The document discusses quantum magnetism and the exchange interaction between identical particles. It explains that the exchange interaction is a quantum mechanical effect that occurs between identical particles and is due to the wave function changing sign or remaining unchanged when particles are exchanged. For fermions like electrons, this interaction is called Pauli repulsion and relates to the Pauli exclusion principle, while for bosons it takes the form of an effective attraction. The exchange interaction is responsible for phenomena like ferromagnetism.

1. QUANTUM MECHANICAL
TREATMENT & EXCHANGE
INTERACTIONS
MUHAMMAD USAMA DAUD
MPHIL PHYSICS (SCHOLOR)

2. QUANTUM MEGNETISM
• QUANTUM MAGNETISM IS A BIT DIFFERENT FROM CLASSICAL MAGNETISM,
• THE KIND YOU SEE WHEN YOU STICK A MAGNET TO A FRIDGE, BECAUSE
INDIVIDUAL ATOMS HAVE A QUALITY CALLED SPIN, WHICH IS QUANTIZED, OR IN
DISCRETE STATES (USUALLY CALLED UP OR DOWN).
• SEEING THE BEHAVIOR OF INDIVIDUAL ATOMS HAS BEEN HARD TO DO,
THOUGH, BECAUSE IT REQUIRED COOLING ATOMS TO EXTREMELY COLD
TEMPERATURES AND FINDING A WAY TO “TRAP” THEM

3. • ACCORDING TO THE CLASSICAL THEORY, MAGNETIC FIELDS ARE CLOUDS OF
ENERGY AROUND MAGNETIC PARTICLES THAT PULL IN OR PUSH AWAY OTHER
MAGNETIC OBJECTS.
• BUT IN THE QUANTUM MECHANICS VIEW, ELECTRONS EMIT UNDETECTABLE,
VIRTUAL PARTICLES THAT TELL OTHER OBJECTS TO MOVE AWAY OR COME
CLOSER.

4. EXCHANGE INTERACTION
THE EXCHANGE INTERACTION (WITH AN EXCHANGE ENERGY AND EXCHANGE
TERM) IS A QUANTUM MECHANICAL EFFECT THAT ONLY OCCURS
BETWEEN IDENTICAL PARTICLES.

5. • THE EFFECT IS DUE TO THE WAVE FUNCTION OF INDISTINGUISHABLE
PARTICLES BEING SUBJECT TO EXCHANGE SYMMETRY,
• THAT IS, EITHER REMAINING UNCHANGED (SYMMETRIC) OR CHANGING SIGN
(ANTISYMMETRIC) WHEN TWO PARTICLES ARE EXCHANGED.
BOTH BOSONS AND FERMIONS CAN EXPERIENCE THE EXCHANGE INTERACTION.
• FOR FERMIONS, THIS INTERACTION IS SOMETIMES CALLED PAULI REPULSION AND
IS RELATED TO THE PAULI EXCLUSION PRINCIPLE.
• FOR BOSONS, THE EXCHANGE INTERACTION TAKES THE FORM OF AN EFFECTIVE
ATTRACTION THAT CAUSES IDENTICAL PARTICLES TO BE FOUND CLOSER
TOGETHER, AS IN BOSE–EINSTEIN CONDENSATION.

6. • THE EXCHANGE INTERACTION ALTERS THE EXPECTATION VALUE OF THE DISTANCE
WHEN THE WAVE FUNCTIONS OF TWO OR MORE INDISTINGUISHABLE PARTICLES
OVERLAP.
• THIS INTERACTION INCREASES (FOR FERMIONS) OR DECREASES (FOR BOSONS)
THE EXPECTATION VALUE OF THE DISTANCE BETWEEN IDENTICAL PARTICLES
(COMPARED TO DISTINGUISHABLE PARTICLES).
• AMONG OTHER CONSEQUENCES, THE EXCHANGE INTERACTION IS RESPONSIBLE
FOR FERROMAGNETISM .
EXCHANGE INTERACTION EFFECTS WERE DISCOVERED INDEPENDENTLY BY
PHYSICISTS WERNER HEISENBERG[2] AND PAUL DIRAC[3] IN 1926

7. • THE EXCHANGE INTERACTION IS SOMETIMES CALLED THE EXCHANGE FORCE.
HOWEVER, IT IS NOT A TRUE FORCE AND SHOULD NOT BE CONFUSED WITH
THE EXCHANGE FORCES PRODUCED BY THE EXCHANGE OF FORCE
CARRIERS, SUCH AS THE ELECTROMAGNETIC FORCE PRODUCED BETWEEN
TWO ELECTRONS BY THE EXCHANGE OF A PHOTON, OR THE STRONG
FORCE BETWEEN TWO QUARKS PRODUCED BY THE EXCHANGE OF A GLUON.

8. • QUANTUM MECHANICAL PARTICLES ARE CLASSIFIED AS BOSONS OR FERMIONS. THE SPIN–
STATISTICS THEOREM OF QUANTUM FIELD THEORY DEMANDS THAT ALL PARTICLES
WITH HALF-INTEGER SPIN BEHAVE AS FERMIONS AND ALL PARTICLES WITH INTEGER SPIN
BEHAVE AS BOSONS.
• MULTIPLE BOSONS MAY OCCUPY THE SAME QUANTUM STATE; HOWEVER, BY THE PAULI
EXCLUSION PRINCIPLE, NO TWO FERMIONS CAN OCCUPY THE SAME STATE.
• SINCE ELECTRONS HAVE SPIN 1/2, THEY ARE FERMIONS. THIS MEANS THAT THE OVERALL
WAVE FUNCTION OF A SYSTEM MUST BE ANTISYMMETRIC WHEN TWO ELECTRONS ARE
EXCHANGED, I.E. INTERCHANGED WITH RESPECT TO BOTH SPATIAL AND SPIN
COORDINATES. FIRST, HOWEVER, EXCHANGE WILL BE EXPLAINED WITH THE NEGLECT OF
SPIN.

9. EXCHANGE OF SPATIAL COORDINATES
• TAKING A HYDROGEN MOLECULE-LIKE SYSTEM (I.E. ONE WITH TWO
ELECTRONS), ONE MAY ATTEMPT TO MODEL THE STATE OF EACH ELECTRON
BY FIRST ASSUMING THE ELECTRONS BEHAVE INDEPENDENTLY, AND TAKING
WAVE FUNCTIONS IN POSITION SPACE.

10. • TREATING THE EXCHANGE INTERACTION IN THE HYDROGEN MOLECULE BY
THE PERTURBATION METHOD, THE OVERALL HAMILTONIAN IS:

11. • HAMILTONIAN BETWEEN TWO ELECTRONS IN ORBITALS ΦA AND ΦB CAN BE
WRITTEN IN TERMS OF THEIR SPIN MOMENTA S(A) AND S(B) .THIS IS NAMED
THE HEISENBERG EXCHANGE HAMILTONIAN OR THE HEISENBERG–DIRAC
HAMILTONIAN IN THE OLDER LITERATURE:
• JAB IS TERMED THE EXCHANGE CONSTANT.

12. IF JAB IS POSITIVE THE EXCHANGE ENERGY
FAVORS ELECTRONS WITH PARALLEL SPINS; THIS
IS A PRIMARY CAUSE OF FERROMAGNETISM.
IF JAB IS NEGATIVE, THE INTERACTION FAVORS
ELECTRONS WITH ANTIPARALLEL SPINS,
POTENTIALLY CAUSING ANTIFERROMAGNETISM.