Chapter 5: Second Quantization Flashcards
Second Quantization
(Overview)
- Misleading, not quantizing anything, but rather changing representations
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Consider: Many-Body Problem in First Quantization
- Hilbert Space
- Basis
- Correct symmetry for particle type
Fock States and Spaces
Second Quantization
(Take-Away)
- s,a account for permutations → ordering irrelevant
- Number of modes in {αi} now important ← new “occupation” representation
- Need to consider what happens with addition/subtraction of mode for both bosons and fermions
Second Quantization
(Bosons: Overview)
- Vacuum state ≡ | 0 >
- Creation/annihilation operators a† , a
- Commutation relations guarantee wavefunction remains symmetric
Second Quantizaion
(Bosons: Action of a,a†)
Second Quantization
(Bosons: General State)
State α occupied by N particles
Second Quantization
(Bosons: Notes)
- | { nα } > form complete basis set for symmetric Fock space
- Order unimportant for bosons
- Creation/annihilation operators act only on specific state
- Total number operator Ntot
Second Quantization
(Bosons: Connection to Wavefunction)
Second Quantization
(Fermions: Overview)
NOTE: Similar to bosons with + → − and a few exceptions
- Commutator not well defined
- Operators anti-commute
Second Quantization
(Fermions: Actions of c, c†)
Second Quantization
(Fermions: General State)
Second Quantization
(Fermions: Notes)
Ordering is important
Second Quantization
(Single-Particle Operator)
Let Ai be single-particle operator
Second Quantization
(Two-Particle Operator)
Let Aij be a two-particle operator
Second Quantization
(Basis Transformation)
- Let φα(x), χν(x) be complete, orthonormal basis sets with (creation) annihilation operators aα(†) , bν(†)
- NOTE: Commutation relation still holds