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Fermions and Bosons

Fermions and Bosons. From the Pauli principle to Bose-Einstein condensate. Structure. Basics One particle in a box Two particles in a box Pauli principle Quantum statistics Bose-Einstein condensate. Basics. Quantum Mechanics. Observable: property of a system (measurable).

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Fermions and Bosons

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

  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 Example for an eigenvalue equation: Schrödinger equation Hamilton operator Energy (observable) Wave function 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(x1,x2)|2 = |Ψ(x2,x1)|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: Ψ(x) |Ψ(x)|² x x Udo Benedikt

  9. One particle in a box For n = 2: Ψ(x) |Ψ(x)|² x x 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 x2  Particles do not influence each other x1 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(x1,x2) = - Ψ(x2,x1) For Fermions: antisymmetric product of “one-particle-solutions” Udo Benedikt

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

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

  20. Two fermions in a box Udo Benedikt

  21. Two fermions in a box nodal plane “Pauli-repulsion” 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(x1,x2) = Ψ(x2,x1) For Bosons: symmetric product of “one-particle-solutions” Udo Benedikt

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

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

  28. Two bosons in a box Udo Benedikt

  29. Two bosons in a box bosons “stick together” nodal plane 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 statistic Bose-Einstein statistic Udo Benedikt

  34. Quantum statistics For T  0 K fFD T = 0 K Fermi-Dirac statistic • Even now excited states are occupied • Highest occupied state  Fermi energy εF • fFD(ε < εF) = 1 and fFD(ε > εF) = 0 •  Electron gas T > 0 K ε/εF Bose-Einstein statistic • Bose-Einstein condensate 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 expanding condensates Two trapped condensates and their ballistic expansion after the magnetic trap has been turned off The two condensates overlap  interference 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] http://cua.mit.edu/ketterle_group/Projects_1997/Projects97.htm [6] http://www.colorado.edu/physics/2000/bec/index.html [7] http://www.mpq.mpg.de/atomlaser/index.html [8] Udo Benedikt, Vorlesungsmitschrift: Theoretische Chemie, 2005 Udo Benedikt

  47. Thanks Dr. Alexander Auer Annemarie Magerl Udo Benedikt

  48. Thanks for your attention Udo Benedikt

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