From Classical to Quantum mechanics

Transcription

1 From Classical to Quantum mcanics Engl & Rid vrij Univrsitit amstrdam Classical wav baviour Ligt is a wav Two-slit xprimnt wit potons (81-85) 1

2 On sourc Intrfrnc sourcs ttp:// ttp:// On narrow slit On wid slit ttp:// ttp:// narrow slits Two-slit xprimnt wit ligt ttp://

3 Non-classical baviour of ligt Non-classical baviour of ligt Two-slit xprimnt wit potons (81-85) Blac body radiation (77-78) Hot Stl and Pyromtr Spctral nrgy dnsity de/df Enrgy dnsity low tmpratur deω ω 3 π c P5.67 x 10-8 T 4 [W/m ] 300 K 600 K 900 K 100 K ω ω B T Poton nrgy [V] 1 V nrgy is Hz T nrgy of potons is quantizd E f ω deω ω π c 3 ω ω B T 1 Y Axis Titl Bacground blacbody radiation deω ω π c 3 ω ω B T 1 T.76 K X Axis Titl ttp:// 3

4 K variations Non-classical baviour of ligt Two-slit xprimnt wit potons (81-85) Blac body radiation (77-78) Potolctric ffct (79-80) Gorg F. Smoot Jon C. Matr Potolctric ffct 1 Potolctric ffct 0 0 Potolctric ffct 3 Potolctric ffct 4 0 Wa ngativ voltag 0 4

5 Potolctric ffct 5 Exprimntal rsults _ Larg ngativ voltag 0 + Stopping potntial (V) Potassium Tungstn Potassium Tungstn / / x x x10 15 Frquncy of ligt (s -1 ) Non-classical baviour of ligt X-rays Two-slit xprimnt wit potons (81-85) Blac body radiation (77-78) Potolctric ffct (79-80) X-ray production _ + Willm Rontgn s laboratory Willm Rontgn potograps ttp:// ttp:// 5

6 Modrn X-ray tub X-ray spctra Targt Bryllium window Focus cup Tungstn filamnt Intnsity Poton nrgy (V) Non-classical baviour of ligt T positiv lctron Two-slit xprimnt wit potons (81-85) Blac body radiation (77-78) Potolctric ffct (79-80) X-ray production Pair production Pair cration discovry Bubbl cambr 6

7 Pair cration simulation Pair production nuclus positron poton lctron Non-classical baviour of ligt Compton s papr Two-slit xprimnt wit potons (81-85) Blac body radiation (77-78) Potolctric ffct (79-80) X-ray production Pair production Compton ffct Compton ffct Assumption: poton as momntum /λ y λ λ cos m c Θ x ϕ ( 1 Θ) cos Θ + γmu cosϕ ' λ λ sin Θ γmu sin ϕ ' λ λ c c + mc + γmc ' λ λ y λ λ cos m c Θ x ϕ ( 1 Θ) 7

8 Non-classical baviour of ligt Two-slit xprimnt wit on poton at a tim Two-slit xprimnt wit potons (81-85) Blac body radiation (77-78) Potolctric ffct (79-80) X-ray production Pair production Compton ffct Two-slit xprimnt wit on poton (81-85) Non-classical baviour of ligt Conclusions for ligt Two-slit xprimnt wit potons (81-85) Blac body radiation (77-78) Potolctric ffct (79-80) X-ray production Pair production Compton ffct Two-slit xprimnt wit on poton (81-85) E f ω E f ω E f ω E f ω p mv λ Singl poton as still a wav caractr Dpnding on t typ of xprimnt ligt can b dscribd as a wav or as particls T ligt particls ar potons. T nrgy of a poton wit an angular frquncy ω is T momntum of a poton wit wav vctor is E ω p Non-classical baviour of particls Non-classical baviour of particls Two-slit xprimnt wit lctrons (81-85) Davisson and Grmr xprimnt (81) Elctron microscop (lctur) Nutron diffraction (lctur) 8

