Fermions and Bosons From the Pauli principle to Bose-Einstein condensate

Fermions and Bosons

From the Pauli principle to Bose-Einstein

condensate

18.10.2006 Udo Benedikt2

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)

Operator: mathematic operation on function

Wave function: describes a system

Eigenvalue equation: unites operator, wave function and observable


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.


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


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



HΨ x EΨ xSchrödinger equation

clever mathematics

Solution

Ψ x 2 sin n π x



For n = 1: Ψ x 2 sin π x

x

Ψ(x)

x

|Ψ(x)|²



For n = 2: Ψ x 2 sin 2 π x

x

Ψ(x)

x

|Ψ(x)|²


Two distinguishable particles in a box

Postulates:

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



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”



For particle 1: n = 1

1 1 1x 2 sin π xFor particle 2: n = 2

2 2 2x 2 sin 2 π x



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





Probability density |Ψ|²


Two fermions in a box

Postulates:

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



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



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



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



1 2 1 1 2 2 1 2 2 1ψ x ,x x x x x



“Pauli-repulsion”

0.70.3

nodal plane







Two bosons in a box

Postulates:

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


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


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


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


Two bosons in a box

1 2 1 1 2 2 1 2 2 1ψ x ,x x x x x


Two bosons in a box

0.3

nodal plane

bosons “stick together”


Two bosons in a box


Two bosons in a box



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.


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


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


Quantum statistics

For high temperatures both statistics merge into Maxwell-Boltzmann statistic


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



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



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



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



What effects can be found?

• Superfluidity

• Superconductivity

• Coherence (interference experiments, atom laser)

Over macroscopic distances



Atom laser

controlled decoupling of a partof the matter wave from the

condensate in the trap



Atom laser

controlled decoupling of a partof the matter wave from the

condensate in the trap



Two trapped condensates and their ballistic expansion after the magnetictrap has been turned off

The two condensates overlap interference

Two expanding condensates



Superconductivity

Electric conductivitywithout resistance



Superfluidity

Superfluid Helium runsout of a bottle fountain


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


Thanks

Dr. Alexander Auer

Annemarie Magerl


Thanks for your attention

Documents

Fermions and Bosons From the Pauli principle to Bose-Einstein condensate