Fermions, Bosons and their Statistics
Peter Hertel
Fundamental Fermions, Bosons and their Statistics particles
Spin and statistics
Many particles Peter Hertel
Gibbs state
Fermions University of Osnabr¨uck,Germany
Zero temperature Lecture presented at APS, Nankai University, China White dwarfs and neutron stars http://www.home.uni-osnabrueck.de/phertel Bosons
Black body radiation March/April 2011 Bose-Einstein condensation • proton p=(uud) and neutron n=(udd) • both have spin 1/2 • excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc. • they have spin 3/2 • p is stable, n → p+e−+¯ν is allowed
Fermions, Bosons and their Statistics Structure fermions
Peter Hertel
Fundamental particles charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
Spin and − + statistics generation 1 e u¯ d νe , ν¯e d¯ u e Many particles − + generation 2 µ c¯ s νµ, ν¯µ ¯s c µ Gibbs state − ¯ + Fermions generation 3 τ ¯t b ντ , ν¯τ b t τ
Zero temperature
White dwarfs • all structure particles have spin 1/2 and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • both have spin 1/2 • excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc. • they have spin 3/2 • p is stable, n → p+e−+¯ν is allowed
Fermions, Bosons and their Statistics Structure fermions
Peter Hertel
Fundamental particles charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
Spin and − + statistics generation 1 e u¯ d νe , ν¯e d¯ u e Many particles − + generation 2 µ c¯ s νµ, ν¯µ ¯s c µ Gibbs state − ¯ + Fermions generation 3 τ ¯t b ντ , ν¯τ b t τ
Zero temperature
White dwarfs • all structure particles have spin 1/2 and neutron stars • proton p=(uud) and neutron n=(udd) Bosons
Black body radiation
Bose-Einstein condensation • excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc. • they have spin 3/2 • p is stable, n → p+e−+¯ν is allowed
Fermions, Bosons and their Statistics Structure fermions
Peter Hertel
Fundamental particles charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
Spin and − + statistics generation 1 e u¯ d νe , ν¯e d¯ u e Many particles − + generation 2 µ c¯ s νµ, ν¯µ ¯s c µ Gibbs state − ¯ + Fermions generation 3 τ ¯t b ντ , ν¯τ b t τ
Zero temperature
White dwarfs • all structure particles have spin 1/2 and neutron stars • proton p=(uud) and neutron n=(udd) Bosons • both have spin 1/2 Black body radiation
Bose-Einstein condensation • they have spin 3/2 • p is stable, n → p+e−+¯ν is allowed
Fermions, Bosons and their Statistics Structure fermions
Peter Hertel
Fundamental particles charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
Spin and − + statistics generation 1 e u¯ d νe , ν¯e d¯ u e Many particles − + generation 2 µ c¯ s νµ, ν¯µ ¯s c µ Gibbs state − ¯ + Fermions generation 3 τ ¯t b ντ , ν¯τ b t τ
Zero temperature
White dwarfs • all structure particles have spin 1/2 and neutron stars • proton p=(uud) and neutron n=(udd) Bosons • both have spin 1/2 Black body radiation • excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc. Bose-Einstein condensation • p is stable, n → p+e−+¯ν is allowed
Fermions, Bosons and their Statistics Structure fermions
Peter Hertel
Fundamental particles charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
Spin and − + statistics generation 1 e u¯ d νe , ν¯e d¯ u e Many particles − + generation 2 µ c¯ s νµ, ν¯µ ¯s c µ Gibbs state − ¯ + Fermions generation 3 τ ¯t b ντ , ν¯τ b t τ
Zero temperature
White dwarfs • all structure particles have spin 1/2 and neutron stars • proton p=(uud) and neutron n=(udd) Bosons • both have spin 1/2 Black body radiation • excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc. Bose-Einstein condensation • they have spin 3/2 Fermions, Bosons and their Statistics Structure fermions
Peter Hertel
Fundamental particles charge -1 -2/3 -1/3 0 +1/3 +2/3 +1
Spin and − + statistics generation 1 e u¯ d νe , ν¯e d¯ u e Many particles − + generation 2 µ c¯ s νµ, ν¯µ ¯s c µ Gibbs state − ¯ + Fermions generation 3 τ ¯t b ντ , ν¯τ b t τ
Zero temperature
White dwarfs • all structure particles have spin 1/2 and neutron stars • proton p=(uud) and neutron n=(udd) Bosons • both have spin 1/2 Black body radiation • excited states ∆++=(uuu)∗, ∆+=(uud)∗ etc. Bose-Einstein condensation • they have spin 3/2 • p is stable, n → p+e−+¯ν is allowed • the photon couples to charged particles only • there are also gluons which mediate strong interactions − − • e.g. n=(udd) → (uud)+W → p+¯νe +e • is there also a graviton G ? • is there also a Higgs boson H ?
Fermions, Bosons and their Statistics Exchange bosons
Peter Hertel
Fundamental particles − − − − W ν¯e e ν¯µµ ν¯τ τ ud¯ cs¯ ¯tb Spin and statistics 0 + − + − + − γ, Z e e µ µ τ τ (¯νe νe ν¯µνµ ν¯τ ντ ) dd¯ uu¯ ¯ss cc¯ bb¯ ¯tt Many particles + + + + ¯ ¯ Gibbs state W e νe µ νµ τ ντ du ¯sc bt
Fermions
Zero + 0 − temperature • W , γ,Z ,W mediate electro-weak interaction White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • there are also gluons which mediate strong interactions − − • e.g. n=(udd) → (uud)+W → p+¯νe +e • is there also a graviton G ? • is there also a Higgs boson H ?
Fermions, Bosons and their Statistics Exchange bosons
Peter Hertel
Fundamental particles − − − − W ν¯e e ν¯µµ ν¯τ τ ud¯ cs¯ ¯tb Spin and statistics 0 + − + − + − γ, Z e e µ µ τ τ (¯νe νe ν¯µνµ ν¯τ ντ ) dd¯ uu¯ ¯ss cc¯ bb¯ ¯tt Many particles + + + + ¯ ¯ Gibbs state W e νe µ νµ τ ντ du ¯sc bt
Fermions
Zero + 0 − temperature • W , γ,Z ,W mediate electro-weak interaction White dwarfs and neutron • the photon couples to charged particles only stars
Bosons
Black body radiation
Bose-Einstein condensation − − • e.g. n=(udd) → (uud)+W → p+¯νe +e • is there also a graviton G ? • is there also a Higgs boson H ?
Fermions, Bosons and their Statistics Exchange bosons
Peter Hertel
Fundamental particles − − − − W ν¯e e ν¯µµ ν¯τ τ ud¯ cs¯ ¯tb Spin and statistics 0 + − + − + − γ, Z e e µ µ τ τ (¯νe νe ν¯µνµ ν¯τ ντ ) dd¯ uu¯ ¯ss cc¯ bb¯ ¯tt Many particles + + + + ¯ ¯ Gibbs state W e νe µ νµ τ ντ du ¯sc bt
Fermions
Zero + 0 − temperature • W , γ,Z ,W mediate electro-weak interaction White dwarfs and neutron • the photon couples to charged particles only stars • there are also gluons which mediate strong interactions Bosons
Black body radiation
Bose-Einstein condensation • is there also a graviton G ? • is there also a Higgs boson H ?
Fermions, Bosons and their Statistics Exchange bosons
Peter Hertel
Fundamental particles − − − − W ν¯e e ν¯µµ ν¯τ τ ud¯ cs¯ ¯tb Spin and statistics 0 + − + − + − γ, Z e e µ µ τ τ (¯νe νe ν¯µνµ ν¯τ ντ ) dd¯ uu¯ ¯ss cc¯ bb¯ ¯tt Many particles + + + + ¯ ¯ Gibbs state W e νe µ νµ τ ντ du ¯sc bt
Fermions
Zero + 0 − temperature • W , γ,Z ,W mediate electro-weak interaction White dwarfs and neutron • the photon couples to charged particles only stars • there are also gluons which mediate strong interactions Bosons − − Black body • e.g. n=(udd) → (uud)+W → p+¯νe +e radiation
Bose-Einstein condensation • is there also a Higgs boson H ?
