Chapter 5: Second Quantization Flashcards

1
Q

Second Quantization

(Overview)

A
  • Misleading, not quantizing anything, but rather changing representations
  • Consider: Many-Body Problem in First Quantization
    • Hilbert Space
    • Basis
    • Correct symmetry for particle type
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2
Q

Fock States and Spaces

A
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3
Q

Second Quantization

(Take-Away)

A
  • 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
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4
Q

Second Quantization

(Bosons: Overview)

A
  • Vacuum state ≡ | 0 >
  • Creation/annihilation operators a , a
  • Commutation relations guarantee wavefunction remains symmetric
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5
Q

Second Quantizaion

(Bosons: Action of a,a)

A
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6
Q

Second Quantization

(Bosons: General State)

A

State α occupied by N particles

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7
Q

Second Quantization

(Bosons: Notes)

A
  • | { 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
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8
Q

Second Quantization

(Bosons: Connection to Wavefunction)

A
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9
Q

Second Quantization

(Fermions: Overview)

A

NOTE: Similar to bosons with + → − and a few exceptions

  • Commutator not well defined
  • Operators anti-commute
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10
Q

Second Quantization

(Fermions: Actions of c, c)

A
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11
Q

Second Quantization

(Fermions: General State)

A
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12
Q

Second Quantization

(Fermions: Notes)

A

Ordering is important

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13
Q

Second Quantization

(Single-Particle Operator)

A

Let Ai be single-particle operator

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14
Q

Second Quantization

(Two-Particle Operator)

A

Let Aij be a two-particle operator

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15
Q

Second Quantization

(Basis Transformation)

A
  • Let φα(x), χν(x) be complete, orthonormal basis sets with (creation) annihilation operators aα(†) , bν(†)
    • NOTE: Commutation relation still holds
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16
Q

Field Operator

(Overview)

A
  • Recall: single-particle wavefunction φαs(r) in mode α with spin s
  • Field operator creates/destroy particle at r
17
Q

Field Operator

(Commutation Relations)

A

(Anti-)Commutators follow as that for (fermions) bosons

  • Bosons
  • Fermions
18
Q

Field Operator

(General State)

A
19
Q

Field Operator

(Take-Away)

A
  • Can always swtich between these two:
    • wavefunction picture ⇐⇒ first quantization
    • state picture ⇐⇒ second quantization
20
Q

Field Operator

(Single-Particle Operator)

A

Let A1 be single-particle operator

21
Q

Field Operator

(Two-Particle Operator)

A

Let A2 be two particle operator

  • NOTE: ψ(r1)ψ(r2)ψ(r2)ψ(r1) ordering removes self-interaction
22
Q

Field Operator

(Hamiltonian)

A
23
Q

Correlations in Non-Interacting Fermi Gas

(Green’s Function)

A

NOTE: j1 is Bessel function

24
Q

Correlations in Non-Interacting Fermi Gas

(Green’s Function: Graph)

A
25
Q

Correlations in Non-Interacting Fermi Gas

(Pair Correlation Function)

A
26
Q

Correlations in Non-Interacting Fermi Gas

(Pair Correlation Function: Graph)

A
27
Q

Quantization of Electric Field

A
  • Until now, went from single-particle quantization → second quantization
  • Can also quantize electromagnetic field directly
    • Decomposes into uncoupled harmonic oscillators
28
Q

Quantization of Electric Field
(Creation/Annihilation Operators, Commutator.
Hamiltonian, Arbitrary State)

A
  • a<em>kλ</em> produces photon in mode k, λ
29
Q

Quantization of Electric Field

(Vector Potential in Heisenberg Picture)

A
30
Q

Quantization of Electric Field

(Canonical Commutation Relations)

A

NOTE: Only valud for equal time

31
Q

Quantization of Electric Field

(Canonical Commutation Relations: Graph)

A

Only non-zero in shaded region

32
Q

Photon Characteristics

(Energy)

A
33
Q

Photon Characteristics

(Momentum)

A
34
Q

Photon Characteristics

(Spin)

A
  • Spin is ±(hbar) along k
  • For massless particles at speed of light, spin is always along axis of propagation with values ±1
35
Q

Photon Characeristics

(Angular Momentum)

A

NOTE: Choosing circularly polarized light basis results in well-defined spin

36
Q

Quantization of Electric Field

(Notes)

A
  • For single photon, expectation value of electric field vanishes
    • Also for state with fixed number of photons
  • Fluctuations of field in vacuum at same position are infinite
    • Not a problem, because any detector measures over finite area/volume