Lecture: Experimental Solid State Physics Today s Outline

Transcription

1 Lecture: Experimental Solid State Pysics Today s Outline Te quantum caracter of particles : Wave-Particles dualism Heisenberg s uncertainty relation Te quantum structure of electrons in atoms

2 Wave-particle dualism Te Wave-Particle Dualism : 1. Particle-caracter of waves

3 Wave-particle dualism: Particle-caracter of waves 1 Planc s Quantum Hypotesis : Rayleig-Jeans: Considers radiation already as modes (standing waves) in a cavity!!! Planc : 1. Energy Transfer is not continuous ω ν 8πν ( ν ) dν 3 c Tdν. Te smallest transferable unit depends on ν (to avoid UV-Catastrope) 3. Te smallest unit is ν ( is up to now only an auxiliary parameter Hilfsgröße ) 4. An oscillator can only be excited n times (n1,,3,...) to an energy of W ν n ν 5. Te oscillation is occupied according to Boltzmann : p( W ) ~ e W T e n ν T ν Planc s Radiation Formula : 3 8πν dν ων ( ν ) dν 3 ν / T c e 1 Birt of Quantum Mecanics & First int to none-continuous nature of ligt

4 Wave-particle dualism: Particle-caracter of waves Potoelectric effect : W.Hallwacs (1895) P.Lenard (19) A.Einstein (195) max E in ν Φ Quantum-lie absorption of ligt ligt as a particle: Poton

5 Wave-particle dualism: Particle-caracter of waves 3 Compton scattering : c m c ( ) φ λs λ + λc sin Compton s λ wit m scattering formula Compton wave lengt of te electron Poton Intensity Scattering of electrons and potons : ligt as a particle Impulse transfer from poton particle as wave Electron is expressed in terms of a wavelengt

6 Wave-particle dualism: Particle-caracter of waves 4 A simple experiment on single potons : Contradiction: Ligt as a wave will be emitted in all directions Ligt as a particle as a certain distribution probability in space Experiment of Taylor : On ig intensities (e.g. lots of potons) te signal on te detectors is uniform! On low intensities (dw / dt << ν / τ wit τdetectorresolution), detectors are measuring single potons Intensity Intensity Time Intensity Time Signals are statistically distributed Time Averaging over time sows te same intensity on all detectors

7 Wave-particle dualism: Particle-caracter of waves 4 A simple experiment on single potons : Contradiction: Ligt as a wave will be emitted in all directions Ligt as a particle as a certain distribution probability in space Experiment of Taylor : On ig intensities (e.g. lots of potons) te signal on te detectors is uniform! Intensity On low intensities (dw / dt << ν / τ wit τdetectorresolution), detectors are measuring single potons Intensity Time Intensity Time Signals are statistically distributed Time Averaging over time sows te same intensity on all detectors

8 Wave-particle dualism Te Wave-Particle Dualism :. Wave-caracter of particles

9 Wave-particle dualism: Wave-caracter of particles Te idea of de Broglie (194) : If ligt as bot caracteristics, wave and particle, ten also classical particles can ave wave caracteristics Tis was not observed before 194! A ind of a new Unifying Teory, lie in mecanics and termodynamics: Te unification of waves and particles! de Broglie: Just a formal assignment of impulse p and wave vector : π p λ λ Every particle as a de-broglie- wavelengt : λ de Broglie p m v m E in Is tis true? Can one proof tis?

10 Wave-particle dualism: Wave-caracter of particles 1 Electrons : Fraunofer diffraction of electrons at an edge, in comparison wit ligt Nobel prize for de Broglie in 199 Example: De-Broglie-wavelengt of electrons after acceleration wit 1V: λ / (m eu) m.1nm MgO λ.1nm Possible Application???

11 Wave-particle dualism: Wave-caracter of particles 1 Electrons : Fraunofer diffraction of electrons at an edge, in comparison wit ligt Nobel prize for de Broglie in 199 Example: De-Broglie-wavelengt of electrons after acceleration wit 1V: λ / (m eu) m.1nm MgO Application: LEED LEED Setup Principle LEED on Si Si(111) Si(1) l n λ de Broglie

12 Wave-particle dualism: Wave-caracter of particles 1 Electrons : Double Slit Experiment wit electrons: Te Bi-Prism of Möllenstedt and Düer Interference structure even for single electrons (e.g. only one electron passes te slits or te wire) Te electron must pass bot sides of te wire (or go troug bot slits on te same time)!

13 Wave-particle dualism: Wave-caracter of particles Neutrons : Predicted by Ruterford (from is scattering experiments) Discovered by J.Cadwic (193) from 9 Be + 4 He 1 C + n Example: De-Broglie-wavelengt of neutrons after acceleration wit 1V: λ / (m eu) m.3nm slow (5meV) neutrons are needed for diffraction on matter Properties of neutron diffraction can penetrate eavy matter up to several cm are sensitive to ligt elements (e.. H) in eavy environment (e.g. metals) sensitive to isotopes

14 Wave-particle dualism: Wave-caracter of particles Neutrons : Even Total Diffraction (lie nown from ligt) is possible Special Neutron Conductors were developed at te HMI (Helmoltz- Zentrum) Multi-layer coating of te conductors is made of alternating layers of pure 58 Ni and 6 Ni in a special order

15 Wave-particle dualism: Wave-caracter of particles 3 Atoms : Example: De-Broglie-wavelengt of α-particles after acceleration wit 1V: λ / (m eu) m.1nm atoms must be even slower (colder) tan neutrons, to observe teir wave nature Depending on te number of protons and neutrons, atoms can be fermions or bosons!

16 Wave-particle dualism: Wave-caracter of particles 3 Atoms : Te Bose-Einstein Condensation Requirement for Bose-Einstein Condensation : De-Broglie wavelengt must be larger tan atom distance Oter (electronic) interactions must be ruled out Estimation of te temperature needed (e.g. 3 Na) : Interaction-free 3 Na vapor: Density must be < 1 14 per cm 3 (solid Na as.6 1 atoms per cm 3 ) Mean atom distance is (1 /m 3 ) -1/ m nm For λ de-broglie nm T must be below 8µK! Only possible in atom traps wit Laser cooling and Evaporation cooling

17 Wave-particle dualism: Wave-caracter of particles 3 Atoms : Te Bose-Einstein Condensation Properties of Bose-Einstein Condensation : All atoms occupy a single quantum state All atoms can be described wit one single wave function Long-range interference of condensates is possible Quantum-mecanical effects appear on a large scale (e.g. supra-fluidity / spin conservation) no Vortex 1 Vortex 8 Vortices 1 Vortices

18 Wave-particle dualism: Wave-caracter of particles Comparison: de-broglie wave lengt of different particles λ de Broglie m E in Electron Neutron He-Core 3Na Ein in ev m 9,11E-31 1,67E-7 6,65E-7 3,8E-6 T E / Bolzmann λ (de-broglie) 131,9 µm 3,75 µm 1,544 µm,644 µm 8,65E-11 ev 1, µk 77,6 Å 1,81 Å,98 Å,379 Å,5 ev 9,1 K 1,3 Å,9 Å,14 Å,6 Å 1 ev 1,16E+6 K,13 Å,3 Å,1 Å,1 Å 1 ev 1,16E+8 K 6,63E-34 J s e 1,6E-19 C Te iger te mass, te lower te λ quantum-mecanical effects are dominating on low masses or low temperatures! At low temperatures : Long-range interference of condensates is possible Quantum-mecanical effects appear on a large scale (e.g. supra-fluidity / spin conservation)

19 Heisenberg s Uncertainty Relation Matter Waves and Wave Functions Heisenberg s Uncertainty Relation

20 Heisenberg s Uncertainty Relation A wave function for a particle : 1. Plain waves Formula for plain waves : Wit : i( ωt x) ψ ( x, t) C e E E ω π ν π Einstein : E ν π π p p de Broglie : λ λ p Formula for a particle : ψ ( x, t) i ( Et px) C e But: Plain wave is not localized lie a particle!

21 Heisenberg s Uncertainty Relation A wave function for a particle :. Wave pacets of plain waves A wave pacet is a superposition of plain waves Example: plain waves (j) See Animation...,5, 1,5 1,,5 Gruppe: Anfang Pase: Anfang ψ ( x, t) ξ,5 cos 1 ξ,5 cos 1 ξ ξ + ξ cos 1 ( ω1t z) ( ω t z) j C mit mit ω ( ω t z) cos t z m j e m i( ω t j j x) ξ, -,5-1, -1,5 -, z

22 Heisenberg s Uncertainty Relation A wave function for a particle :. Wave pacets of plain waves A wave pacet is a superposition of plain waves Example: plain waves (j) Now we ave new parameters : Pase velocity: Group velocity: λ ω vp T v Gr ω j>1 v P ω ψ ( x, t) mean mean j C j e i( ω j t j x) Te group velocity is te velocity of te particle! For better localization we need to superpose an infinite number of plain waves (j )...

23 Heisenberg s Uncertainty Relation A wave function for a particle : 3. Wave pacets of number of waves Te sum converts to an integral : wit Te group velocity converts to a Differential : Now we ave to calculate Ψ!!! ( ) m m m p E 1 1 ) ( ω ω + x t i d e C t x ) ( ) ( ), ( ω ψ v m p m d d v Gr ω

24 Heisenberg s Uncertainty Relation A wave function for a particle : 3. Wave pacets of number of waves Te sum converts to an integral : ψ ( x, t) + C( ) e i( ω t x) d is small, compared to (see animation before) C() will not cange muc C() C( ) Taylor-Evolution for ω: ω ω + dω/d (- ) +... ψ ( x, t) C( Integration gives : ψ ( x, t) C( Ψ ~ sin(x) / x! + i( ω t x) i( ω ' t x) ξ ) e e ) e i( ω t x) sin dξ [ ( ω' t x) ] ω' t x