Fermions, Bosons and their Statistics Exchange bosons
Peter Hertel
Fundamental particles − − − − W ν¯e e ν¯µµ ν¯τ τ ud¯ cs¯ ¯tb Spin and statistics 0 + − + − + − γ, Z e e µ µ τ τ (¯νe νe ν¯µνµ ν¯τ ντ ) dd¯ uu¯ ¯ss cc¯ bb¯ ¯tt Many particles + + + + ¯ ¯ Gibbs state W e νe µ νµ τ ντ du ¯sc bt
Fermions
Zero + 0 − temperature • W , γ,Z ,W mediate electro-weak interaction White dwarfs and neutron • the photon couples to charged particles only stars • there are also gluons which mediate strong interactions Bosons − − Black body • e.g. n=(udd) → (uud)+W → p+¯νe +e radiation • is there also a graviton G ? Bose-Einstein condensation Fermions, Bosons and their Statistics Exchange bosons
Peter Hertel
Fundamental particles − − − − W ν¯e e ν¯µµ ν¯τ τ ud¯ cs¯ ¯tb Spin and statistics 0 + − + − + − γ, Z e e µ µ τ τ (¯νe νe ν¯µνµ ν¯τ ντ ) dd¯ uu¯ ¯ss cc¯ bb¯ ¯tt Many particles + + + + ¯ ¯ Gibbs state W e νe µ νµ τ ντ du ¯sc bt
Fermions
Zero + 0 − temperature • W , γ,Z ,W mediate electro-weak interaction White dwarfs and neutron • the photon couples to charged particles only stars • there are also gluons which mediate strong interactions Bosons − − Black body • e.g. n=(udd) → (uud)+W → p+¯νe +e radiation • is there also a graviton G ? Bose-Einstein condensation • is there also a Higgs boson H ? Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation The Aleph detector (opened) at Cern • or helicity 1/2 and -1/2 • massless structure fermions (neutrinos) have either helicity 1/2 or -1/2
•
• this explains parity violation • spin and statistics are related
Fermions, Bosons and their Statistics Spin 1/2 fermions
Peter Hertel
Fundamental particles • massive structure fermion (quarks, electrons) have spin Spin and 1/2 statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • massless structure fermions (neutrinos) have either helicity 1/2 or -1/2
•
• this explains parity violation • spin and statistics are related
Fermions, Bosons and their Statistics Spin 1/2 fermions
Peter Hertel
Fundamental particles • massive structure fermion (quarks, electrons) have spin Spin and 1/2 statistics
Many particles • or helicity 1/2 and -1/2
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation •
• this explains parity violation • spin and statistics are related
Fermions, Bosons and their Statistics Spin 1/2 fermions
Peter Hertel
Fundamental particles • massive structure fermion (quarks, electrons) have spin Spin and 1/2 statistics
Many particles • or helicity 1/2 and -1/2 Gibbs state • massless structure fermions (neutrinos) have either helicity Fermions 1/2 or -1/2 Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • this explains parity violation • spin and statistics are related
Fermions, Bosons and their Statistics Spin 1/2 fermions
Peter Hertel
Fundamental particles • massive structure fermion (quarks, electrons) have spin Spin and 1/2 statistics
Many particles • or helicity 1/2 and -1/2 Gibbs state • massless structure fermions (neutrinos) have either helicity Fermions 1/2 or -1/2 Zero temperature
White dwarfs and neutron stars •
Bosons
Black body radiation
Bose-Einstein condensation • spin and statistics are related
Fermions, Bosons and their Statistics Spin 1/2 fermions
Peter Hertel
Fundamental particles • massive structure fermion (quarks, electrons) have spin Spin and 1/2 statistics
Many particles • or helicity 1/2 and -1/2 Gibbs state • massless structure fermions (neutrinos) have either helicity Fermions 1/2 or -1/2 Zero temperature
White dwarfs and neutron stars •
Bosons
Black body radiation • this explains parity violation Bose-Einstein condensation Fermions, Bosons and their Statistics Spin 1/2 fermions
Peter Hertel
Fundamental particles • massive structure fermion (quarks, electrons) have spin Spin and 1/2 statistics
Many particles • or helicity 1/2 and -1/2 Gibbs state • massless structure fermions (neutrinos) have either helicity Fermions 1/2 or -1/2 Zero temperature
White dwarfs and neutron stars •
Bosons
Black body radiation • this explains parity violation Bose-Einstein condensation • spin and statistics are related • a 360◦ rotation yields 2πis ψ¯ = e ψ = φRψ
• φR = +1 for integer spin (bosons), and -1 for half-integer spin (fermions) • exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1) • Pauli exclusion principle
Fermions, Bosons and their Statistics Spin and statisics
Peter Hertel
Fundamental • rotation of a particle with spin s is described by particles i Spin and α · S statistics ψR = e ~ ψ
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • φR = +1 for integer spin (bosons), and -1 for half-integer spin (fermions) • exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1) • Pauli exclusion principle
Fermions, Bosons and their Statistics Spin and statisics
Peter Hertel
Fundamental • rotation of a particle with spin s is described by particles i Spin and α · S statistics ψR = e ~ ψ Many particles • a 360◦ rotation yields Gibbs state ¯ 2πis Fermions ψ = e ψ = φRψ
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • exchanging two identicle particles must not change |ψ|2
ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1)
• spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1) • Pauli exclusion principle
Fermions, Bosons and their Statistics Spin and statisics
Peter Hertel
Fundamental • rotation of a particle with spin s is described by particles i Spin and α · S statistics ψR = e ~ ψ Many particles • a 360◦ rotation yields Gibbs state ¯ 2πis Fermions ψ = e ψ = φRψ Zero • φR = +1 for integer spin (bosons), and -1 for half-integer temperature spin (fermions) White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • spin-statistics theorem says φR = φX
• for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1) • Pauli exclusion principle
Fermions, Bosons and their Statistics Spin and statisics
Peter Hertel
Fundamental • rotation of a particle with spin s is described by particles i Spin and α · S statistics ψR = e ~ ψ Many particles • a 360◦ rotation yields Gibbs state ¯ 2πis Fermions ψ = e ψ = φRψ Zero • φR = +1 for integer spin (bosons), and -1 for half-integer temperature spin (fermions) White dwarfs and neutron 2 stars • exchanging two identicle particles must not change |ψ| Bosons ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1) Black body radiation
Bose-Einstein condensation • for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1) • Pauli exclusion principle
Fermions, Bosons and their Statistics Spin and statisics
Peter Hertel
Fundamental • rotation of a particle with spin s is described by particles i Spin and α · S statistics ψR = e ~ ψ Many particles • a 360◦ rotation yields Gibbs state ¯ 2πis Fermions ψ = e ψ = φRψ Zero • φR = +1 for integer spin (bosons), and -1 for half-integer temperature spin (fermions) White dwarfs and neutron 2 stars • exchanging two identicle particles must not change |ψ| Bosons ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1) Black body radiation • spin-statistics theorem says φR = φX
Bose-Einstein condensation • Pauli exclusion principle
Fermions, Bosons and their Statistics Spin and statisics
Peter Hertel
Fundamental • rotation of a particle with spin s is described by particles i Spin and α · S statistics ψR = e ~ ψ Many particles • a 360◦ rotation yields Gibbs state ¯ 2πis Fermions ψ = e ψ = φRψ Zero • φR = +1 for integer spin (bosons), and -1 for half-integer temperature spin (fermions) White dwarfs and neutron 2 stars • exchanging two identicle particles must not change |ψ| Bosons ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1) Black body radiation • spin-statistics theorem says φR = φX
Bose-Einstein • for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1) condensation Fermions, Bosons and their Statistics Spin and statisics
Peter Hertel
Fundamental • rotation of a particle with spin s is described by particles i Spin and α · S statistics ψR = e ~ ψ Many particles • a 360◦ rotation yields Gibbs state ¯ 2πis Fermions ψ = e ψ = φRψ Zero • φR = +1 for integer spin (bosons), and -1 for half-integer temperature spin (fermions) White dwarfs and neutron 2 stars • exchanging two identicle particles must not change |ψ| Bosons ψ(σ1x1, σ2x2) = φXψ(σ2x2, σ1x1) Black body radiation • spin-statistics theorem says φR = φX
Bose-Einstein • for fermions: ψ(σ1x1, σ2x2) = −ψ(σ2x2, σ1x1) condensation • Pauli exclusion principle Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation
Wolfgang Pauli, Austrian/Swiss Physicist, Nobel prize 1945 • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator • fermions: (X , Y ) = {X , Y } = XY + YX , the anti-commutator • N = A∗A is number operator • Ω is vacuum • create n particles 1 ∗ n ψn = √ (A ) Ω n! • Nψn = n ψn • bosons: n = 0, 1, 2,... • fermions: n = 0, 1
Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator • fermions: (X , Y ) = {X , Y } = XY + YX , the anti-commutator • N = A∗A is number operator • Ω is vacuum • create n particles 1 ∗ n ψn = √ (A ) Ω n! • Nψn = n ψn • bosons: n = 0, 1, 2,... • fermions: n = 0, 1
Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • fermions: (X , Y ) = {X , Y } = XY + YX , the anti-commutator • N = A∗A is number operator • Ω is vacuum • create n particles 1 ∗ n ψn = √ (A ) Ω n! • Nψn = n ψn • bosons: n = 0, 1, 2,... • fermions: n = 0, 1
Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I Spin and statistics • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • N = A∗A is number operator • Ω is vacuum • create n particles 1 ∗ n ψn = √ (A ) Ω n! • Nψn = n ψn • bosons: n = 0, 1, 2,... • fermions: n = 0, 1
Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I Spin and statistics • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator Many particles • fermions: (X , Y ) = {X , Y } = XY + YX , the Gibbs state anti-commutator Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • Ω is vacuum • create n particles 1 ∗ n ψn = √ (A ) Ω n! • Nψn = n ψn • bosons: n = 0, 1, 2,... • fermions: n = 0, 1
Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I Spin and statistics • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator Many particles • fermions: (X , Y ) = {X , Y } = XY + YX , the Gibbs state anti-commutator Fermions ∗ Zero • N = A A is number operator temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • create n particles 1 ∗ n ψn = √ (A ) Ω n! • Nψn = n ψn • bosons: n = 0, 1, 2,... • fermions: n = 0, 1
Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I Spin and statistics • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator Many particles • fermions: (X , Y ) = {X , Y } = XY + YX , the Gibbs state anti-commutator Fermions ∗ Zero • N = A A is number operator temperature • Ω is vacuum White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • Nψn = n ψn • bosons: n = 0, 1, 2,... • fermions: n = 0, 1
Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I Spin and statistics • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator Many particles • fermions: (X , Y ) = {X , Y } = XY + YX , the Gibbs state anti-commutator Fermions ∗ Zero • N = A A is number operator temperature • Ω is vacuum White dwarfs and neutron stars • create n particles Bosons 1 ∗ n ψn = √ (A ) Ω Black body n! radiation
Bose-Einstein condensation • bosons: n = 0, 1, 2,... • fermions: n = 0, 1
Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I Spin and statistics • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator Many particles • fermions: (X , Y ) = {X , Y } = XY + YX , the Gibbs state anti-commutator Fermions ∗ Zero • N = A A is number operator temperature • Ω is vacuum White dwarfs and neutron stars • create n particles Bosons 1 ∗ n ψn = √ (A ) Ω Black body n! radiation • Nψn = n ψn Bose-Einstein condensation • fermions: n = 0, 1
Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I Spin and statistics • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator Many particles • fermions: (X , Y ) = {X , Y } = XY + YX , the Gibbs state anti-commutator Fermions ∗ Zero • N = A A is number operator temperature • Ω is vacuum White dwarfs and neutron stars • create n particles Bosons 1 ∗ n ψn = √ (A ) Ω Black body n! radiation • Nψn = n ψn Bose-Einstein condensation • bosons: n = 0, 1, 2,... Fermions, Bosons and their Statistics Arbitrary number of particles
Peter Hertel • A∗ creates a particle, A = (A∗)† annihilates a particle Fundamental particles • (A, A) = 0, (A∗, A∗) = 0, (A, A∗) = I Spin and statistics • bosons: (X , Y ) = [X , Y ] = XY − YX , the commutator Many particles • fermions: (X , Y ) = {X , Y } = XY + YX , the Gibbs state anti-commutator Fermions ∗ Zero • N = A A is number operator temperature • Ω is vacuum White dwarfs and neutron stars • create n particles Bosons 1 ∗ n ψn = √ (A ) Ω Black body n! radiation • Nψn = n ψn Bose-Einstein condensation • bosons: n = 0, 1, 2,... • fermions: n = 0, 1 ∗ • creators Aj and annihilators Aj ∗ ∗ ∗ • (Aj , Ak ) = (Aj , Ak ) = 0 and (Aj , Ak ) = δjk • for fermions: ∗ ∗ ∗ ∗ • Aj Ak Ω = −Ak Aj Ω ∗ • Nj = Aj Aj has eigenvalues 0 and 1, i.e. • a state is occupied at most once
• note that [Nj , Nk ] = 0
Fermions, Bosons and their Statistics Many states
Peter Hertel
Fundamental particles Spin and • statistics j labels one-particle states
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation ∗ ∗ ∗ • (Aj , Ak ) = (Aj , Ak ) = 0 and (Aj , Ak ) = δjk • for fermions: ∗ ∗ ∗ ∗ • Aj Ak Ω = −Ak Aj Ω ∗ • Nj = Aj Aj has eigenvalues 0 and 1, i.e. • a state is occupied at most once
• note that [Nj , Nk ] = 0
Fermions, Bosons and their Statistics Many states
Peter Hertel
Fundamental particles Spin and • statistics j labels one-particle states ∗ Many particles • creators Aj and annihilators Aj Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • for fermions: ∗ ∗ ∗ ∗ • Aj Ak Ω = −Ak Aj Ω ∗ • Nj = Aj Aj has eigenvalues 0 and 1, i.e. • a state is occupied at most once
• note that [Nj , Nk ] = 0
Fermions, Bosons and their Statistics Many states
Peter Hertel
Fundamental particles Spin and • statistics j labels one-particle states ∗ Many particles • creators Aj and annihilators Aj Gibbs state ∗ ∗ ∗ • (Aj , Ak ) = (Aj , Ak ) = 0 and (Aj , Ak ) = δjk Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation ∗ ∗ ∗ ∗ • Aj Ak Ω = −Ak Aj Ω ∗ • Nj = Aj Aj has eigenvalues 0 and 1, i.e. • a state is occupied at most once
• note that [Nj , Nk ] = 0
Fermions, Bosons and their Statistics Many states
Peter Hertel
Fundamental particles Spin and • statistics j labels one-particle states ∗ Many particles • creators Aj and annihilators Aj Gibbs state ∗ ∗ ∗ • (Aj , Ak ) = (Aj , Ak ) = 0 and (Aj , Ak ) = δjk Fermions • for fermions: Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation ∗ • Nj = Aj Aj has eigenvalues 0 and 1, i.e. • a state is occupied at most once
• note that [Nj , Nk ] = 0
Fermions, Bosons and their Statistics Many states
Peter Hertel
Fundamental particles Spin and • statistics j labels one-particle states ∗ Many particles • creators Aj and annihilators Aj Gibbs state ∗ ∗ ∗ • (Aj , Ak ) = (Aj , Ak ) = 0 and (Aj , Ak ) = δjk Fermions • for fermions: Zero temperature ∗ ∗ ∗ ∗ • Aj Ak Ω = −Ak Aj Ω White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • a state is occupied at most once
• note that [Nj , Nk ] = 0
Fermions, Bosons and their Statistics Many states
Peter Hertel
Fundamental particles Spin and • statistics j labels one-particle states ∗ Many particles • creators Aj and annihilators Aj Gibbs state ∗ ∗ ∗ • (Aj , Ak ) = (Aj , Ak ) = 0 and (Aj , Ak ) = δjk Fermions • for fermions: Zero temperature ∗ ∗ ∗ ∗ • Aj Ak Ω = −Ak Aj Ω White dwarfs and neutron • N = A∗A has eigenvalues 0 and 1, i.e. stars j j j
Bosons
Black body radiation
Bose-Einstein condensation • note that [Nj , Nk ] = 0
Fermions, Bosons and their Statistics Many states
Peter Hertel
Fundamental particles Spin and • statistics j labels one-particle states ∗ Many particles • creators Aj and annihilators Aj Gibbs state ∗ ∗ ∗ • (Aj , Ak ) = (Aj , Ak ) = 0 and (Aj , Ak ) = δjk Fermions • for fermions: Zero temperature ∗ ∗ ∗ ∗ • Aj Ak Ω = −Ak Aj Ω White dwarfs and neutron • N = A∗A has eigenvalues 0 and 1, i.e. stars j j j Bosons • a state is occupied at most once Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Many states
Peter Hertel
Fundamental particles Spin and • statistics j labels one-particle states ∗ Many particles • creators Aj and annihilators Aj Gibbs state ∗ ∗ ∗ • (Aj , Ak ) = (Aj , Ak ) = 0 and (Aj , Ak ) = δjk Fermions • for fermions: Zero temperature ∗ ∗ ∗ ∗ • Aj Ak Ω = −Ak Aj Ω White dwarfs and neutron • N = A∗A has eigenvalues 0 and 1, i.e. stars j j j Bosons • a state is occupied at most once Black body radiation • note that [Nj , Nk ] = 0 Bose-Einstein condensation • if interaction term can be neglected P • H = j j Nj • examples are quasi-free electrons (hopping model) or phonons P • N = j Nj is the particle number operator • note that the number of particles in a system is not fixed (open system)
Fermions, Bosons and their Statistics Energy
Peter Hertel
Fundamental particles Spin and P ∗ P ∗ ∗ statistics • H = j j Aj Aj + jklm Vjklm Aj Ak Al Am Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation P • H = j j Nj • examples are quasi-free electrons (hopping model) or phonons P • N = j Nj is the particle number operator • note that the number of particles in a system is not fixed (open system)
