ls2_unit_3

ls2_unit_3 - THE INTERACTION OF RADIATION AND MATTER:...

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THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 26 R. Victor Jones, March 9, 2000 III. R EVIEW OF B ASIC Q UANTUM M ECHANICS : T WO -L EVEL Q UANTUM S YSTEMS : The literature of quantum optics and laser spectroscopy abounds with discussions of the two-level (two-state) system. This emphasis comes about because the interaction of such systems with the electromagnetic field may be treated in great detail to obtain valuable analytic results and, hopefully, the analysis of two-level systems generates insights that may be extended to more realistic situations. Fortunately, there are several important instances in which the application of the two-level model provides a very good approximation to a more complete theory. In the following, we label the upper level of the system by the letter a and the lower by the letter b . From Equation [ I-12a ] we write, specifically, the wave function of the two level system as ψ r r , t ( ) = r r ψ t ( ) = C a t ( ) u a r r ( ) exp i ω a t ( ) + C b t ( ) u b r r ( ) exp i ω b t ( ) [ III-1 ] where we know from Equation [ I-12b ] that the time varying coefficients satisfy, in general, the following equations: ˙ C a t ( ) = i h C a t ( ) E a H 1 t ( ) E a + C b t ( ) E a H 1 t ( ) E b exp i ω ab t ( ) { } [ III-2a ] ˙ C b t ( ) = i h C b t ( ) E b H 1 t ( ) E b + C a t ( ) E b H 1 t ( ) E a exp i ω ba t ( ) { } [ III-2b ] If we take the interaction to be the electric dipole interaction with an applied electric field we may write H 1 t ( ) = r p S r E r R S , t ( ) = e r r r E r R S , t ( ) [ III-3a ]
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THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 27 R. Victor Jones, March 9, 2000 where r R S denotes the position of the center of the two-level system or atom. 14 Thus we write E i H 1 t ( ) E j = V ij = E i e r r E j r E r R S , t ( ) [ III-3b ] In all but the most bizarre circumstances we may use persuasive symmetry arguments to reason that E i e r r E i 0 Thus Equations [ III-2 ] reduce to ˙ C a t ( ) = i h C b t ( ) V ab exp i ω ab t ( ) [ III-4a ] ˙ C b t ( ) = i h C a t ( ) V ab exp i ω ab t ( ) [ III-4b ] where V ab = E a e r r E b r E r R S , t ( ) = −℘ E r R S , t ( ) . R ABI F LOPPING -- WITHOUT D AMPING : For an oscillatory applied field V ab = −℘ E o cos ω r t = 1 2 E o exp i ω r t ( ) + c . c . [ III-5 ] we see, in the rotating-wave approximation , that ˙ C a t ( ) = i 2 h E o C b t ( ) exp i ω ab −ω r ( ) t [ ] = i 2 Ω o R C b t ( ) exp i ω ab −ω r ( ) t [ ] [ III-6a ] 14 The use of this form of interaction needs considerable elaboration, but we defer that discussion until later.
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THE INTERACTION OF RADIATION AND MATTER: SEMICLASSICAL THEORY PAGE 28 R. Victor Jones, March 9, 2000 ˙ C b t ( ) = i 2 h E o C a t ( ) exp i ω ab −ω r ( ) t [ ] = i 2 Ω o R C a t ( ) exp i ω ab −ω r ( ) t [ ] [ III-6b ] where Ω o R ≡℘ E o h
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This note was uploaded on 02/29/2012 for the course PHYSICS 216 taught by Professor Staff during the Fall '11 term at BU.

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ls2_unit_3 - THE INTERACTION OF RADIATION AND MATTER:...

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