9 Non-classical baviour of particls Two-slit xprimnt for lctrons Two-slit xprimnt wit lctrons (81-85) Two-slit : quantitativ analysis Two-slit : intrfrnc Amplitud Two-slit : dstructiv intrfrnc Two-slit : constructiv intrfrnc 9

10 Elctrons+potons Non-classical baviour of particls Ψ n lctrons Two-slit xprimnt wit lctrons (81-85) Davisson and Grmr xprimnt (81) potons Davisson original papr Dtail of t original papr Davisson-Grmr xprimnt Exprimntal rsults θ d 10

11 Davisson-Grmr xprimnt Davisson-Grmr xprimnt 3 X θ θ d d Elctron scattring at t surfac Non-classical baviour of particls θ Two-slit xprimnt wit lctrons (81-85) Davisson and Grmr xprimnt (81) Elctron microscop (lctur) d d sinθ nλ Scanning lctron microscop Non-classical baviour of particls λ mv Two-slit xprimnt wit lctrons (81-85) Davisson and Grmr xprimnt (81) Elctron microscop (lctur) Nutron diffraction (lctur) 11

12 Elctron scattring in t bul Nutron diffraction θ Nutrons av also a wav caractr D D cosθ nλ Conclusions for particls Two-slit xprimnt wit lctrons (81-85) Davisson and Grmr xprimnt (81) Elctron microscop (lctur) Nutron diffraction (lctur) ω E p Gussing Scrödingr s quation for a fr particl Wav-particl duality Wat av w larnd so far? From wav to particl From particl to wav Scrödingr quation p mv λ E ν ω On would also li to av 1 p E m vparticl m ω m λ p E ν p E ω ( ) 1 p ω E m vparticl m m 1

13 Wat ar w looing for? A wav so tat: T intnsity of t wav is proportional to t dnsity of particls ω m Ψ n Tis wav sould b solution of a wav quation u t u x V u u α β t x ω V αω β ω m Nota bn: tis is only possibl wit a complx wav i(x-ωt). It dos not wor wit sin or cosin functions How to coos α and β? Insrt i(x-ωt) into α ( i) β m u u i m x t u u α β t x ( ) α( ω) β( )( ) β ix ωt ix ( ωt) ix ( ωt) i i i ω m Ψ Ψ i m x t Hisnbrg s uncrtainty principl As i(x-ωt) is a complx wav w cannot simply ta t squar, but Ψ n Ψ ( x, t) Fr particl wav function i A ( x ωt ) is a solution of Scrödingr s quation Ψ Ψ i m x t ( x ωt ) i( x ωt ) i Ψ( x, t) A A A Many i ( x ) ωt ar ncssary to localiz a wav function x x 1 Ψ constant Tis is not ralistic for a particl 13

14 Hisnbrg s Uncrtainty Principl Hisnbrg s Uncrtainty Principl x 1 ( ) x p x x p x E x t 3 tims quantum pysics Elctron tunnling p x Hisnbrg s uncrtainty principl Spin of t lctron Atoms cannot b vanisingly small Many i ( x ) ωt ar ncssary to localiz a wav function V g d ω m Vg d m p mv v m m particl T vlocity of a particl is givn by t group vlocity V g d ω m Vg d m p mv v m m particl 14

15 T group vlocity is compltly diffrnt from t pas vlocity V pas ω ω m ω Vpas m p mv v m m particl V g d ω m Vg d m p mv v m m particl Gussing Scrödingr s quation for a particl in a potntial Potntial nrgy Total nrgy F E total E intic + U x v ½mv +½Kx x x Forc F-Kx U ½Kx F U ½Kx Wor ÛFdx - Potntial nrgy U x x How to introduc potntial nrgy? Stationary stats Insrt i(x-ωt) into Ψ Ψ i m x t givs W loo for solutions of t tim-dpndnt Scrödingr quation Kintic nrgy m ω Kintic + nrgy Total nrgy in t form Ψ Ψ + U( x)ψ i m x t Ψ ( x, t) Φ( x) Et i and find m U x + ( ) ω Ψ Ψ + U( x)ψ i m x t Φ + U( x) Φ EΦ m x Tis is t tim-indpndnt Scrödingr quation for stationary stats (standing wavs) 15