Fermions and Bosons From the Pauli principle to
Fermions and Bosons From the Pauli principle to Bose-Einstein condensate 18. 10. 2006 Udo Benedikt
Structure l l l 2 Basics One particle in a box Two particles in a box Pauli principle Quantum statistics Bose-Einstein condensate 18. 10. 2006 Udo Benedikt
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 3 18. 10. 2006 Udo Benedikt
Basics Example for an eigenvalue equation: Schrödinger equation Hamilton operator Wave function Energy (observable) The wave function Ψ itself has no physical importance, but the probability density of the particle is given by |Ψ|². 4 18. 10. 2006 Udo Benedikt
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 5 18. 10. 2006 Udo Benedikt
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 6 18. 10. 2006 Udo Benedikt
One particle in a box Schrödinger equation clever mathematics Solution 7 18. 10. 2006 Udo Benedikt
One particle in a box For n = 1: Ψ(x) |Ψ(x)|² x 8 18. 10. 2006 x Udo Benedikt
One particle in a box For n = 2: Ψ(x) |Ψ(x)|² x x 9 18. 10. 2006 Udo Benedikt
Two distinguishable particles in a box Postulates: • Distinguishable particles • Box length = 1 • Infinite potential walls • Particles do not interact with each other 10 18. 10. 2006 Udo Benedikt
Two distinguishable particles in a box Wanted! Dead or alive Wave function for the system Suggestion Hartree product Product of “one-particle-solutions” 11 18. 10. 2006 Udo Benedikt
Two distinguishable particles in a box For particle 1: n = 1 For particle 2: n = 2 12 18. 10. 2006 Udo Benedikt
Two distinguishable particles in a box x 2 Particles do not influence each other 13 x 1 18. 10. 2006 Udo Benedikt
Two distinguishable particles in a box 14 18. 10. 2006 Udo Benedikt
Two distinguishable particles in a box Probability density |Ψ|² 15 18. 10. 2006 Udo Benedikt
Two fermions in a box Postulates: • Indistinguishable fermions • Box length = 1 • Infinite potential walls • Antisymmetric wave function 16 18. 10. 2006 Udo Benedikt
Two fermions in a box Fermions: Ψ(x 1, x 2) = - Ψ(x 2, x 1) For Fermions: antisymmetric product of “one-particle-solutions” 17 18. 10. 2006 Udo Benedikt
Two fermions in a box For fermion 2: n = 2 For fermion 1: n = 1 18 18. 10. 2006 Udo Benedikt
Two fermions in a box For fermion 2: n = 1 For fermion 1: n = 2 19 18. 10. 2006 Udo Benedikt
Two fermions in a box 20 18. 10. 2006 Udo Benedikt
Two fermions in a box nodal plane “Pauli-repulsion” 21 18. 10. 2006 Udo Benedikt
Two fermions in a box 22 18. 10. 2006 Udo Benedikt
Two fermions in a box Probability density |Ψ|² 23 18. 10. 2006 Udo Benedikt
Two bosons in a box Postulates: • Indistinguishable bosons • Box length = 1 • Infinite potential walls • Symmetric wave function 24 18. 10. 2006 Udo Benedikt
Two bosons in a box Bosons: Ψ(x 1, x 2) = Ψ(x 2, x 1) For Bosons: symmetric product of “one-particle-solutions” 25 18. 10. 2006 Udo Benedikt
Two bosons in a box For boson 2: n = 2 For boson 1: n = 1 26 18. 10. 2006 Udo Benedikt
Two bosons in a box For boson 2: n = 1 For boson 1: n = 2 27 18. 10. 2006 Udo Benedikt
Two bosons in a box 28 18. 10. 2006 Udo Benedikt
Two bosons in a box bosons “stick together” 29 nodal plane 18. 10. 2006 Udo Benedikt
Two bosons in a box 30 18. 10. 2006 Udo Benedikt
Two bosons in a box Probability density |Ψ|² 31 18. 10. 2006 Udo Benedikt
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. 32 18. 10. 2006 Udo Benedikt
Quantum statistics Generally: Describes probabilities of occupation of different quantum states Fermi-Dirac statistic 33 18. 10. 2006 Bose-Einstein statistic Udo Benedikt
Quantum statistics For T 0 K f. FD T=0 K Fermi-Dirac statistic • Even now excited states are occupied • Highest occupied state Fermi energy εF • f. FD(ε < εF) = 1 and f. FD(ε > εF) = 0 Electron gas T>0 K ε/εF Bose-Einstein statistic • Bose-Einstein condensate 34 18. 10. 2006 Udo Benedikt
Quantum statistics For high temperatures both statistics merge into Maxwell-Boltzmann statistic 35 18. 10. 2006 Udo Benedikt
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 36 18. 10. 2006 Udo Benedikt
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 37 18. 10. 2006 Udo Benedikt
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 38 18. 10. 2006 Udo Benedikt
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 n. K 39 18. 10. 2006 Udo Benedikt
Bose-Einstein condensate (BEC) What effects can be found? • Superfluidity • Superconductivity • Coherence (interference experiments, atom laser) Over macroscopic distances 40 18. 10. 2006 Udo Benedikt
Bose-Einstein condensate (BEC) Atom laser controlled decoupling of a part of the matter wave from the condensate in the trap 41 18. 10. 2006 Udo Benedikt
Bose-Einstein condensate (BEC) Atom laser controlled decoupling of a part of the matter wave from the condensate in the trap 42 18. 10. 2006 Udo Benedikt
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 43 18. 10. 2006 Udo Benedikt
Bose-Einstein condensate (BEC) Superconductivity Electric conductivity without resistance 44 18. 10. 2006 Udo Benedikt
Bose-Einstein condensate (BEC) Superfluidity Superfluid Helium runs out of a bottle fountain 45 18. 10. 2006 Udo Benedikt
Literature [1] Bransden, B. H. , Joachain, C. J. , Quantum Mechanics, 2 nd edition, Prentice-Hall, Harlow, England, 2000 [2] Atkins, P. W. , Friedman, R. S. , Molecular Quantum Mechanics, 3 rd 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/Projects 97. 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 46 18. 10. 2006 Udo Benedikt
Thanks Dr. Alexander Auer Annemarie Magerl 47 18. 10. 2006 Udo Benedikt
Thanks for your attention 48 18. 10. 2006 Udo Benedikt
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