Chapter 5 Particles and Waves Particle wave dualism for objects primarily known as particles Introduction to Schrödinger equation
Recall: Summary wave particle dualism Electromagnetic waves have particle properties (photoeffect, Comptoneffect etc.) the particle is called photon (γ) rest mass m 0 =0, energy E=hν, momentum p=k localisation in space and time limited by uncertainty relation: p i x i h/2 and E t h/2 Probability to find a photon w r,t r,t 2 and w r r 2 derived by wave equation in 3D: 1 2 r,t c 2 t 2 r,t Linear superposition principle, interference etc 1 2 3... w 2 1 2 3... 2 1 2 2 2 3 2... 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 2
5.1 Experimental evidence 5.1.1 Debye Sherrer diffraction experiment shown in lectures: Debye-Sherrer diffraction from polycristalline carbon e-beam α screen cathode +U target low high electron energy remember diffraction of light: d sin α = m λ hence: λ decreases with particle energy (c.f. the photon case: λ=h/p) 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 3
5.1.2 Low energy electron diffraction (LEED) from single cristal surface 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 4
LEED devices once an today 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 5
low energy electron diffraction (LEED) from single cristal Cu surface 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 6
LEED: 100eV Si(111)7x7 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 7
5.1.3 Standing electron waves on Ag(111) with STM low temperature STM: electron waves around point defects 1. M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature 363, 524 (1993) 2. Y. Hasegawa and Ph. Avouris, Phys. Rev. Lett. 71, 1071 (1993) 3. J. Li, W. D. Schneider, and R. Berndt, Phys. Rev. B. 56, 7656 (1997) electron waves at defects: changing energy 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 8
5.1.4 Scattering experiments with atomic beams 5.1.4.1 Experiment oven target detector detector θ θ 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 9
5.1.4.2Particle diffraction: rainbow scattering I(è)è 7/3 rainbow θ = 0 o 10 o 20 o 30 o 40 o 50 o 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 10
rainbow scattering: Cs on Hg rainbow 0 o 30 o 60 o 90 o 120 o 150 o 180 o θ 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 11
5.1.5 He atom diffraction from grating W. Schöllkopf and J.P. Toennies, J. (MPI f. Strömungsforschung, Göttingen) Chem. Phys. 104, 1155 (1996) Die Abbildung zeigt das Beugungsbild eines Heliumatomstrahls. Es kann mit der aus der Optik bekannten Kirchhoff-schen Beugungstheorie verstanden werden. Das verwendete Transmissionsgitter wurde hergestellt von Tim Savas und Hank Smith, MIT, Mass., USA. 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 12
5.1.6 Diffraction of Fulleren Molecules a large molecule (football): and yet diffraction can be observed: Home page of Prof. Dr. Anton Zeilinger Institut für Experimentalphysik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria (local copy) C:\science\rasmol\rw32b2a.exe D:\wwwhertel\physik3\material\chapter5\fullerene\c60a.pdb 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 13
5.2 De Broglie - 5.2.1De Broglie wavelength In 1923, as part of his Ph. D. Thesis, Louis de Broglie (he was of the French aristocracy - hence the title "Prince") argued that since light could be seen to behave under some conditions as particles (photoelectric effect) and other times as waves (diffraction), we can also consider that matter has the same ambiguity of possessing both particle and wave properties. p k p h/ with p 2mE (non relativistic) h p h 2mE and for electrons: 12.3 Å E/eV This is an empirical relation!!! Very well veryfied by a multitude of experiments All quantum mechanics is based on this relation! Many applications (e.g. in structural analysis) 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 14
5.2.2 De Broglie wave length for relativistic particles however, for electrons we usually have to calculate relativistic E 2 c 2 p 2 m 0 2 c 4 with E 2 E kin E 0 2 p 1 c E 2 E 2 0 1 c E2 kin 2E kin E 0 2m 0 E kin 1 E kin 2m 0 c 2 2m 0 c 2 1 MeV 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 15
5.2.3 Matter waves Probability amplitude r,t Probability to find particle: w r,t r,t 2 Interference effect just as for optical waves e.g. double slit: w r 1 2 2 w 1 w 2 interference term r,t has, however, in comparison to el.mag. rad. no similairly obvious explanation (just probability amplitude, matter waves) We have the same situation as for photons: if we try to trace the path of a particle, the interference is lost! General rule: interference phenoma are observed when different pathways (differentcontributions to the full matter field) are by principle indistiguishable. In contrast: if they can be distiguished - even if only in principle - there is no interference!! 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 16
5.2.4 Uncertainty relation One classical example (Heisenberg): Try to observe an electron with a light microscope resolution of microscope (diffraction limited according to Abbé) illumination by photon objective lens e - x á Äp x x sin for precise observation we have to keep small and large: that implies already a modification of the experimental situation. Photon transfers momentum onto the electron: p x p ph sin h sin p x x h in general the Heisenberg uncertainty relation is valid p x x h 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 17
5.2.5 Insert: theory of microscope according to Abbe total magnification= mag oc mag ob = (s eye /f oc ) (t/f ob ) Abbé explains the primary picture by diffraction f oc ocular f ob t real intermediate picture objective lens focal plane object object with radius x produces a circular diffraction pattern next object must be separated by: x sin resolution in an immersion microscope x n sin 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 18
5.2.6 Uncertainty relation continued Also any other method to localise the electron leads to the same restriction E.g. electron through a slit diffraction sin min x e - x x p p x α p x p sin p x p h/p x p x x h p x x h and E t h exact value depends on definition of 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 19
5.2.7 Limit of the classical trajectory Thus, the classical picture of a well defined trajectory for a particle p classical quantum mechanics x x t and p p t ends here: both quantities are only defined to within the limits of the uncertainty relatio p x x h x terminology: wave packet note: the spreading of the wave packet (Auseinanderlaufen) quantum mechanics only makes predictions about probability (amplitude) w r,t r,t 2 which represents the statistical ensemble 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 20
5.2.8 The simplest example: freely moving particle plane wave with p, E p2 2m k p/ E/ p2 2m k2 2m k (dispersion relation) r,t C exp i t kr C exp i k 2 2m t kr C exp i E t pr note: w r,t r,t 2 C 2 independent of space and time! i.e.: if particle has well defined momentum (energy) it cannot be localised 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 21
5.3 Wave packets Now, we want to describe real particles, i.e. objects which have a certain localisation - at the same time illustratev gr First a few definitions: r,t : matter wave (Materie-Wellen), probability amplitude w r,t r,t 2 probability to find a particle at time t and position r more precisely: probability to find particle in t... t dt and d 3 r is given by w r, t dtd 3 r mass density: r,t m w r,t m r,t 2 charge density: el r,t e 0 w r,t e 0 r,t 2 phase velocity (here 1D): places where the phase is constant i.e. where d dt t kx 0 k dx dt dx dt v ph k k E p 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 22
5.3 Wave packets 5.3.1 Phase and group velocity v ph k k E p and in nonrelativisticapproximation: in contrast to electromagnetic waves, absolute phase (and absolute energy) as well and phase velocity ù/k are arbitrary and without physical relevance E is only defined modulo a constant only Ø(x,t) ² is measurable however, phase differences (and energy differences) can be measured. Another thing is the so called group velocity: with p k we have E p2 2m 2 k 2 2m v gr d dk 1 d dk de dp m p v E p2 2m v ph p 2m v 2 k2 dispersion relation: 2m a very meaningfull result Next, wewantto describerealparticles, i.e. objects whichhaveacertain localisationand a certain velocity(group velocity) v gr 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 23
5.3.2 A very simple wave packet Two similar frequencies (energies) 1 and 2 (with k 1 and k 2 ) x, t exp i 1 t k 1 x exp i 2 t k 2 x with 2 exp i exp i 1 2 2 1 2 2 1 2 2 v gr t k 1 k 2 2 t=0 x t k 1 k 2 x 2 0 1 2 2 exp i and k and k 0 1 2 2 t k 1 k 2 2 note: in QM the wave amplitude is (in general) really complex! x, t exp i 0 t k 0 x cos 2 t k 2 x x x x t>0 beat (Schwebung) with beat frequency 2 weseethatthe maximumof the wave is found at t k 2 2 x 0 wave travels with the speed v gr x t k d dk 22.11.01 FU - Physik III - WS 2000/2001 I.V. Hertel 24