Fermions Bosons Some Simple Quantum Mechanics From Professor
Fermions & Bosons Some Simple Quantum Mechanics! From Professor Udo Benedikt University of California at Berkeley
Fermions & Bosons Some Simple Quantum Mechanics!
Outline 3 1. One particle in a box 2. Two particles in a box 3. The Pauli principle 4. Quantum statistics 5. Bose-Einstein condensate Udo Benedikt
One particle in a box Assumptions: • Side of box is L= 1 • The box is limited by infinite potential walls The particle cannot be outside the box or on the walls 4 Udo Benedikt
One particle in a box The Schrödinger equation: Solution 5 Mathematics Udo Benedikt
One particle in a box For n = 1: Ψ(x) |Ψ(x)|² x 6 x Udo Benedikt x
One particle in a box For n = 2: Ψ(x) |Ψ(x)|² x 7 x Udo Benedikt x
Two “classical” (distinguishable) particles in a box Assumptions: • Distinguishable particles • Box length L = 1 • Infinite potential walls • Particles don’t interact with 8 each other Udo Benedikt
Two “classical” (distinguishable) particles in a box Wave Function for the system Suggestion: Hartree product Product of “one-particle-solutions” 9 Udo Benedikt
Two distinguishable particles in a box For particle 2: n=2 2(x 2) x 2 For particle 1: n=1 1(x 1) 10 x 1 Udo Benedikt
Two distinguishable particles in a box x 2 Particles do not influence each other 11 x 1 Udo Benedikt
Two distinguishable particles in a box 12 Udo Benedikt
Two distinguishable particles in a box Probability density |Ψ|² 13 Udo Benedikt
Two fermions in a box Assumptions: • Indistinguishable fermions • Box length = 1 • Infinite potential walls • Antisymmetric wave function 14 Udo Benedikt
Two fermions in a box Fermions Two Particle Wave Function must be Antisymmetric Ψ(x 1, x 2) = - Ψ(x 2, x 1) An antisymmetric product of 2 “ 1 -particle-solutions”: 15 Udo Benedikt
Two fermions in a box For fermion 2: n = 2 16 For fermion 1: n = 1 Udo Benedikt
Two fermions in a box For fermion 2: n = 1 For fermion 1: n = 2 17 Udo Benedikt
Two fermions in a box 18 Udo Benedikt
Two fermions in a box nodal plane “Pauli-repulsion” 19 Udo Benedikt
Two fermions in a box 20 Udo Benedikt
Two fermions in a box Probability density |Ψ|² 21 Udo Benedikt
Two bosons in a box Assumptions • Indistinguishable bosons • Box length = 1 • Infinite potential walls • Symmetric wave function 22 Udo Benedikt
Two bosons in a box Bosons Two Particle Wave Function Must be Symmetric Ψ(x 1, x 2) = Ψ(x 2, x 1) A symmetric product of 2 “ 1 -particle-solutions” 23 Udo Benedikt
Two bosons in a box For boson 2: n = 2 24 For boson 1: n = 1 Udo Benedikt
Two bosons in a box For boson 2: n = 1 For boson 1: n = 2 25 18. 10. 2006 Udo Benedikt
Two bosons in a box 26 18. 10. 2006 Udo Benedikt
Two bosons in a box bosons “stick together” 27 18. 10. 2006 nodal plane Udo Benedikt
Two bosons in a box 28 18. 10. 2006 Udo Benedikt
Two bosons in a box Probability Density |Ψ|² 29 18. 10. 2006 Udo Benedikt
Pauli Principle The total wave function must be antisymmetric under the interchange of any pair of identical Fermions & symmetric under the interchange of any pair of identical Bosons. 30 18. 10. 2006 Udo Benedikt
Pauli Principle Two Fermions: No two fermions can occupy the same single particle state. 31 18. 10. 2006 Udo Benedikt
Quantum statistics Generally: Describes probabilities of occupation of different quantum states Fermi-Dirac Statistics 32 18. 10. 2006 Bose-Einstein Statistics Udo Benedikt
Quantum statistics For T 0 K Fermi-Dirac statistics now excited states are occupied • Highest occupied state Fermi energy εF • f. FD(ε < εF) = 1 and f. FD(ε > εF) = 0 Electron gas f. FD T=0 K • Even T>0 K ε/εF Bose-Einstein statistics • Bose-Einstein 33 condensate 18. 10. 2006 Udo Benedikt
Quantum statistics 34 For high temperatures both statistics merge into Maxwell-Boltzmann statistics 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 35 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 & sodium atoms • 2001 these three scientists were awarded the Nobel prize in physics 36 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 packets becomes bigger so that they can overlap condensation starts 37 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 38 18. 10. 2006 Udo Benedikt
Bose-Einstein condensate (BEC) What effects can be found? • Superfluidity • Superconductivity • Coherence (interference experiments, atom laser) Over macroscopic distances 39 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 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) Two expanding condensates Two trapped condensates and their ballistic expansion after the magnetic trap has been turned off The two condensates overlap interference 42 18. 10. 2006 Udo Benedikt
Bose-Einstein condensate (BEC) Superconductivity Electric conductivity without resistance 43 18. 10. 2006 Udo Benedikt
Bose-Einstein condensate (BEC) Superfluidity Superfluid Helium runs out of a bottle fountain 44 18. 10. 2006 Udo Benedikt
Literature 45 [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 18. 10. 2006 Udo Benedikt
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