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Schr ö dinger, Heisenberg, Interaction Pictures

Schr ö dinger, 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

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Schr ö dinger, Heisenberg, Interaction Pictures

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  1. 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

  2. Schrödinger, Heisenberg, Interaction Pictures • Operators in Heisenberg picture

  3. Schrödinger, Heisenberg, Interaction Pictures • Heisenberg equation of motion

  4. Schrödinger, Heisenberg, Interaction Pictures • Operators in interaction picture • Split Hamiltonian H = Ho + H1 Ho is time-independent

  5. Schrödinger, Heisenberg, Interaction Pictures • Operators in interaction picture: Exercise: Prove that • Schrödinger picture • Interaction picture • Heisenberg picture

  6. Schrödinger, Heisenberg, Interaction Pictures • Time evolution operator • Integrate to obtain implicit form for U

  7. Schrödinger, Heisenberg, Interaction Pictures • Solve by iteration

  8. t’ t’ t’=t’’ t’=t’’ t’’<t’ t’=t t’=t dt’ dt’’ t’ t’’ t’’ t’’ t’’=t t’’=t Schrödinger, Heisenberg, Interaction Pictures • Rearrange the term

  9. Schrödinger, Heisenberg, Interaction Pictures • Time evolution operator as a time-ordered product • Utility of time evolution operator in evaluating expectation value

  10. Y(r1s1, r2s2) 2 particles in a 1-D box Y(r1s1r2s2) = -Y(r2s2 r1s1) r2s2 r1s1 Y(r1s1= r2s2, r2s2) = 0 Occupation Number Formalism • Spin Statistics Theorem Fermion wave function must be anti-symmetric wrt particle exchange

  11. 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 ??

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

  13. Occupation Number Formalism • Fermion Creation and Annihilation Operators • Boson Creation and Annihilation Operators • Fermion example

  14. 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

  15. 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 |10> and |11> for i=j=1; i=1,j=2 and comment

  16. 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

  17. 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

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

  19. j i 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 i ℓ i j k ℓ k k j ℓ i j i j ℓ Left out – Right out – Left in – Right in k

  20. Field Operators • Field operators defined by

  21. Field Operators • Commutation relations

  22. Field Operators • Hamiltonian in field operator form • Heisenberg equation of motion for field operators

  23. Field Operators • One-body part

  24. Field Operators • Two-body part

  25. Field Operators • Two-body part continued • ½ factors included

  26. Wick’s Theorem • Time-dependence of Fermion operators in interaction picture

  27. eF j i j i j i eF eF Wick’s Theorem • Time-ordered products of 1-body part of Hamiltonian i j eF

  28. Wick’s Theorem • Time-dependence of particle and hole operators in interaction picture • Time-ordered product of two operators (M 155)

  29. Wick’s Theorem • Time-ordered products of more than two operators (M 155)

  30. Wick’s Theorem • Time-ordered products of more than two operators (M 155)

  31. Wick’s Theorem • Time-ordered products of more than two operators (M 155)

  32. Wick’s Theorem • Longer products of operators (M 155)

  33. Wick’s Theorem • Longer products of operators (M 155)

  34. Wick’s Theorem • Longer products of operators (M 155)

  35. Wick’s Theorem • Longer products of operators (M 155)

  36. Wick’s Theorem • Diagrammatic interpretation of results t2 particle destroyed t1 particle created t2 particle destroyed t1 particle created t2 t1

  37. Wick’s Theorem • Diagrammatic interpretation of results Scattering by V Fermion loops

  38. Wick’s Theorem • Diagrammatic interpretation of results

  39. Wick’s Theorem • Products with more than one intermediate time

  40. Wick’s Theorem • Products with more than one intermediate time

  41. Wick’s Theorem • Products with more than one intermediate time

  42. Wick’s Theorem • Normal ordered product of operators (M 364, F 83) • Contraction (contracted product) of operators

  43. Wick’s 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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