Fermions, Bosons and their Statistics Energy
Peter Hertel
Fundamental particles Spin and P ∗ P ∗ ∗ statistics • H = j j Aj Aj + jklm Vjklm Aj Ak Al Am Many particles • if interaction term can be neglected Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • examples are quasi-free electrons (hopping model) or phonons P • N = j Nj is the particle number operator • note that the number of particles in a system is not fixed (open system)
Fermions, Bosons and their Statistics Energy
Peter Hertel
Fundamental particles Spin and P ∗ P ∗ ∗ statistics • H = j j Aj Aj + jklm Vjklm Aj Ak Al Am Many particles • if interaction term can be neglected Gibbs state • H = P N Fermions j j j
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation P • N = j Nj is the particle number operator • note that the number of particles in a system is not fixed (open system)
Fermions, Bosons and their Statistics Energy
Peter Hertel
Fundamental particles Spin and P ∗ P ∗ ∗ statistics • H = j j Aj Aj + jklm Vjklm Aj Ak Al Am Many particles • if interaction term can be neglected Gibbs state • H = P N Fermions j j j
Zero • examples are quasi-free electrons (hopping model) or temperature phonons White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • note that the number of particles in a system is not fixed (open system)
Fermions, Bosons and their Statistics Energy
Peter Hertel
Fundamental particles Spin and P ∗ P ∗ ∗ statistics • H = j j Aj Aj + jklm Vjklm Aj Ak Al Am Many particles • if interaction term can be neglected Gibbs state • H = P N Fermions j j j
Zero • examples are quasi-free electrons (hopping model) or temperature phonons White dwarfs and neutron • P stars N = j Nj is the particle number operator
Bosons
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Energy
Peter Hertel
Fundamental particles Spin and P ∗ P ∗ ∗ statistics • H = j j Aj Aj + jklm Vjklm Aj Ak Al Am Many particles • if interaction term can be neglected Gibbs state • H = P N Fermions j j j
Zero • examples are quasi-free electrons (hopping model) or temperature phonons White dwarfs and neutron • P stars N = j Nj is the particle number operator Bosons • note that the number of particles in a system is not fixed Black body (open system) radiation
Bose-Einstein condensation • free energy F defined by tr G = 1 • which gives (µN − H)/kBT F = −kBT ln tr e • temperature T defined by tr GH = U = H¯ • chemical potential µ defined by tr GN = N¯ • generalization to more than one species of particles is obvious
Fermions, Bosons and their Statistics Free energy
Peter Hertel
Fundamental particles
Spin and • The equilibrium (Gibbs) state of an open system is statistics (F − H + µN)/kBT Many particles G = e
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • which gives (µN − H)/kBT F = −kBT ln tr e • temperature T defined by tr GH = U = H¯ • chemical potential µ defined by tr GN = N¯ • generalization to more than one species of particles is obvious
Fermions, Bosons and their Statistics Free energy
Peter Hertel
Fundamental particles
Spin and • The equilibrium (Gibbs) state of an open system is statistics (F − H + µN)/kBT Many particles G = e Gibbs state • free energy F defined by tr G = 1 Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • temperature T defined by tr GH = U = H¯ • chemical potential µ defined by tr GN = N¯ • generalization to more than one species of particles is obvious
Fermions, Bosons and their Statistics Free energy
Peter Hertel
Fundamental particles
Spin and • The equilibrium (Gibbs) state of an open system is statistics (F − H + µN)/kBT Many particles G = e Gibbs state • free energy F defined by tr G = 1 Fermions • which gives Zero temperature (µN − H)/k T F = −k T ln tr e B White dwarfs B and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • chemical potential µ defined by tr GN = N¯ • generalization to more than one species of particles is obvious
Fermions, Bosons and their Statistics Free energy
Peter Hertel
Fundamental particles
Spin and • The equilibrium (Gibbs) state of an open system is statistics (F − H + µN)/kBT Many particles G = e Gibbs state • free energy F defined by tr G = 1 Fermions • which gives Zero temperature (µN − H)/k T F = −k T ln tr e B White dwarfs B and neutron ¯ stars • temperature T defined by tr GH = U = H
Bosons
Black body radiation
Bose-Einstein condensation • generalization to more than one species of particles is obvious
Fermions, Bosons and their Statistics Free energy
Peter Hertel
Fundamental particles
Spin and • The equilibrium (Gibbs) state of an open system is statistics (F − H + µN)/kBT Many particles G = e Gibbs state • free energy F defined by tr G = 1 Fermions • which gives Zero temperature (µN − H)/k T F = −k T ln tr e B White dwarfs B and neutron ¯ stars • temperature T defined by tr GH = U = H Bosons • chemical potential µ defined by tr GN = N¯ Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Free energy
Peter Hertel
Fundamental particles
Spin and • The equilibrium (Gibbs) state of an open system is statistics (F − H + µN)/kBT Many particles G = e Gibbs state • free energy F defined by tr G = 1 Fermions • which gives Zero temperature (µN − H)/k T F = −k T ln tr e B White dwarfs B and neutron ¯ stars • temperature T defined by tr GH = U = H Bosons • chemical potential µ defined by tr GN = N¯ Black body radiation • generalization to more than one species of particles is
Bose-Einstein obvious condensation Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation
Josiah Willard Gibbs, US American physicist, 1839-1903 • work out the trace, i. e. sum over eigenvalues n = 0, 1 (µN − H)/k T Y (µ − )/k T tr e B = 1 + e j B j • free energy X (µ − j )/kBT F = −kBT ln 1 + e j • in [, + d] there are V z() d single particle states • one may write Z (µ − )/kBT F = −kBTV d z() ln 1 + e
Fermions, Bosons and their Statistics Fermi-Dirac statistics
Peter Hertel • because particle number operators commute Fundamental (µN − H)/k T Y (µ − )N /k T particles e B = e j j B Spin and statistics j
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • free energy X (µ − j )/kBT F = −kBT ln 1 + e j • in [, + d] there are V z() d single particle states • one may write Z (µ − )/kBT F = −kBTV d z() ln 1 + e
Fermions, Bosons and their Statistics Fermi-Dirac statistics
Peter Hertel • because particle number operators commute Fundamental (µN − H)/k T Y (µ − )N /k T particles e B = e j j B Spin and statistics j Many particles • work out the trace, i. e. sum over eigenvalues n = 0, 1 Gibbs state (µN − H)/kBT Y (µ − j )/kBT Fermions tr e = 1 + e
Zero j temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • in [, + d] there are V z() d single particle states • one may write Z (µ − )/kBT F = −kBTV d z() ln 1 + e
Fermions, Bosons and their Statistics Fermi-Dirac statistics
Peter Hertel • because particle number operators commute Fundamental (µN − H)/k T Y (µ − )N /k T particles e B = e j j B Spin and statistics j Many particles • work out the trace, i. e. sum over eigenvalues n = 0, 1 Gibbs state (µN − H)/kBT Y (µ − j )/kBT Fermions tr e = 1 + e
Zero j temperature • free energy White dwarfs and neutron X (µ − j )/kBT stars F = −kBT ln 1 + e Bosons j Black body radiation
Bose-Einstein condensation • one may write Z (µ − )/kBT F = −kBTV d z() ln 1 + e
Fermions, Bosons and their Statistics Fermi-Dirac statistics
Peter Hertel • because particle number operators commute Fundamental (µN − H)/k T Y (µ − )N /k T particles e B = e j j B Spin and statistics j Many particles • work out the trace, i. e. sum over eigenvalues n = 0, 1 Gibbs state (µN − H)/kBT Y (µ − j )/kBT Fermions tr e = 1 + e
Zero j temperature • free energy White dwarfs and neutron X (µ − j )/kBT stars F = −kBT ln 1 + e Bosons j Black body • in [, + d] there are V z() d single particle states radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Fermi-Dirac statistics
Peter Hertel • because particle number operators commute Fundamental (µN − H)/k T Y (µ − )N /k T particles e B = e j j B Spin and statistics j Many particles • work out the trace, i. e. sum over eigenvalues n = 0, 1 Gibbs state (µN − H)/kBT Y (µ − j )/kBT Fermions tr e = 1 + e
Zero j temperature • free energy White dwarfs and neutron X (µ − j )/kBT stars F = −kBT ln 1 + e Bosons j Black body • in [, + d] there are V z() d single particle states radiation
Bose-Einstein • one may write condensation Z (µ − )/kBT F = −kBTV d z() ln 1 + e Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation
Enrico Fermi, Italian/USA physicist, 1901-1954 Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation
Paul Dirac, Britisch physicist, 1902-1984 • = p2/2m • gives X Z = V d z() j 1 2m3/2 Z √ = V d π2 ~2
Fermions, Bosons and their Statistics Non-relativistic fermions
Peter Hertel
Fundamental particles • sum over states (factor 2 for spin) Spin and statistics X Z d3x d3p Many particles = 2 (2π~)3 Gibbs state j
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • gives X Z = V d z() j 1 2m3/2 Z √ = V d π2 ~2
Fermions, Bosons and their Statistics Non-relativistic fermions
Peter Hertel
Fundamental particles • sum over states (factor 2 for spin) Spin and statistics X Z d3x d3p Many particles = 2 (2π~)3 Gibbs state j Fermions • = p2/2m Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Non-relativistic fermions
Peter Hertel
Fundamental particles • sum over states (factor 2 for spin) Spin and statistics X Z d3x d3p Many particles = 2 (2π~)3 Gibbs state j Fermions • = p2/2m Zero temperature • gives Z White dwarfs X and neutron = V d z() stars j Bosons 3/2 Black body 1 2m Z √ radiation = V d π2 2 Bose-Einstein ~ condensation • particle density is 1 ∂F n(T , µ) = − V ∂µ Z 1 = d z() ( − µ)/k T e B + 1 • pressure is ∂F p = − ∂V Z (µ − )/kBT = kBT d z() ln 1 + e 2 Z = d z() 3 ( − µ)/k T e B + 1
Fermions, Bosons and their Statistics Particle density and pressure Peter Hertel • recall Z Fundamental (µ − )/kBT particles F = −kBTV d z() ln 1 + e Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • pressure is ∂F p = − ∂V Z (µ − )/kBT = kBT d z() ln 1 + e 2 Z = d z() 3 ( − µ)/k T e B + 1
Fermions, Bosons and their Statistics Particle density and pressure Peter Hertel • recall Z Fundamental (µ − )/kBT particles F = −kBTV d z() ln 1 + e Spin and statistics • particle density is Many particles 1 ∂F n(T , µ) = − Gibbs state V ∂µ Fermions Z 1 Zero = d z() temperature ( − µ)/k T e B + 1 White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Particle density and pressure Peter Hertel • recall Z Fundamental (µ − )/kBT particles F = −kBTV d z() ln 1 + e Spin and statistics • particle density is Many particles 1 ∂F n(T , µ) = − Gibbs state V ∂µ Fermions Z 1 Zero = d z() temperature ( − µ)/k T e B + 1 White dwarfs and neutron • pressure is stars ∂F Bosons p = − ∂V Black body radiation Z (µ − )/k T = k T d z() ln 1 + e B Bose-Einstein B condensation 2 Z = d z() 3 ( − µ)/k T e B + 1 • particle density Z µ n(0, µ) = d z() −∞ • pressure 2 Z µ p(0, µ) = d z() 3 −∞ • if particle densityn ¯ is given, then Z F n¯ = n(0, F) = d z() −∞ defines Fermi energy
Fermions, Bosons and their Statistics Zero temperature
Peter Hertel
Fundamental • if T → 0 particles 1 = θ(µ − ) Spin and ( − µ)/k T statistics e B + 1 Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • pressure 2 Z µ p(0, µ) = d z() 3 −∞ • if particle densityn ¯ is given, then Z F n¯ = n(0, F) = d z() −∞ defines Fermi energy
Fermions, Bosons and their Statistics Zero temperature
Peter Hertel
Fundamental • if T → 0 particles 1 = θ(µ − ) Spin and ( − µ)/k T statistics e B + 1 Many particles • particle density Gibbs state Z µ Fermions n(0, µ) = d z() Zero −∞ temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • if particle densityn ¯ is given, then Z F n¯ = n(0, F) = d z() −∞ defines Fermi energy
Fermions, Bosons and their Statistics Zero temperature
Peter Hertel
Fundamental • if T → 0 particles 1 = θ(µ − ) Spin and ( − µ)/k T statistics e B + 1 Many particles • particle density Gibbs state Z µ Fermions n(0, µ) = d z() Zero −∞ temperature • pressure White dwarfs 2 Z µ and neutron p(0, µ) = d z() stars 3 −∞ Bosons
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Zero temperature
Peter Hertel
Fundamental • if T → 0 particles 1 = θ(µ − ) Spin and ( − µ)/k T statistics e B + 1 Many particles • particle density Gibbs state Z µ Fermions n(0, µ) = d z() Zero −∞ temperature • pressure White dwarfs 2 Z µ and neutron p(0, µ) = d z() stars 3 −∞ Bosons • if particle densityn ¯ is given, then Black body F radiation Z n¯ = n(0, F) = d z() Bose-Einstein condensation −∞ defines Fermi energy • from small to huge molecules • dielectrics, conductors, semiconductors, ferromagnets • stability of normal matter • stars can be stable even at T = 0
Fermions, Bosons and their Statistics Atoms and so forth
Peter Hertel
Fundamental particles
Spin and statistics Many particles • Periodic system Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • dielectrics, conductors, semiconductors, ferromagnets • stability of normal matter • stars can be stable even at T = 0
Fermions, Bosons and their Statistics Atoms and so forth
Peter Hertel
Fundamental particles
Spin and statistics Many particles • Periodic system Gibbs state
Fermions • from small to huge molecules
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • stability of normal matter • stars can be stable even at T = 0
Fermions, Bosons and their Statistics Atoms and so forth
Peter Hertel
Fundamental particles
Spin and statistics Many particles • Periodic system Gibbs state
Fermions • from small to huge molecules Zero • dielectrics, conductors, semiconductors, ferromagnets temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • stars can be stable even at T = 0
Fermions, Bosons and their Statistics Atoms and so forth
Peter Hertel
Fundamental particles
Spin and statistics Many particles • Periodic system Gibbs state
Fermions • from small to huge molecules Zero • dielectrics, conductors, semiconductors, ferromagnets temperature
White dwarfs • stability of normal matter and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Atoms and so forth
Peter Hertel
Fundamental particles
Spin and statistics Many particles • Periodic system Gibbs state
Fermions • from small to huge molecules Zero • dielectrics, conductors, semiconductors, ferromagnets temperature
White dwarfs • stability of normal matter and neutron stars • stars can be stable even at T = 0 Bosons
Black body radiation
Bose-Einstein condensation • gravitational force per unit volume ρ(r)M(r) f (r) = −G r 2 • pressure gradient and force must balance p 0(r) = f (r) • relate pressure with mass density
Fermions, Bosons and their Statistics Hydrodynamic equilibrium
Peter Hertel
Fundamental particles
Spin and statistics • The mass within a sphere of radius R is Z r Many particles M(r) = 4π ds s2ρ(s) Gibbs state 0 Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • pressure gradient and force must balance p 0(r) = f (r) • relate pressure with mass density
Fermions, Bosons and their Statistics Hydrodynamic equilibrium
Peter Hertel
Fundamental particles
Spin and statistics • The mass within a sphere of radius R is Z r Many particles M(r) = 4π ds s2ρ(s) Gibbs state 0 Fermions • gravitational force per unit volume Zero temperature ρ(r)M(r) f (r) = −G 2 White dwarfs r and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • relate pressure with mass density
Fermions, Bosons and their Statistics Hydrodynamic equilibrium
Peter Hertel
Fundamental particles
Spin and statistics • The mass within a sphere of radius R is Z r Many particles M(r) = 4π ds s2ρ(s) Gibbs state 0 Fermions • gravitational force per unit volume Zero temperature ρ(r)M(r) f (r) = −G 2 White dwarfs r and neutron stars • pressure gradient and force must balance 0 Bosons p (r) = f (r) Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Hydrodynamic equilibrium
Peter Hertel
Fundamental particles
Spin and statistics • The mass within a sphere of radius R is Z r Many particles M(r) = 4π ds s2ρ(s) Gibbs state 0 Fermions • gravitational force per unit volume Zero temperature ρ(r)M(r) f (r) = −G 2 White dwarfs r and neutron stars • pressure gradient and force must balance 0 Bosons p (r) = f (r) Black body radiation • relate pressure with mass density
Bose-Einstein condensation • recall 1 2m3/2 √ z() = π2 ~2 • particle density 2 2m3/2 n(0, µ) = µ3/2 3π2 ~2 • pressure 4 2m3/2 p(0, µ) = µ5/2 15π2 ~2 • eliminate chemical potential 2 p = a ~ n5/3 where a = 1.2058 m • since one electron is related with two nucleons 2 ρ 5/3 p = a ~ me 2mp
Fermions, Bosons and their Statistics Equation of state Peter Hertel • electron gas at T = 0
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • particle density 2 2m3/2 n(0, µ) = µ3/2 3π2 ~2 • pressure 4 2m3/2 p(0, µ) = µ5/2 15π2 ~2 • eliminate chemical potential 2 p = a ~ n5/3 where a = 1.2058 m • since one electron is related with two nucleons 2 ρ 5/3 p = a ~ me 2mp
Fermions, Bosons and their Statistics Equation of state Peter Hertel • electron gas at T = 0
