Fermions and Bosons From the Pauli principle to Bose-Einstein condensate

Udo Benedikt 2 Structure Basics One particle in a box Two particles in a box Pauli principle Quantum statistics Bose-Einstein condensate

Udo Benedikt 3 Basics Quantum Mechanics Observable: property of a system (measurable) Operator: mathematic operation on function Wave function: describes a system Eigenvalue equation: unites operator, wave function and observable

Udo Benedikt 4 Basics Schrödinger equation Hamilton operator Wave function Energy (observable) Example for an eigenvalue equation: The wave function Ψ itself has no physical importance, but the probability density of the particle is given by |Ψ|².

Udo Benedikt 5 Basics Operator : interchanges two particles in wave function ε = -1  antisymmetric wave function  Fermions ε = 1  symmetric wave function  Bosons Generally: |Ψ(x 1,x 2 )| 2 = |Ψ(x 2,x 1 )| 2

Udo Benedikt 6 One particle in a box Postulates: Length of the box is 1 Box is limited by infinite potential walls  particle cannot be outside the box or on the walls

Udo Benedikt 7 One particle in a box Schrödinger equation clever mathematics Solution

Udo Benedikt 8 One particle in a box For n = 1: x Ψ(x) x |Ψ(x)|²

Udo Benedikt 9 One particle in a box For n = 2: x Ψ(x) x |Ψ(x)|²

Udo Benedikt 10 Two distinguishable particles in a box Postulates: Distinguishable particles Box length = 1 Infinite potential walls Particles do not interact with each other

Udo Benedikt 11 Two distinguishable particles in a box Wanted! Dead or alive Wave function for the system Suggestion Hartree product Product of “one-particle-solutions”

Udo Benedikt 12 Two distinguishable particles in a box For particle 1: n = 1 For particle 2: n = 2

Udo Benedikt 13 Two distinguishable particles in a box x1x1 x2x2  Particles do not influence each other

Udo Benedikt 14 Two distinguishable particles in a box

Udo Benedikt 15 Two distinguishable particles in a box Probability density |Ψ|²

Udo Benedikt 16 Two fermions in a box Postulates: Indistinguishable fermions Box length = 1 Infinite potential walls Antisymmetric wave function

Udo Benedikt 17 Two fermions in a box Fermions: Ψ(x 1,x 2 ) = - Ψ(x 2,x 1 ) For Fermions: antisymmetric product of “one-particle-solutions”

Udo Benedikt 18 Two fermions in a box For fermion 1: n = 1 For fermion 2: n = 2

Udo Benedikt 19 Two fermions in a box For fermion 1: n = 2 For fermion 2: n = 1

Udo Benedikt 20 Two fermions in a box

Udo Benedikt 21 Two fermions in a box “Pauli-repulsion” nodal plane

Udo Benedikt 22 Two fermions in a box

Udo Benedikt 23 Two fermions in a box Probability density |Ψ|²

Udo Benedikt 24 Two bosons in a box Postulates: Indistinguishable bosons Box length = 1 Infinite potential walls Symmetric wave function

Udo Benedikt 25 Two bosons in a box Bosons: Ψ(x 1,x 2 ) = Ψ(x 2,x 1 ) For Bosons: symmetric product of “one-particle-solutions”

Udo Benedikt 26 Two bosons in a box For boson 1: n = 1 For boson 2: n = 2

Udo Benedikt 27 Two bosons in a box For boson 1: n = 2 For boson 2: n = 1

Udo Benedikt 28 Two bosons in a box

Udo Benedikt 29 Two bosons in a box nodal plane bosons “stick together”

Udo Benedikt 30 Two bosons in a box

Udo Benedikt 31 Two bosons in a box Probability density |Ψ|²

Udo Benedikt 32 Pauli principle The total wave function must be antisymmetric under the interchange of any pair of identical fermions and symmetrical under the interchange of any pair of identical bosons. Fermions:  No two fermions can occupy the same state.

Udo Benedikt 33 Quantum statistics Generally: Describes probabilities of occupation of different quantum states Fermi-Dirac statisticBose-Einstein statistic

Udo Benedikt 34 Quantum statistics For T  0 K Fermi-Dirac statistic Bose-Einstein statistic Even now excited states are occupied Highest occupied state  Fermi energy ε F f FD (ε ε F ) = 0  Electron gas Bose-Einstein condensate ε/εFε/εF f FD T = 0 K T > 0 K

Udo Benedikt 35 Quantum statistics  For high temperatures both statistics merge into Maxwell-Boltzmann statistic

Udo Benedikt 36 Bose-Einstein condensate (BEC) What is it? Extreme aggregate state of a system of indistinguishable particles, that are all in the same state  bosons Macroscopic quantum objects in which the individual atoms are completely delocalized Same probability density everywhere  One wave function for the whole system

Udo Benedikt 37 Bose-Einstein condensate (BEC) Who discovered it? Theoretically predicted by Satyendra Nath Bose and Albert Einstein in 1924 First practical realizations by Eric A. Cornell, Wolfgang Ketterle and Carl E. Wieman in 1995  condensation of a gas of rubidium and sodium atoms 2001 these three scientists were awarded with the Nobel price in physics

Udo Benedikt 38 Bose-Einstein condensate (BEC) How does it work? Condensation occurs when a critical density is reached  Trapping and chilling of bosons  Wavelength of the wave packages becomes bigger so that they can overlap  condensation starts

Udo Benedikt 39 Bose-Einstein condensate (BEC) How to get it? Laser cooling until T ~ 100 μK  particles are slowed down to several cm/s Particles caught in magnetic trap Further chilling through evaporative cooling until T ~ 50 nK

Udo Benedikt 40 Bose-Einstein condensate (BEC) What effects can be found? Superfluidity Superconductivity Coherence (interference experiments, atom laser)  Over macroscopic distances

Udo Benedikt 41 Bose-Einstein condensate (BEC) Atom laser controlled decoupling of a part of the matter wave from the condensate in the trap

Udo Benedikt 42 Bose-Einstein condensate (BEC) Atom laser controlled decoupling of a part of the matter wave from the condensate in the trap

Udo Benedikt 43 Bose-Einstein condensate (BEC) Two trapped condensates and their ballistic expansion after the magnetic trap has been turned off The two condensates overlap  interference Two expanding condensates

Udo Benedikt 44 Bose-Einstein condensate (BEC) Superconductivity  Electric conductivity without resistance

Udo Benedikt 45 Bose-Einstein condensate (BEC) Superfluidity Superfluid Helium runs out of a bottle  fountain

Udo Benedikt 46 Literature [1] Bransden,B.H., Joachain,C.J., Quantum Mechanics, 2nd edition, Prentice-Hall, Harlow,England, 2000 [2] Atkins,P.W., Friedman,R.S., Molecular Quantum Mechanics, 3rd edition, Oxford University Press, Oxford, 1997 [3] Göpel,W., Wiemhöfer,H.D., Statistische Thermodynamik, Spektum Akademischer Verlag, Heidelberg,Berlin, 2000 [4] Bammel,K., Faszination Physik, Spektum Akademischer Verlag, Heidelberg,Berlin, 2004 [5] [6] [7] [8] Udo Benedikt, Vorlesungsmitschrift: Theoretische Chemie, 2005

Udo Benedikt 47 Thanks Dr. Alexander Auer Annemarie Magerl

Udo Benedikt 48 Thanks for your attention