He eigenkets are approximately orthogonal only when

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he eigenkets are approximately orthogonal only when α-β is large! g. The “displacement operator:” There are a growing and significant set of applications where it is useful to express the coherent states directly in terms of the vacuum state 0 . If we use the number state generating rule n = a n n ! 0 -- i.e. Equation [ I-27 ] -- the coherent state may be written in the form α = exp - 1 2 α 2 α a n n ! 0 n = exp α a - 1 2 α 2 0 [ III-62 ] If we make us of the Baker-Hausdorff theorem, 19 we may easily show that 19 The Baker-Hausdorff theorem or identity may be stated as exp A + B { } = exp A { } exp B { } exp - 1 2 A , B [ ] { } when A , A , B [ ] [ ] = B , A , B [ ] [ ] = 0 . For a proof, see, for example, Charles P. Slichter’s Principles of Magnetic Resonance , Appendix A or William Louisell’s Radiation and Noise in Quantum Electronics .
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 32 R. Victor Jones, May 2, 2000 32 α = A α ( 29 0 = exp α a - α a 0 [ III-63 ] so that A α ( 29 may be interpreted as a creation operator which generates a coherent state from the vacuum. (Its adjoint operator A α ( 29 = A ( 29 is a destruction operator which destroys a state). In some treatments A α ( 29 is described as the “displacement operator” (written D α ( 29 ) 20 and the coherent states are called the “displaced states of the vacuum.” 21 To explore this point of view (and to give some meaning to the phase of the coherent state eigenvalue), we may express α in a two-dimensional, dimensionless “phase space” representation. To that end, following Equation [ I-16 ], we write the dimensionless coordinate as θ = 2 m ϖ h 1 2 q = a exp i γ [ ] + a exp - i γ [ ] [ III-64a ] and the dimensionless momentum as π = 2 m h ϖ 1 2 p = a exp i γ+π 2 ( 29 [ ] + a exp - i γ+π 2 ( 29 [ ] [ III-64b ] so that θ , π [ ] = 2 i a , a = 2 i [ III-64c ] 20 We can (or rather you will) show that D α ( 29 a D α ( 29 = a + α and D α ( 29 a D α ( 29 = a + α 21 See Elements of Quantum Optics , Pierre Meystre and Murray Sargent III, Spinger-Verlag (1991), ISBN 0-387- 54190-X.
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T HE I NTERACTION OF R ADIATION AND M ATTER : Q UANTUM T HEORY P AGE 33 R. Victor Jones, May 2, 2000 33 and since these variables are canonical 22 ∆θ ( 29 2 π ( 29 2 1 [ III-64d ] Since a = 1 2 θ- i π ( 29 exp - i γ [ ] a = 1 2 θ + i π ( 29 exp i γ [ ] [ III-65 ] the mode field (see Equation [II-24a]) b r E r r , t ( 29 = i ˆ e E a exp i r k r r - i ϖ t [ ] - a exp - i r k r r + i ϖ t [ ] [ III-66a ] becomes r E r r , t ( 29 = - ˆ e E π cos r k r r t ( 29 + θ sin r k r r t ( 29 { } [ III-66b ] Since p has a coordinate space representation - i h d d q = - i h ϖ 2 ( 29 1 2 d d θ and q has a momentum representation i h d d p = i h 2 ϖ ( 29 1 2 d d π , 23 α a - α a = α r a - a + i α i a + a =- α r d d θ i d d π [ ] [ III-67a ] 22 Of course, in general A ( 29 2 B ( 29 2 1 2 A , B [ ] 2 where A ( 29 2 = A 2 - A 2 23 If this unfamiliar, see Equations [ I-20 ] and [ I-22 ] in the lecture notes entitled The Interaction of Radiation and Matter: Semiclassical Theory.
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