, the diffractin pattern can be bserved. All matters have wave-lie prperties. 0 m

10 The Electrn Micrscpe by Knll and Rusa in 933 Optical micrscpe Transmissin electrn M. electrns Lenses? Magnetic fcusing 93 Rusa a micrscpe that can magnify very small details with high reslving pwer due t the use f electrns as the surce f illuminatin, magnifying at levels up t,000,000 times.

11 Rusa s electrn micrscpe in 934 JEOL 00F

12 TEM is a tl fr diffractin and imaging with spectrscpy in materials science and engineering t characterize the micrstructures f materials, including mrphlgy, crystal structure, and cmpsitins, which are essential fr understanding the prcessing, structure and prperties. Si wire A chain f endhedral Gd-metallfullerenes (Gd@C8) is encapsulated in a SWNT. 3 nm nm Science 90, 80 (000).

13 Nanlithgraphy : the transference f a desired pattern nt a substrate Late 970s : began t cnfine the lateral mtin in the plane f DEG using lithgraphic techniques Mdern IC technlgy E-beam lithgraphy the mst ppular technique fr nan-meter structures The characteristic wavelength fr electrn at energies 0~30eV : λ 34 h Js. 0 m me ( -30 )( g J ) The ultimate reslutin ~ 0-0 nm (inelastic lss prcesses as high energy electrns either lse energy t r are reflected by the substrate) Current ppularity f e-beam lithgraphy can be ascribed t the fact that it can be perfrmed with relatively simple mdificatin t a SEM thrugh cmputer cntrl f x, y psitin f the electrn beam.

14 substrate cating PMMA n the substrate (ply-methyl-methacrylate) Electrn beam Expsuring PMMA by e-beam Develping (MIBK+IPA) Methylisbutyletne (slvent)+ isprpal alchl slutin) Depsiting metal Lift-ff prcess (Acetne)

15 Wave grups and Dispersin Tward a Wave descriptin f Matter Particle Large prbability t be fund in a small regin f space at a specific time t m v Wave representatin Wave grup r summed cllectin f waves with different wavelengths: amplitudes and relative phases chsen t prduce cnstructive interference in small regin. v g v assciate grup velcity with velcity f particle Hw can we btain a lcalized wave grup?

16 verlap f tw waves ( x, t) A cs( x t) ( x, t) A cs( x t) y y π λ πf wavenumber angular frequency Out f phase: destructive interference In phase: cnstructive interference superpsitin f tw waves y ( x, t) y ( x, t) + y ( x, t) + + A cs x t cs using trignmetric identity x t

17 A snapsht ( ) ( ) t x cs t x A cs t, y x,, + + Averages Differences Within the envelp, wave has velcity ( ) ( ) v p + + The envelp (grup f cmpnents) has velcity v g

18 Other characteristics f the envelp (grup f cmpnents) x π t π where x is a limited extent in space f an envelp where t is a limited duratin in time f an envelp In general, pulses f any ind have this uncertainty relatin. (reciprcity relatin) x The uncertainty in spatial extent f a pulse is inversely prprtinal t the range f wave number maing up the pulse. t The uncertainty in tempral extent f a pulse is inversely prprtinal t the range f wave frequency maing up the pulse. Bth cannt becme arbitrary small, but as ne decreases the ther increases.

19 (a) (b) time increasing (d) (e) v grup velcity g d d () time increasing (c) (d) (f) (g) phase velcity v p

20 A pulse Cnstructing a truly lcalized pulse requires a summatin f an infinite number f harmnic waves w/. cntinuusly varying wavelengths and amplitude. f(x, t) Lcalized wave grup π ( ) ( x t e i ) d A Amplitude f cntributing wave at wavenumber

21 Peridic functin Nn-peridic

22 Dispersin The individual harmnic waves that frm a pulse travel at different phase velcities and cause an riginally sharp pulse t change shape and becme spread ut, r dispersed. In a dispersive medium the grup velcity can be less r greater than the phase velcity, depending n the sign f dv p /d. d v g vp + d Nn-dispersive? Media in which phase velcity des nt vary with λ. Such as vacuum fr EM waves. d v p d

23 Ψ(x) ~ Ψ ~ x ix i ( ) e d Ψ( ) Ψ(x) e dx

24 A matter wave pacet m v v g h de Brglie s pstulate : λ, P π P E, πf λ h h E d de vp, vg P d dp f E h Free particles in the nn-relativistic regime mv E de mvdv E, P mv vp v, vg v P dp mdv Free particles in the relativistic regime v E γmc P γmv E P p c v v g de dp mdv mdv ( ) 3/ mvdv v c ( ) / ( ) ( v c + v c mdv v c ) mvdv ( v c + v c ) The grup velcity f the matter wave the velcity f the particle. v 3/

25 Furier Integrals f(x) π a ( ) e ix d A spatially lcalized wave grup. a() π f ( x) e ix dx Amplitude f the wave with wave number. V(t) π g ( ) e i t d The strength f signal as a functin f time. g ( ) V(t) e i t dt π The spectrum f the signal.

