ls3_unit_5 - THE INTERACTION OF RADIATION AND MATTER:...

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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 42 R. Victor Jones, April 18, 2000 V. PHOTON ABSORPTION AND EMISSION "POOR MAN'S" SECOND QUANTIZATION OF MATERIAL SYSTEM: In treating the complete quantum mechanical problem, it is useful to recast the material (atomic) Hamiltonian in terms of an appropriate set of creation and destruction operators. To that end we make the following definition H A x = h ϖ x x [ V-1 ] Using the ubiquitous identity operation x x x = 1 , we may write the material Hamiltonian in second quantized form -- viz. H A = x x x H A y y y = x h ϖ x x y y y x = h ϖ x x x x [ V-2 ] In general, the operator x y b x b y applied to any state z yields x y z = b x b y z = x δ yz [ V-3 ] -- i.e. the operator changes a state z to a state x if the state is y otherwise it produces zero. In other words, the operator destroys the state y and creates a state x . The second quantization viewpoint is particularly useful in treating the interaction of a two-level material system with the radiation field. This case, is most conveniently formulate in two- vector notation with the use of Pauli spin matrices -- viz. a = 1 0 and b = 0 1 [ V-4a ] a = 1 0 [ ] and b = 0 1 [ ] [ V-4b ]
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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 43 R. Victor Jones, April 18, 2000 a b = 1 0 0 1 [ ] = 0 1 0 0 = σ + b a = 0 1 1 0 [ ] = 0 0 1 0 = σ - [ V-4c ] a a = 1 0 1 0 [ ] = 1 0 0 0 b b = 0 1 0 1 [ ] = 0 0 0 1 [ V-4d ] Consequently, the atomic Hamiltonian may be written H A = h ϖ a 1 0 0 0 + h ϖ b 0 0 0 1 = 1 2 h ϖ a b ( 29 1 0 0 - 1 + 1 2 h ϖ a b ( 29 1 0 0 1 [ V-5a ] and if we neglect the mean energy of the states H A 1 2 h ϖ ab 1 0 0 - 1 = 1 2 h ϖ ab σ z [ V-5b ]
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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 44 R. Victor Jones, April 18, 2000 and the electric dipole interaction Hamiltonian becomes H ED = e r D r E T 0 ( 29 = a a + b b [ ] e r D [ ] a a + b b [ ] r E T 0 ( 29 = a a e r D [ ] b b + b b e r D [ ] a a { } r E T 0 ( 29 = e a r D b σ + + e b r D a σ - { } r E T 0 ( 29 = - r ℘σ + + r ℘ σ - { } r E T 0 ( 29 [ V-6 ] From Equation [ II-24a ] in this lecture set we can write r E T 0 ( 29 ˆ e l { }σ E l { } i a l { }σ t ( 29 exp i r k l { } r r A [ ] - i a l { } σ t ( 29 exp - i r k l { } r r A [ ] σ= 1 2 l { } [ V-7 ] where E l {} = h ϖ l { } 2 ε 0 V is the so called the electric field per photon and r r A is the location of the center of the atom under consideration. Thus Equation [ V-6 ] may be written quite generally for a two level atom as H ED = e a r D b σ + + e b r D a σ - { } × ˆ e l { } s E l {} i a l
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This note was uploaded on 01/31/2011 for the course PHYSICS 108 taught by Professor Staff during the Winter '08 term at UC Davis.

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ls3_unit_5 - THE INTERACTION OF RADIATION AND MATTER:...

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