Fundamental • recall particles 1 2m3/2 √ Spin and z() = statistics π2 ~2 Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • pressure 4 2m3/2 p(0, µ) = µ5/2 15π2 ~2 • eliminate chemical potential 2 p = a ~ n5/3 where a = 1.2058 m • since one electron is related with two nucleons 2 ρ 5/3 p = a ~ me 2mp
Fermions, Bosons and their Statistics Equation of state Peter Hertel • electron gas at T = 0
Fundamental • recall particles 1 2m3/2 √ Spin and z() = statistics π2 ~2 Many particles • particle density Gibbs state 3/2 2 2m 3/2 Fermions n(0, µ) = µ 3π2 2 Zero ~ temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • eliminate chemical potential 2 p = a ~ n5/3 where a = 1.2058 m • since one electron is related with two nucleons 2 ρ 5/3 p = a ~ me 2mp
Fermions, Bosons and their Statistics Equation of state Peter Hertel • electron gas at T = 0
Fundamental • recall particles 1 2m3/2 √ Spin and z() = statistics π2 ~2 Many particles • particle density Gibbs state 3/2 2 2m 3/2 Fermions n(0, µ) = µ 3π2 2 Zero ~ temperature • pressure 3/2 White dwarfs 4 2m 5/2 and neutron p(0, µ) = µ stars 15π2 ~2 Bosons
Black body radiation
Bose-Einstein condensation • since one electron is related with two nucleons 2 ρ 5/3 p = a ~ me 2mp
Fermions, Bosons and their Statistics Equation of state Peter Hertel • electron gas at T = 0
Fundamental • recall particles 1 2m3/2 √ Spin and z() = statistics π2 ~2 Many particles • particle density Gibbs state 3/2 2 2m 3/2 Fermions n(0, µ) = µ 3π2 2 Zero ~ temperature • pressure 3/2 White dwarfs 4 2m 5/2 and neutron p(0, µ) = µ stars 15π2 ~2 Bosons • eliminate chemical potential Black body 2 radiation p = a ~ n5/3 where a = 1.2058 Bose-Einstein m condensation Fermions, Bosons and their Statistics Equation of state Peter Hertel • electron gas at T = 0
Fundamental • recall particles 1 2m3/2 √ Spin and z() = statistics π2 ~2 Many particles • particle density Gibbs state 3/2 2 2m 3/2 Fermions n(0, µ) = µ 3π2 2 Zero ~ temperature • pressure 3/2 White dwarfs 4 2m 5/2 and neutron p(0, µ) = µ stars 15π2 ~2 Bosons • eliminate chemical potential Black body 2 radiation p = a ~ n5/3 where a = 1.2058 Bose-Einstein m condensation • since one electron is related with two nucleons 2 ρ 5/3 p = a ~ me 2mp Fermions, 1 Bosons and their Statistics 0.9
Peter Hertel 0.8
Fundamental particles 0.7
Spin and 0.6 statistics
Many particles 0.5 Gibbs state 0.4 Fermions
Zero 0.3 temperature
White dwarfs 0.2 and neutron stars 0.1 Bosons 0 Black body 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 radiation
Bose-Einstein Pressure p (decreasing) and mass M (increasing) of a white dwarf vs. distance r condensation from the center in natural units. We have assumed zero temperature, two nucleons per electron which are treated non-relativistically. r = 1 corresponds to 6500 km, M = 1 to 0.85 sun masses. Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation
Crab nebula, the cloud of debris of the 1054 supernova Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation
There is a neutron star (pulsar) at its center. Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation
Jagadris Chandra Bose, Indian physicist, 1858-1937 • such as the photon • such as (qq¯) states - mesons • or (ee) - Cooper pairs • or (ppnn), the helium nucleus or helium atom • or phonons, the quanta of lattice vibrations
Fermions, Bosons and their Statistics Bosons
Peter Hertel
Fundamental particles
Spin and statistics
Many particles • bosons have integer spin or integer helicity
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • such as (qq¯) states - mesons • or (ee) - Cooper pairs • or (ppnn), the helium nucleus or helium atom • or phonons, the quanta of lattice vibrations
Fermions, Bosons and their Statistics Bosons
Peter Hertel
Fundamental particles
Spin and statistics
Many particles • bosons have integer spin or integer helicity Gibbs state • such as the photon Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • or (ee) - Cooper pairs • or (ppnn), the helium nucleus or helium atom • or phonons, the quanta of lattice vibrations
Fermions, Bosons and their Statistics Bosons
Peter Hertel
Fundamental particles
Spin and statistics
Many particles • bosons have integer spin or integer helicity Gibbs state • such as the photon Fermions • such as (qq¯) states - mesons Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • or (ppnn), the helium nucleus or helium atom • or phonons, the quanta of lattice vibrations
Fermions, Bosons and their Statistics Bosons
Peter Hertel
Fundamental particles
Spin and statistics
Many particles • bosons have integer spin or integer helicity Gibbs state • such as the photon Fermions • such as (qq¯) states - mesons Zero temperature • or (ee) - Cooper pairs White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • or phonons, the quanta of lattice vibrations
Fermions, Bosons and their Statistics Bosons
Peter Hertel
Fundamental particles
Spin and statistics
Many particles • bosons have integer spin or integer helicity Gibbs state • such as the photon Fermions • such as (qq¯) states - mesons Zero temperature • or (ee) - Cooper pairs White dwarfs and neutron • or (ppnn), the helium nucleus or helium atom stars
Bosons
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Bosons
Peter Hertel
Fundamental particles
Spin and statistics
Many particles • bosons have integer spin or integer helicity Gibbs state • such as the photon Fermions • such as (qq¯) states - mesons Zero temperature • or (ee) - Cooper pairs White dwarfs and neutron • or (ppnn), the helium nucleus or helium atom stars
Bosons • or phonons, the quanta of lattice vibrations
Black body radiation
Bose-Einstein condensation ∗ ∗ • [Aj , Ak ] = [Aj , Ak ] = 0 - commutators! ∗ • [Aj , Ak ] = δjk I ∗ • Nj = Aj , Aj - number of particles in state j • eigenvalues are n = 0, 1, 2,... • work out the trace, i. e. sum over eigenvalues n = 0, 1, 2,... P • N = j Nj - total number of indistinguishable particles
Fermions, Bosons and their Statistics Many bosons
Peter Hertel
Fundamental particles
Spin and statistics • recall the spin/statistics theorem
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation ∗ • [Aj , Ak ] = δjk I ∗ • Nj = Aj , Aj - number of particles in state j • eigenvalues are n = 0, 1, 2,... • work out the trace, i. e. sum over eigenvalues n = 0, 1, 2,... P • N = j Nj - total number of indistinguishable particles
Fermions, Bosons and their Statistics Many bosons
Peter Hertel
Fundamental particles
Spin and statistics • recall the spin/statistics theorem
Many particles ∗ ∗ • [Aj , Ak ] = [Aj , Ak ] = 0 - commutators! Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation ∗ • Nj = Aj , Aj - number of particles in state j • eigenvalues are n = 0, 1, 2,... • work out the trace, i. e. sum over eigenvalues n = 0, 1, 2,... P • N = j Nj - total number of indistinguishable particles
Fermions, Bosons and their Statistics Many bosons
Peter Hertel
Fundamental particles
Spin and statistics • recall the spin/statistics theorem
Many particles ∗ ∗ • [Aj , Ak ] = [Aj , Ak ] = 0 - commutators! Gibbs state ∗ • [Aj , A ] = δjk I Fermions k
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • eigenvalues are n = 0, 1, 2,... • work out the trace, i. e. sum over eigenvalues n = 0, 1, 2,... P • N = j Nj - total number of indistinguishable particles
Fermions, Bosons and their Statistics Many bosons
Peter Hertel
Fundamental particles
Spin and statistics • recall the spin/statistics theorem
Many particles ∗ ∗ • [Aj , Ak ] = [Aj , Ak ] = 0 - commutators! Gibbs state ∗ • [Aj , A ] = δjk I Fermions k ∗ Zero • Nj = Aj , Aj - number of particles in state j temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • work out the trace, i. e. sum over eigenvalues n = 0, 1, 2,... P • N = j Nj - total number of indistinguishable particles
Fermions, Bosons and their Statistics Many bosons
Peter Hertel
Fundamental particles
Spin and statistics • recall the spin/statistics theorem
Many particles ∗ ∗ • [Aj , Ak ] = [Aj , Ak ] = 0 - commutators! Gibbs state ∗ • [Aj , A ] = δjk I Fermions k ∗ Zero • Nj = Aj , Aj - number of particles in state j temperature • eigenvalues are n = 0, 1, 2,... White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation P • N = j Nj - total number of indistinguishable particles
Fermions, Bosons and their Statistics Many bosons
Peter Hertel
Fundamental particles
Spin and statistics • recall the spin/statistics theorem
Many particles ∗ ∗ • [Aj , Ak ] = [Aj , Ak ] = 0 - commutators! Gibbs state ∗ • [Aj , A ] = δjk I Fermions k ∗ Zero • Nj = Aj , Aj - number of particles in state j temperature • eigenvalues are n = 0, 1, 2,... White dwarfs and neutron stars • work out the trace, i. e. sum over eigenvalues
Bosons n = 0, 1, 2,...