25 Heisenberg s Uncertainty Relation A wave function for a particle : Relation to Fourier Transformation We ave done noting else tan a Fourier Transformation : Our integral : Fourier Transformation : ψ ( x, t) + C( ) e i( ω t x) d + iωt f ( t e dt F( ω ) ) But: Our f(ω)c() was constant inside te interval : Te resulting function ψ ( x, t) C( ) e i( ω t x) sin [ ( ω' t x) ] ω' t is te Fourier-Transformed of te Box-Function C()! x C() See also Diffraction at a slit and Huygens Principle

26 Heisenberg s Uncertainty Relation A wave function for a particle : 3. Wave pacets of number of waves Example of Re[Ψ(x,t)] Ψ itself is not a pysical value! (it can get imaginary) Interpretation of Ψ: Ψ gives te spatial probability distribution to find te particle in dx : W ( x, t) dx wit + x ψ ( x, t) ψ ( x, t) dx dx 1 Origin-Animation: wave-nature particle-nature (Normalization) Consider, tat is an impulse difference, ten x ~ 1/!!!!!!!!!!!!!!!!!!! Re[Ψ(x,t)] [arbitrary units] [Ψ(x,t)] [arbitrary units] 1,,5, -,5-1, 1,,8,6,4,, F1 Re(Ψ(x,t)) for F x [arbitrary units] [Ψ(x,t)] for x [arbitrary units]

27 Heisenberg s Uncertainty Relation Heisenberg s Uncertainty Relation : Remember: Ψ(x,t) is just constructed from infinite plain waves in an interval of ± around, eac wit te impulse p ten represents an impulse uncertainty witin Ψ(x,t) Considering, tat a particle (wic is represented by Ψ(x,t)) is limited by te first null of Ψ(x,t): x π / Te particle widt is x x x x π π x π Wit p or p : p x x π x p As tis is a minimal condition, it follows : x p or t E Tis is a direct result from te wave nature of matter! [Ψ(x,t)] [arbitrary units] 1,,8,6,4,, F x [Ψ(x,t)] for 1.5 x π / x [arbitrary units]

28 Heisenberg s Uncertainty Relation Heisenberg s Uncertainty Relation : Consequences Is te reason for te existence of potons : min. time t to syncronize an oscillator wit an electric field is 1 period: T 1/ν energy wat belongs to tat time is E t E / t /ν -1 ν Tere is no E! Harmonic Oscillator as a minimal energy > Fluctuations in vacuum (Dar Energy???) Virtual particles (Lamb-Sift / Casimir-Effect) Radioactive decay (W and Z Bosons) E in E pot E Heisenberg Tere is a minimal orbit radius for electrons in a Coulomb potential : E Klass E Kin + E Pot p m e e 4π ε r wit r p E 1 m e r e 4π ε r Min r Min 4πε m e e r Min is nearly Bor s Radius, except of instead of!

29 Te quantum structure of electrons in atoms Te quantum structure of atoms

30 Te quantum structure of electrons in atoms Kircoff & Bunsen (1859): Atoms can only emit or absorb discrete energies! Several series of te H-Atom were nown : - Lyman (196) VUV - Balmer (1885) visible - Pascen (198) IR - Bracett (19) IR - Pfund (194) far IR Empirical teory of Balmer : {Ry Rydberg constant} 1 1 ν Ry λ n 1 n n 1 1 n 3, 4,... Applies also for te oter series

31 Te quantum structure of electrons in atoms Te atom model of Niels Bor µ v 1 From te condition centripetal force Coulomb force... r 4π ε Z e... te radius is given : r {µ Reduced Mass of electron and nucleus} 4π ε µ v So far: Every radius r is possible! Niels Bor: Electron is Standing Wave around te nucleus: π r n λ debroglie Wit λ debroglie /(µ v) te discrete radii are: Remember: We got Bor s radius already from Heisenberg s Uncertainty Relation! Te discrete energy levels are: E n 1 8 µ Z e ε 4 r 1 n n ε π µ Z e Everyting is fine??? Bor s model explains te discrete levels and te measured data very well but only for single-electron atoms (H, He +, Li +,...) Wy te electrons on teir orbits (accelerated motion) do not radiate? Later observed: fine structure More quantum-mecanical teory is needed... r Bor n Z Z e r

32 Summary Wave-Particle Dualism : Planc: Quantum-lie absorption of ligt de Broglie: A wavelengt can be assigned also to particles Heisenberg s Uncertainty Realtion : Superposition of plain waves gives localized wave functions Tese wave functions sow an inerent uncertainty Te Quantum Structure of Electrons in Atoms : Bor s atom model explains data for H-lie atoms very well But: Tere are some open questions (no radiation from electrons in orbits, fine structure)...