Download Identical Particles

Identical Particles
(Many Body Physics)
Two particle systems
• Two particle systems:
V(x1,x2) describe interactions between two particles.
Principles of identical particles
• If two particles are identical, for example, two
particles are electrons ( they have the same
physical quantities), they are indistinguishable.
• Systems are symmetric respecting two particles.
• Even God can not tell which is which. Namely,
there is no way to label the particles.
• Any single particle measurable quantities will be
the same. No Matter which particle is measured.
Mathematical Statement of identical
Particles
• The wavefunction
• Wavefunction must be symmetric or antisymmetric
• Spin and Statistics:
Particles with half integer spins: antisymmetric (Fermion)
Particles with integer spins: Symmetric (Boson)
Free particles without interaction
• Single particle states:
• Nonidentical particles: Wavefunction
• Identical Partilces:
Pauli Exclusion principles
• For non-interacting fermions
• identical Fermions: if n1=n2, Wavefunction
• Two identical fermions can not occupy the same
state. A state can only host one identical fermion.
Extension including spin
• Without Spin-orbital coupling: Wavefunction
includes orbital and spin parts. It is simply a
direct product of the two
For Boson: Both orbital and spin parts of wavefunction must have the same
symmetry.
For Fermion: The orbital and spin parts must have opposite symmetry.
Example: two electrons (spin ½)
• Spin part: if two electrons form a singlet:
• If two electrons form a triplet:
Exchange Energy
• Consider
Exchange Energy
• The second term is absent if particles are distinguishable.
• Exchange energy is an important force in forming chemical
bonds.
Example:
Symmetric: produce attractive interaction (Covalent bond)
Antisymmetryic: Produce repulsive interaction
Periodic Tables
Free electron gas (Metal)
• Consider a rectangular solid:
• Let’s consider 2N>>1 electrons in the system, each state can host two electrons
( consider electron is spin ½). Let’s consider kF be the largest momentum to
be occupied
Fermi Wavevector
Total energy:
Quantum statistics
• Partition functions:
Probability: Consider a system with single particle states: {dn,En}: where
dn is degeneracy of the single particle states, En the energy of single particle:
Non-identical particles
Fermions
Bosons
Quantum statistics
• Grand Canonical Ensemble
Non-identical particles
Fermions
Bosons
Non-identical particles
Fermions
Bosons
Quantum statistics
• Density distribution:
Non-identical particles
Fermions
Bosons
Blackbody Radiation
• Photon: