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18.10.2006 Udo Benedikt2
Structure
Basics One particle in a box Two particles in a box Pauli principle Quantum statistics Bose-Einstein condensate
18.10.2006 Udo Benedikt3
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
18.10.2006 Udo Benedikt4
Basics
HΨ EΨ
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.
18.10.2006 Udo Benedikt5
Basics
P
1 2 2 1 1 2PΨ x ,x Ψ x ,x εΨ x ,x
ε 1
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
18.10.2006 Udo Benedikt6
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
18.10.2006 Udo Benedikt7
One particle in a box
HΨ x EΨ xSchrödinger equation
clever mathematics
Solution
Ψ x 2 sin n π x
18.10.2006 Udo Benedikt10
Two distinguishable particles in a box
Postulates:
• Distinguishable particles• Box length = 1• Infinite potential walls• Particles do not interact with each other
18.10.2006 Udo Benedikt11
Two distinguishable particles in a box
1 2 1 1 2 2Ψ x ,x x x
Wanted! Dead or alive
Wave function for the system
Suggestion
Hartree product
Product of “one-particle-solutions”
18.10.2006 Udo Benedikt12
Two distinguishable particles in a box
For particle 1: n = 1
1 1 1x 2 sin π xFor particle 2: n = 2
2 2 2x 2 sin 2 π x
18.10.2006 Udo Benedikt13
Two distinguishable particles in a box
1 2 1 2Ψ x ,x =2 sin x π sin 2 π x
x1
x2
1 2 1 1 2 2Ψ x ,x x x
Particles do not influence each other
18.10.2006 Udo Benedikt16
Two fermions in a box
Postulates:
• Indistinguishable fermions• Box length = 1• Infinite potential walls• Antisymmetric wave function
18.10.2006 Udo Benedikt17
Two fermions in a box
1 2 1 1 2 2 1 2 2 1ψ x ,x x x x x
Fermions: Ψ(x1,x2) = - Ψ(x2,x1)
For Fermions: antisymmetric product of “one-particle-solutions”
1 2 2 1 1 2 2 1 1 1 2 2
1 2
PΨ x ,x Ψ x ,x x x x x
Ψ x ,x
18.10.2006 Udo Benedikt18
Two fermions in a box
For fermion 1: n = 1 1 1 1x 2 sin π x
For fermion 2: n = 2
2 2 2x 2 sin 2 π x 1 2 1 1 2 2f x ,x x x
18.10.2006 Udo Benedikt19
Two fermions in a box
1 2 1 2 2 1f x ,x x x
For fermion 1: n = 2
1 2 2x 2 sin π x For fermion 2: n = 1
2 1 1x 2 sin 2 π x
18.10.2006 Udo Benedikt24
Two bosons in a box
Postulates:
• Indistinguishable bosons• Box length = 1• Infinite potential walls• Symmetric wave function
18.10.2006 Udo Benedikt25
Two bosons in a box
1 2 1 1 2 2 1 2 2 1ψ x ,x x x x x
Bosons: Ψ(x1,x2) = Ψ(x2,x1)
For Bosons: symmetric product of “one-particle-solutions”
1 2 2 1 1 2 2 1 1 1 2 2
1 2
PΨ x ,x Ψ x ,x x x x x
Ψ x ,x
18.10.2006 Udo Benedikt26
Two bosons in a box
For boson 1: n = 1 1 1 1x 2 sin π x
For boson 2: n = 2
2 2 2x 2 sin 2 π x 1 2 1 1 2 2f x ,x x x
18.10.2006 Udo Benedikt27
Two bosons in a box
1 2 1 2 2 1f x ,x x x
For boson 1: n = 2
1 2 2x 2 sin π x For boson 2: n = 1
2 1 1x 2 sin 2 π x
18.10.2006 Udo Benedikt32
Pauli principle
The total wave function must be antisymmetric under the interchange of any pair of identical fermions andsymmetrical under the interchange of any pair of identical bosons.
1 1 1 1 2 1 1 1 2 1ψ x ,x x x x x 0 Fermions:
No two fermions can occupy the same state.
18.10.2006 Udo Benedikt33
Quantum statistics
1
iFDf exp 1
kT
Generally: Describes probabilities of occupation of different quantum states
Fermi-Dirac statistic Bose-Einstein statistic
1
iBEf exp 1
kT
18.10.2006 Udo Benedikt34
Quantum statistics
For T 0 K
Fermi-Dirac statistic
Bose-Einstein statistic
• Even now excited states are occupied
• Highest occupied state Fermi energy εF
• fFD(ε < εF) = 1 and fFD(ε > εF) = 0
Electron gas
• Bose-Einstein condensate
1
1 ε/εF
fFD T = 0 K
T > 0 K
18.10.2006 Udo Benedikt35
Quantum statistics
For high temperatures both statistics merge into Maxwell-Boltzmann statistic
18.10.2006 Udo Benedikt36
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
18.10.2006 Udo Benedikt37
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
18.10.2006 Udo Benedikt38
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
18.10.2006 Udo Benedikt39
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
18.10.2006 Udo Benedikt40
Bose-Einstein condensate (BEC)
What effects can be found?
• Superfluidity
• Superconductivity
• Coherence (interference experiments, atom laser)
Over macroscopic distances
18.10.2006 Udo Benedikt41
Bose-Einstein condensate (BEC)
Atom laser
controlled decoupling of a partof the matter wave from the
condensate in the trap
18.10.2006 Udo Benedikt42
Bose-Einstein condensate (BEC)
Atom laser
controlled decoupling of a partof the matter wave from the
condensate in the trap
18.10.2006 Udo Benedikt43
Bose-Einstein condensate (BEC)
Two trapped condensates and their ballistic expansion after the magnetictrap has been turned off
The two condensates overlap interference
Two expanding condensates
18.10.2006 Udo Benedikt44
Bose-Einstein condensate (BEC)
Superconductivity
Electric conductivitywithout resistance
18.10.2006 Udo Benedikt45
Bose-Einstein condensate (BEC)
Superfluidity
Superfluid Helium runsout of a bottle fountain
18.10.2006 Udo Benedikt46
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