Ivancevic_Applied-Diff-Geom

992 Applied Differential Geometry: A Modern Introductionby the detection (or annihilation) of a particle. Consequently, in a coherentstate, one has exactly the same probability to detect a second particle.Note, this condition is necessary for the coherent state’s Poisson detectionstatistics. Compare this to a single–particle’s Fock state: Once one particleis detected, we have zero probability of detecting another.Now, recall that a Bose–Einstein condensate (BEC) is a collection ofboson atoms that are all in the same quantum state. An approximatetheoretical description of its properties can be derived by assuming theBEC is in a coherent state. However, unlike photons, atoms interact witheach other so it now appears that it is more likely to be one of the squeezedcoherent states (see [Breitenbach et. al. (1997)]). In quantum field theoryand string theory, a generalization of coherent states to the case of infinitelymany degrees of freedom is used to define a vacuum state with a differentvacuum expectation value from the original vacuum.6.1.7 Dirac’s < bra | ket > Transition AmplitudeNow, we are ready to move–on into the realm of quantum mechanics. Recallthat [Dirac (1982)] described behavior of quantum systems in terms ofcomplex–valued ket–vectors |A > living in the Hilbert space H, and theirduals, bra–covectors < B| (i.e., 1–forms) living in the dual Hilbert spaceH ∗ . 2 The Hermitian inner product of kets and bras, the bra–ket < B|A >,is a complex number, which is the evaluation of the ket |A > by the bra< B|. This complex number, say re iθ represents the system’s transition2 Recall that a norm on a complex vector space H is a mapping from H into thecomplex numbers, ‖·‖ : H → C; h ↦→ ‖h‖, such that the following set of norm–axiomshold:(N1) ‖h‖ ≥ 0 for all h ∈ H and ‖h‖ = 0 implies h = 0 (positive definiteness);(N2) ‖λ h‖ = |λ| ‖h‖ for all h ∈ H and λ ∈ C (homogeneity); and(N3) ‖h 1 + h 2 ‖ ≤ ‖h 1 ‖ + ‖h 2 ‖ for all h 1 , h 2 ∈ H (triangle inequality). The pair(H, ‖·‖) is called a normed space.A Hermitian inner product on a complex vector space H is a mapping 〈·, ·〉 : H×H → Csuch that the following set of inner–product–axioms hold:(IP1) 〈h h 1 + h 2 〉 = 〈h h 1 + h h 2 〉 ;(IP2) 〈α h, h 1 〉 = α 〈 h, h 1 〉 ;(IP3) 〈h 1 , h 2 〉 = 〈h 1 , h 2 〉 (so 〈h, h〉 is real);(IP4) 〈h, h〉 ≥ 0 and 〈h, h〉 = 0 provided h = 0.The standard inner product on the product space C n = C × · · · × C is defined by〈z, w〉 = P ni=1 z iw i , and axioms (IP1)–(IP4) are readily checked. Also C n is a normedspace with ‖z‖ 2 = P ni=1 |z i| 2 . The pair (H, 〈·, ·〉) is called an inner product space.Let (H, ‖·‖) be a normed space. If the corresponding metric d is complete, we say(H, ‖·‖) is a Banach space. If (H, ‖·‖) is an inner product space whose correspondingmetric is complete, we say (H, ‖·‖) is a Hilbert space.