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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

Fermions and Bosons

From the Pauli principle to Bose-Einstein condensate

structure
Structure
  • Basics
  • One particle in a box
  • Two particles in a box
  • Pauli principle
  • Quantum statistics
  • Bose-Einstein condensate

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basics
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

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basics1
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.

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basics2
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

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one particle in a box
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

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one particle in a box1
One particle in a box

Schrödinger equation

clever mathematics

Solution

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one particle in a box2
One particle in a box

For n = 1:

Ψ(x)

|Ψ(x)|²

x

x

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one particle in a box3
One particle in a box

For n = 2:

Ψ(x)

|Ψ(x)|²

x

x

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two distinguishable particles in a box
Two distinguishable particles in a box

Postulates:

  • Distinguishable particles
  • Box length = 1
  • Infinite potential walls
  • Particles do not interact
  • with each other

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two distinguishable particles in a box1
Two distinguishable particles in a box

Wanted! Dead or alive

Wave function for the system

Suggestion

Hartree product

Product of “one-particle-solutions”

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two distinguishable particles in a box2
Two distinguishable particles in a box

For particle 1: n = 1

For particle 2: n = 2

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two distinguishable particles in a box3
Two distinguishable particles in a box

x2

 Particles do not

influence each other

x1

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two distinguishable particles in a box5
Two distinguishable particles in a box

Probability density |Ψ|²

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two fermions in a box
Two fermions in a box

Postulates:

  • Indistinguishable fermions
  • Box length = 1
  • Infinite potential walls
  • Antisymmetric wave function

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two fermions in a box1
Two fermions in a box

Fermions: Ψ(x1,x2) = - Ψ(x2,x1)

For Fermions: antisymmetric product of “one-particle-solutions”

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two fermions in a box2
Two fermions in a box

For fermion 2: n = 2

For fermion 1: n = 1

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two fermions in a box3
Two fermions in a box

For fermion 2: n = 1

For fermion 1: n = 2

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two fermions in a box5
Two fermions in a box

nodal plane

“Pauli-repulsion”

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two fermions in a box7
Two fermions in a box

Probability density |Ψ|²

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two bosons in a box
Two bosons in a box

Postulates:

  • Indistinguishable bosons
  • Box length = 1
  • Infinite potential walls
  • Symmetric wave function

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two bosons in a box1
Two bosons in a box

Bosons: Ψ(x1,x2) = Ψ(x2,x1)

For Bosons: symmetric product of “one-particle-solutions”

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two bosons in a box2
Two bosons in a box

For boson 2: n = 2

For boson 1: n = 1

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two bosons in a box3
Two bosons in a box

For boson 2: n = 1

For boson 1: n = 2

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two bosons in a box5
Two bosons in a box

bosons “stick together”

nodal plane

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two bosons in a box7
Two bosons in a box

Probability density |Ψ|²

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pauli principle
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.

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quantum statistics
Quantum statistics

Generally: Describes probabilities of occupation of

different quantum states

Fermi-Dirac statistic

Bose-Einstein statistic

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quantum statistics1
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

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quantum statistics2
Quantum statistics
  • For high temperatures both statistics merge

into Maxwell-Boltzmann statistic

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bose einstein condensate bec
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

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bose einstein condensate bec1
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

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bose einstein condensate bec2
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

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bose einstein condensate bec3
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

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bose einstein condensate bec4
Bose-Einstein condensate (BEC)

What effects can be found?

  • Superfluidity
  • Superconductivity
  • Coherence (interference experiments, atom laser)
  •  Over macroscopic distances

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bose einstein condensate bec5
Bose-Einstein condensate (BEC)

Atom laser

controlled decoupling of a part

of the matter wave from the

condensate in the trap

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bose einstein condensate bec6
Bose-Einstein condensate (BEC)

Atom laser

controlled decoupling of a part

of the matter wave from the

condensate in the trap

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bose einstein condensate bec7
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

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bose einstein condensate bec8
Bose-Einstein condensate (BEC)

Superconductivity

 Electric conductivity

without resistance

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bose einstein condensate bec9
Bose-Einstein condensate (BEC)

Superfluidity

Superfluid Helium runs

out of a bottle

 fountain

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literature
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

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thanks
Thanks

Dr. Alexander Auer

Annemarie Magerl

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