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Many bosons
Peter Hertel
Fundamental particles
Spin and statistics • recall the spin/statistics theorem
Many particles ∗ ∗ • [Aj , Ak ] = [Aj , Ak ] = 0 - commutators! Gibbs state ∗ • [Aj , A ] = δjk I Fermions k ∗ Zero • Nj = Aj , Aj - number of particles in state j temperature • eigenvalues are n = 0, 1, 2,... White dwarfs and neutron stars • work out the trace, i. e. sum over eigenvalues
Bosons n = 0, 1, 2,... P Black body • N = Nj - total number of indistinguishable particles radiation j
Bose-Einstein condensation • free energy is (µN − H)/kBT F = −kBT ln tr e
• Nj commute, therefore (µN − H)/k T Y (µ − )N /k T e B = e j j B j • work out the trace, i. e. sum over eigenvalues n = 0, 1, 2,... ∞ n (µ − )N /k T X (µ − )/k T tr e j j B = e j B n=0 1 = (µ − )/k T 1 − e j B
Fermions, Bosons and their Statistics Free energy
Peter Hertel • energy is H = P A∗A + ... Fundamental j j j j particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • Nj commute, therefore (µN − H)/k T Y (µ − )N /k T e B = e j j B j • work out the trace, i. e. sum over eigenvalues n = 0, 1, 2,... ∞ n (µ − )N /k T X (µ − )/k T tr e j j B = e j B n=0 1 = (µ − )/k T 1 − e j B
Fermions, Bosons and their Statistics Free energy
Peter Hertel • energy is H = P A∗A + ... Fundamental j j j j particles • free energy is Spin and statistics (µN − H)/kBT F = −kBT ln tr e Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • work out the trace, i. e. sum over eigenvalues n = 0, 1, 2,... ∞ n (µ − )N /k T X (µ − )/k T tr e j j B = e j B n=0 1 = (µ − )/k T 1 − e j B
Fermions, Bosons and their Statistics Free energy
Peter Hertel • energy is H = P A∗A + ... Fundamental j j j j particles • free energy is Spin and statistics (µN − H)/kBT F = −kBT ln tr e Many particles
Gibbs state • Nj commute, therefore Fermions (µN − H)/k T Y (µ − )N /k T e B = e j j B Zero temperature j
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Free energy
Peter Hertel • energy is H = P A∗A + ... Fundamental j j j j particles • free energy is Spin and statistics (µN − H)/kBT F = −kBT ln tr e Many particles
Gibbs state • Nj commute, therefore Fermions (µN − H)/k T Y (µ − )N /k T e B = e j j B Zero temperature j White dwarfs • work out the trace, i. e. sum over eigenvalues and neutron stars n = 0, 1, 2,... Bosons ∞ n (µ − j )Nj /kBT X (µ − j )/kBT Black body tr e = e radiation n=0 Bose-Einstein 1 condensation = (µ − )/k T 1 − e j B • compare with the free energy of a fermi gas Z (µ − )/kBT F = −kBTV d z() ln 1 + e
• recall spin/statistics theorem: again the +/− difference
Fermions, Bosons and their Statistics Free energy ctd.
Peter Hertel
Fundamental particles
Spin and • the free energy of a boson gas is statistics X (µ − j )/kBT Many particles F = kBT ln 1 − e Gibbs state j Fermions Z (µ − )/kBT Zero = kBTV d z() ln 1 − e temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • recall spin/statistics theorem: again the +/− difference
Fermions, Bosons and their Statistics Free energy ctd.
Peter Hertel
Fundamental particles
Spin and • the free energy of a boson gas is statistics X (µ − j )/kBT Many particles F = kBT ln 1 − e Gibbs state j Fermions Z (µ − )/kBT Zero = kBTV d z() ln 1 − e temperature White dwarfs • compare with the free energy of a fermi gas and neutron stars Z (µ − )/kBT Bosons F = −kBTV d z() ln 1 + e
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Free energy ctd.
Peter Hertel
Fundamental particles
Spin and • the free energy of a boson gas is statistics X (µ − j )/kBT Many particles F = kBT ln 1 − e Gibbs state j Fermions Z (µ − )/kBT Zero = kBTV d z() ln 1 − e temperature White dwarfs • compare with the free energy of a fermi gas and neutron stars Z (µ − )/kBT Bosons F = −kBTV d z() ln 1 + e
Black body radiation • recall spin/statistics theorem: again the +/− difference Bose-Einstein condensation • µ = 0 • black body free energy is Z ∞ 2 dω ω −~ω/kBT F = kBTV 2 3 ln 1 − e 0 π c • internal energy U • U = F + TS = F − T ∂F /∂T • Planck’s formula Z ∞ dω ω2 ω U = V ~ π2c3 ω/k T 0 e ~ B − 1
Fermions, Bosons and their Statistics Black body radiation
Peter Hertel • photon energy is ω = pc Fundamental ~ particles • density of states can be read off from Spin and 3 3 2 statistics Z d x d p Z dω ω 2 = V Many particles (2π~)3 π2c3 Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • black body free energy is Z ∞ 2 dω ω −~ω/kBT F = kBTV 2 3 ln 1 − e 0 π c • internal energy U • U = F + TS = F − T ∂F /∂T • Planck’s formula Z ∞ dω ω2 ω U = V ~ π2c3 ω/k T 0 e ~ B − 1
Fermions, Bosons and their Statistics Black body radiation
Peter Hertel • photon energy is ω = pc Fundamental ~ particles • density of states can be read off from Spin and 3 3 2 statistics Z d x d p Z dω ω 2 = V Many particles (2π~)3 π2c3 Gibbs state • µ = 0 Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • internal energy U • U = F + TS = F − T ∂F /∂T • Planck’s formula Z ∞ dω ω2 ω U = V ~ π2c3 ω/k T 0 e ~ B − 1
Fermions, Bosons and their Statistics Black body radiation
Peter Hertel • photon energy is ω = pc Fundamental ~ particles • density of states can be read off from Spin and 3 3 2 statistics Z d x d p Z dω ω 2 = V Many particles (2π~)3 π2c3 Gibbs state • µ = 0 Fermions • black body free energy is Zero temperature Z ∞ 2 dω ω −~ω/kBT White dwarfs F = kBTV 2 3 ln 1 − e and neutron 0 π c stars
Bosons
Black body radiation
Bose-Einstein condensation • U = F + TS = F − T ∂F /∂T • Planck’s formula Z ∞ dω ω2 ω U = V ~ π2c3 ω/k T 0 e ~ B − 1
Fermions, Bosons and their Statistics Black body radiation
Peter Hertel • photon energy is ω = pc Fundamental ~ particles • density of states can be read off from Spin and 3 3 2 statistics Z d x d p Z dω ω 2 = V Many particles (2π~)3 π2c3 Gibbs state • µ = 0 Fermions • black body free energy is Zero temperature Z ∞ 2 dω ω −~ω/kBT White dwarfs F = kBTV 2 3 ln 1 − e and neutron 0 π c stars • internal energy U Bosons
Black body radiation
Bose-Einstein condensation • Planck’s formula Z ∞ dω ω2 ω U = V ~ π2c3 ω/k T 0 e ~ B − 1
Fermions, Bosons and their Statistics Black body radiation
Peter Hertel • photon energy is ω = pc Fundamental ~ particles • density of states can be read off from Spin and 3 3 2 statistics Z d x d p Z dω ω 2 = V Many particles (2π~)3 π2c3 Gibbs state • µ = 0 Fermions • black body free energy is Zero temperature Z ∞ 2 dω ω −~ω/kBT White dwarfs F = kBTV 2 3 ln 1 − e and neutron 0 π c stars • internal energy U Bosons
Black body • U = F + TS = F − T ∂F /∂T radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Black body radiation
Peter Hertel • photon energy is ω = pc Fundamental ~ particles • density of states can be read off from Spin and 3 3 2 statistics Z d x d p Z dω ω 2 = V Many particles (2π~)3 π2c3 Gibbs state • µ = 0 Fermions • black body free energy is Zero temperature Z ∞ 2 dω ω −~ω/kBT White dwarfs F = kBTV 2 3 ln 1 − e and neutron 0 π c stars • internal energy U Bosons
