Field Quantisation: Creation Operators and Particle Number
Treating each mode of a field as a harmonic oscillator makes its quantum number a count of particles, so particles become excitations rather than objects. Creation and annihilation operators then explain why identical particles are identical and why particle number need not be conserved.
What a learner can do afterwards
- Maps the modes of a classical field onto independent oscillators and reads the quantum numbers as occupation numbers
- Explains why exchange symmetry is automatic once states are built by creation operators
- Says what changes when a process creates or destroys particles
1 · Read
Treat each mode of a field as a harmonic oscillator with levels E of n = (n + 1 over 2) hw, for n = 0, 1, 2, and on. The number n counts identical packets of energy hw added above the ground state: E0 is 1 over 2 hw, E1 is 3 over 2 hw, E2 is 5 over 2 hw. Even the ground state keeps half a quantum.
Read energies by counting. An oscillator with hw = 0.20 eV has ground energy 0.5 times 0.20 eV, which is 0.10 eV. With hw = 0.50 eV, level n = 2 is 2.5 times 0.50 eV, which is 1.25 eV: ground 0.25 eV plus two quanta of 0.50 eV each.
Equal spacing is the key that turns levels into particle counts. Saying a mode sits at level n is the same as saying n quanta live there, and those quanta are what we call particles. A photon is one quantum in a field mode, and each field mode is an independent oscillator whose quantum number is an occupation number.
Creation operators raise n by one and annihilation operators lower it, so processes can create or destroy particles and number need not be conserved. Because states are built by creators, exchange symmetry comes free: identical particles are identical since only counts, never labels, are stored.
Field modes are oscillators, their level numbers are particle counts, and creators change those counts freely.
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