Schrdinger, Heisenberg, Interaction Pictures

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Schrödinger, Heisenberg, Interaction Pictures

• Experiment measurable quantities (variables)
• Quantum mechanics operators (variables) and
  state functions
• Classical mechanics variables carry time
  dependence
• State at time, t, determined by initial
  conditions
• Schrödinger mechanics operators
  time-independent
• State function carries time-dependence
• Expectation value
• Heisenberg mechanics operators carry time
  dependence
• State function is time-independent
• Time evolution operator

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Schrödinger, Heisenberg, Interaction Pictures

• Operators in Heisenberg picture

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Schrödinger, Heisenberg, Interaction Pictures

• Heisenberg equation of motion

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Schrödinger, Heisenberg, Interaction Pictures

• Operators in interaction picture
• Split Hamiltonian H Ho H1 H is
  time-independent

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Schrödinger, Heisenberg, Interaction Pictures

• Operators in interaction picture
• Exercise Prove that

• Schrödinger picture
• Interaction picture
• Heisenberg picture

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Schrödinger, Heisenberg, Interaction Pictures

• Time evolution operator
• Integrate to obtain implicit form for U

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Schrödinger, Heisenberg, Interaction Pictures

• Solve by iteration

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Schrödinger, Heisenberg, Interaction Pictures

• Rearrange the term

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Schrödinger, Heisenberg, Interaction Pictures

• Time evolution operator as a time-ordered product
• Utility of time evolution operator in evaluating
  expectation value

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Occupation Number Formalism

• Spin Statistics Theorem
• Fermion wave function must be anti-symmetric wrt
  particle exchange

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Occupation Number Formalism

• Slater determinant of (orthonormal) orbitals for
  N particles
• N! terms in wavefunction, N! nonzero terms in
  norm (orthogonality)
• P is permutation operator
• Number of particles is fixed
• Matrix elements evaluated by Slater Rules
• Configuration Interaction methods (esp. in
  molecular quantum chemistry)
• How to accommodate systems with different
  particle numbers, scattering, time-dependent
  phenomena ??

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Occupation Number Formalism

• Basis functions e.g. eigenfunctions of mean-field
  Hamiltonian (M 123, F 12)

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Occupation Number Formalism

• Fermion Creation and Annihilation Operators
• Boson Creation and Annihilation Operators
• Fermion example

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Occupation Number Formalism

• Factors outside Fermion kets enforce Pauli
  Exclusion Principle
• No more than one Fermion in a state
• Identical particle exchange accompanied by sign
  change
• Sequence of operations below permutes two
  particles
• Accompanied by a change of sign
• Also works when particles are not in adjacent
  orbitals

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Occupation Number Formalism

• Fermion anti-commutation rules

Number operator counts particles in a particular
ket (as an eigenvalue) Exercise (1) Prove
Fermion anti-commutation rules using defining
relations (2) Apply to 10gt
and 11gt for ij1 i1,j2 and comment
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Occupation Number Formalism

• Fermion particle, , and hole, , operators
• Virtual (empty, unoccupied) states
• particle annihilation operator
  destroys Fermion above eF
• particle creation operator creates
  Fermion above eF
• Filled (occupied) states
• hole annihilation operator creates
  Fermion below eF
• hole creation operator destroys
  Fermion below eF
• Fermi vacuum state
• all states filled below eF
• Commutation relations from Fermion relations

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Occupation Number Formalism

• Number of holes or particles is not specified
• Relevant expectation value is wrt Fermi vacuum
• Specify states by creation/annihilation of
  particles or holes wrt

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Occupation Number Formalism

• Coordinate notation
• Matrix Mechanics
• Occupation Number (Second Quantized) form
• One body (KE EN) and two body (EE) terms

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Occupation Number Formalism

• Potential scattering of electron (particle) or
  hole
• Electron-electron scattering Electron-hole
  scattering Hole-hole scattering
• Scattering electron-hole pair creation

i
j
l
i
j
i
l
k
k
k
j
l
i
j
l
i
j
Left out Right out Left in Right in
k
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Field Operators

• Field operators defined by

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Field Operators

• Commutation relations

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Field Operators

• Hamiltonian in field operator form
• Heisenberg equation of motion for field operators

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Field Operators

• One-body part

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Field Operators

• Two-body part

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Field Operators

• Two-body part continued
• ½ factors included

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Wicks Theorem

• Time-dependence of Fermion operators in
  interaction picture

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Wicks Theorem

• Time-ordered products

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Wicks Theorem

• Time-dependence of particle and hole operators in
  interaction picture
• Time-ordered products

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Wicks Theorem

• Longer products of operators (M 155)
• This is rather tedious and fortunately Wicks
  Theorem comes to the rescue

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Wicks Theorem

• Normal ordered product of operators (M 364, F 83)
• Contraction (contracted product) of operators

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Wicks Theorem

• Contraction (contracted product) of operators
• For more operators (F 83) all possible pairwise
  contractions of operators
• Uncontracted, all singly contracted, all doubly
  contracted,
• Take matrix element over Fermi vacuum
• All terms zero except fully contracted products

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Wicks Theorem

• More Examples of Applications of Wicks Theorem

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Wicks Theorem

• More Examples of Applications of Wicks Theorem