Black body • U = F + TS = F − T ∂F /∂T radiation • Planck’s formula Bose-Einstein ∞ 2 condensation Z dω ω ω U = V ~ π2c3 ω/k T 0 e ~ B − 1 Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation
Max Planck, German physicist, 1858-1947 • valid only if µ < min j = 0 • −∂F /∂µ = N¯ is average particle number • the average particle densityn ¯ is therefore Z ∞ 1 n¯(T , µ) = d z() ( − µ)/k T 0 e B − 1 • n¯(T , 0) is maximal density: Z ∞ 1 n¯ (T ) = d z() max /k T 0 e B − 1
Fermions, Bosons and their Statistics Particle density
Peter Hertel
Fundamental particles • recall ∞ Spin and X n(µ − j )/kBT 1 statistics e = (µ − j )/kBT Many particles n=0 1 − e Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • −∂F /∂µ = N¯ is average particle number • the average particle densityn ¯ is therefore Z ∞ 1 n¯(T , µ) = d z() ( − µ)/k T 0 e B − 1 • n¯(T , 0) is maximal density: Z ∞ 1 n¯ (T ) = d z() max /k T 0 e B − 1
Fermions, Bosons and their Statistics Particle density
Peter Hertel
Fundamental particles • recall ∞ Spin and X n(µ − j )/kBT 1 statistics e = (µ − j )/kBT Many particles n=0 1 − e Gibbs state • valid only if µ < min j = 0 Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • the average particle densityn ¯ is therefore Z ∞ 1 n¯(T , µ) = d z() ( − µ)/k T 0 e B − 1 • n¯(T , 0) is maximal density: Z ∞ 1 n¯ (T ) = d z() max /k T 0 e B − 1
Fermions, Bosons and their Statistics Particle density
Peter Hertel
Fundamental particles • recall ∞ Spin and X n(µ − j )/kBT 1 statistics e = (µ − j )/kBT Many particles n=0 1 − e Gibbs state • valid only if µ < min j = 0 Fermions • −∂F /∂µ = N¯ is average particle number Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • n¯(T , 0) is maximal density: Z ∞ 1 n¯ (T ) = d z() max /k T 0 e B − 1
Fermions, Bosons and their Statistics Particle density
Peter Hertel
Fundamental particles • recall ∞ Spin and X n(µ − j )/kBT 1 statistics e = (µ − j )/kBT Many particles n=0 1 − e Gibbs state • valid only if µ < min j = 0 Fermions • −∂F /∂µ = N¯ is average particle number Zero temperature • the average particle densityn ¯ is therefore White dwarfs Z ∞ 1 and neutron stars n¯(T , µ) = d z() 0 ( − µ)/kBT Bosons e − 1
Black body radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Particle density
Peter Hertel
Fundamental particles • recall ∞ Spin and X n(µ − j )/kBT 1 statistics e = (µ − j )/kBT Many particles n=0 1 − e Gibbs state • valid only if µ < min j = 0 Fermions • −∂F /∂µ = N¯ is average particle number Zero temperature • the average particle densityn ¯ is therefore White dwarfs Z ∞ 1 and neutron stars n¯(T , µ) = d z() 0 ( − µ)/kBT Bosons e − 1
Black body • n¯(T , 0) is maximal density: radiation Z ∞ 1 Bose-Einstein n¯max(T ) = d z() condensation /k T 0 e B − 1 • ifn ¯ is given, there is a temperature Tc such that
n¯max(Tc) =n ¯ • for even lower temperature, only a fraction of the particles are in thermal equilibrium • the rest is in the multiply occupied ground state • supra-fluidity • superconductivity • light
Fermions, Bosons and their Statistics Bose-Einstein condensation
Peter Hertel
Fundamental particles Spin and • If T decreases, so doesn ¯max(T ) statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • for even lower temperature, only a fraction of the particles are in thermal equilibrium • the rest is in the multiply occupied ground state • supra-fluidity • superconductivity • light
Fermions, Bosons and their Statistics Bose-Einstein condensation
Peter Hertel
Fundamental particles Spin and • If T decreases, so doesn ¯max(T ) statistics • Many particles ifn ¯ is given, there is a temperature Tc such that Gibbs state n¯max(Tc) =n ¯ Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • the rest is in the multiply occupied ground state • supra-fluidity • superconductivity • light
Fermions, Bosons and their Statistics Bose-Einstein condensation
Peter Hertel
Fundamental particles Spin and • If T decreases, so doesn ¯max(T ) statistics • Many particles ifn ¯ is given, there is a temperature Tc such that Gibbs state n¯max(Tc) =n ¯ Fermions • for even lower temperature, only a fraction of the particles Zero temperature are in thermal equilibrium
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • supra-fluidity • superconductivity • light
Fermions, Bosons and their Statistics Bose-Einstein condensation
Peter Hertel
Fundamental particles Spin and • If T decreases, so doesn ¯max(T ) statistics • Many particles ifn ¯ is given, there is a temperature Tc such that Gibbs state n¯max(Tc) =n ¯ Fermions • for even lower temperature, only a fraction of the particles Zero temperature are in thermal equilibrium White dwarfs • the rest is in the multiply occupied ground state and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation • superconductivity • light
Fermions, Bosons and their Statistics Bose-Einstein condensation
Peter Hertel
Fundamental particles Spin and • If T decreases, so doesn ¯max(T ) statistics • Many particles ifn ¯ is given, there is a temperature Tc such that Gibbs state n¯max(Tc) =n ¯ Fermions • for even lower temperature, only a fraction of the particles Zero temperature are in thermal equilibrium White dwarfs • the rest is in the multiply occupied ground state and neutron stars • supra-fluidity Bosons
Black body radiation
Bose-Einstein condensation • light
Fermions, Bosons and their Statistics Bose-Einstein condensation
Peter Hertel
Fundamental particles Spin and • If T decreases, so doesn ¯max(T ) statistics • Many particles ifn ¯ is given, there is a temperature Tc such that Gibbs state n¯max(Tc) =n ¯ Fermions • for even lower temperature, only a fraction of the particles Zero temperature are in thermal equilibrium White dwarfs • the rest is in the multiply occupied ground state and neutron stars • supra-fluidity Bosons • Black body superconductivity radiation
Bose-Einstein condensation Fermions, Bosons and their Statistics Bose-Einstein condensation
Peter Hertel
Fundamental particles Spin and • If T decreases, so doesn ¯max(T ) statistics • Many particles ifn ¯ is given, there is a temperature Tc such that Gibbs state n¯max(Tc) =n ¯ Fermions • for even lower temperature, only a fraction of the particles Zero temperature are in thermal equilibrium White dwarfs • the rest is in the multiply occupied ground state and neutron stars • supra-fluidity Bosons • Black body superconductivity radiation • light Bose-Einstein condensation Fermions, Bosons and their Statistics
Peter Hertel
Fundamental particles
Spin and statistics
Many particles
Gibbs state
Fermions
Zero temperature
White dwarfs and neutron stars
Bosons
Black body radiation
Bose-Einstein condensation
Albert Einstein, German physicist, 1879-1955