Fermions and Bosons

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# Fermions and Bosons - PowerPoint PPT Presentation

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

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

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

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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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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

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 box

Schrödinger equation

clever mathematics

Solution

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One particle in a box

For n = 1:

Ψ(x)

|Ψ(x)|²

x

x

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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

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 box

Wave function for the system

Suggestion

Hartree product

Product of “one-particle-solutions”

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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 box

x2

 Particles do not

influence each other

x1

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Two distinguishable particles in a box

Probability density |Ψ|²

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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 box

Fermions: Ψ(x1,x2) = - Ψ(x2,x1)

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

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Two fermions in a box

For fermion 2: n = 2

For fermion 1: n = 1

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Two fermions in a box

For fermion 2: n = 1

For fermion 1: n = 2

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Two fermions in a box

nodal plane

“Pauli-repulsion”

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Two fermions in a box

Probability density |Ψ|²

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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 box

Bosons: Ψ(x1,x2) = Ψ(x2,x1)

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

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Two bosons in a box

For boson 2: n = 2

For boson 1: n = 1

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Two bosons in a box

For boson 2: n = 1

For boson 1: n = 2

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Two bosons in a box

bosons “stick together”

nodal plane

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Two bosons in a box

Probability density |Ψ|²

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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

Generally: Describes probabilities of occupation of

different quantum states

Fermi-Dirac statistic

Bose-Einstein statistic

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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 statistics
• For high temperatures both statistics merge

into Maxwell-Boltzmann statistic

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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 (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 (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 (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 (BEC)

What effects can be found?

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

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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 (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 (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 (BEC)

Superconductivity

 Electric conductivity

without resistance

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Bose-Einstein condensate (BEC)

Superfluidity

Superfluid Helium runs

out of a bottle

 fountain

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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

[7] http://www.mpq.mpg.de/atomlaser/index.html

[8] Udo Benedikt, Vorlesungsmitschrift: Theoretische Chemie, 2005

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Thanks

Dr. Alexander Auer

Annemarie Magerl

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