26 A truncated sinusidal wave ( ) t V(t) g t d e i π (a ) Spectrum t t t d e e T T i i π () < < therwise 0 t V t t T T e i ( ) ( ) ( ) ( ) ( ) π π T i T i T T i i e e i e t ( ) ( ) π T sin g( T π Uncertainty relatin π π T T t

27 A square wave f(x) π a ( ) e ix d π + x e i d a ( ) 0 < < + therwise x e i π i π + ( x ) ( x ) sin e i x π Uncertainty relatin 4π x 4π

28 The Heisenberg Uncertainty Principle in 97 It is impssible t determine simultaneusly with unlimited precisin the psitin and mmentum f a particle. If a measurement f psitin x is made with an uncertainty x and a simultaneus measurement f mmentum P x is made within an uncertainty P x, then the precisin f measurement is inherently limited by Heisenberg P x x h/ (mmentum-psitin uncertainty) Similarly, E t h/ (energy-time uncertainty) Physical rigin? Explre fundamental cnsequences f uncertainty principle thrugh idealized experiments

29 illuminating electrn t detect it (phtn cllides w/. the electrn) Measuring prcess

30 Basic idea electrn at rest, struc by single phtn scattered phtn cllected thrugh lens f micrscpe at angle α scattered electrn carries mmentum alng the x-axis The maximum change in mmentum f phtn after the scattering P phtn x P e x p sinθ h sinθ λ h sinθ λ cnservatin f mmentum What abut the lcatin alng x-directin f electrn? diffractin pattern x e λ sinθ reslving pwer f a micrscpe in a reasnable agreement e e h λ Px x sinθ h λ sinθ A measure f the minimum uncntrllable disturbance.

31 Can we d better? Increase aperture θ P e x h sinθ λ Decrease wavelength λ h sinθ P e x λ reduce diffractin limit n reslutin x e λ sinθ increase phtn energy & mmentum x e λ sinθ Prcess f measurement itself disturbs system in such a way as t mae it impssible t nw x and P x with precisin better than given by uncertainty principle. Nwadays it is mre widely accepted that quantum uncertainty (lac f determinism) is intrinsic t the thery.

32 97 Heisenberg and Bhr first shwed hw essential the cncept f prbability is t unin f wave and particle descriptins f matter and radiatin. Prbabilistic view is the fundamental ne in quantum physics. Unlie The basic laws are deterministic in classical physics. The wave-particle duality Principle f cmplementarity by Bhr The wave and particle mdels are cmplementary. If a measurement prves the wave character f radiatin r matter, then it is impssible t prve the particle character in the same measurement, and cnversely. Which mdel we use is determined by the nature f the measurement. The lin between wave and particle mdel is prvided by a prbability interpretatin f wave-particle duality.

33 Radiatin Pynting vectr r r Wave mdel ( ) ( ) ( ) [ ] S r, t E r, t B r, t watt/m Maxwell s E.M. thery Time average f S intensity I Particle mdel intensity I Nhf Einstein s µ r r r r c E µ N : average number f phtns per unit time crssing unit area perpendicular t the prpagating directin. N ~ E prbability f phtn number density a statistical view Brn Matter wave? A wave functin fr a particle w/. mmentum P and energy E. r Ψ r, t : satisfying the Schrödinger wave equatin Ψ ( ) Late 96 by Brn prbability per unit vlume f finding a particle at a given place and time

34 Tae bullet as prjectiles fr duble slit experiment electrn When cnditin, λ matter ~ d(slit spacing), is satisfied and there are plenty f numbers f incident electrns typical interference pattern fr duble slits appears

35 Duble slit diffractin Yung s duble slit diffractin Interference f tw light waves frm slits as surces Cnstructive interference : r need L>>d d sinθ nλ Lcatin f maximum n screen Fr small angle θ, y sinθ L nλ d Separatin between maxima n screen y nλ L d valid fr any wave incident n duble slits

36 Neutrns: A Zeilinger et al., Reviews f Mdern Physics 60, 067 (988). He atms: O Carnal and J Mlyne, Physical Review Letters 66, 689 (99). C 60 mlecules: M Arndt et al., Nature 40, 680 (999). With multiple-slit grating Fringe visibility decrease as mlecules are heated. L. Hacermüller et al. Nature 47, 7 (004). Withut grating Interference patterns can nt be explained classically - clear demnstratin f matter waves

37 Duble slit diffractin fr matter L0cm Type KE [ev[ ev] m c [MeV] λ [nm] d [nm] y y [cm] Electrn Prtn Neutrn Pin Pin Duble slit diffractin fr phtn λ [nm] E [ev[ ev] d [µm][ m] y y [cm]

38 Duble slits prbability phtn r particle (λ /c) (λ h/p) S P r ( r, t) P + P r Ψ ( r, t) + Ψ ( r, t) r Single slit Prbability, P S S S P ( r, t) Ψ ( r, t) + Ψ ( r, t) Prbability density ψ Prbability density amplitude r Ψ r r ( r, t) beying superpsitin f matter waves ( r r,, t t) r? Ψ ( t), ( t ) P Ψ rr r, r t

39 Classical idea f a particle Smething with definite mass and trajectry prvided by laws f mechanics, given sme initial cnditins. Fr example: we can nw thrugh which slit each particle passed In real subatmic wrld, bserve tw-slit diffractin pattern True even if nly ne particle is near the duble slit at a time. Pattern nt created by interference between particles If we mae measurements t identify which slit each particle passes thrugh we destry the interference pattern! Need a new mechanics that incrprates bth wave and particle nature f subatmic bjects