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where If> is u in case 1 and v in case 2. The first term on the right-hand side of (9.98) represents an incoming wave from infinity, and the second term, the scattered outgoing wave. So XI and XR are related by the scattering matrix S (Le., XR = SXI). So there lies the problem of using natural modes in a quantum theory. The Sommerfeld radiation condition satisfied by natural modes is only part of the asymptotic boundary condition at infinity (9.98) that defines the set of all possible solutions for the radiation field. To make a complete set out of natural modes, we must supplement them with other modes that, to put it crudely, satisfy the remaining required asymptotic boundary conditions (9.98). This is, as far as I know, still an open problem in
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\5See Section 9.2 or 10 of[607J.
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the general three-dimensional case. There is a one-dimensional solution [172, 173], though. In a nutshell, this solution consists of dividing the modes-of-the-universe boundary problem into two separate boundary problems, one of which yields the natural modes and the other, their required complement to make a complete set. For more on this one-dimensional case, see the recommended reading.
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The description of dissipative systems directly by a Lagrangian or even a Hamiltonian is a polemic topic in the literature. In the Newtonian formulation of classical mechanics, friction forces are dealt with just like any other force. The variational formulation of mechanics, however, requires that the forces be derivable from a work function (they must be "monogenic" rather than "polygenic" forces; see, e.g., [386]). As quantum mechanics is derived from the latter formulation rather than from Newton's, it inherited this requirement. In the case of the Hamiltonian formulation of quantum mechanics, the requirement is even more stringent: The Hamiltonian must be the total energy of the system (see Secs. 27 and 28 of [159]). This is the main reason for the usual treatment of dissipation in quantum mechanics where the environment is included to make the total system closed and nondissipative. Feynman's 1948 "proof" of Maxwell equations 16 has been reinterpreted as a derivation that the most general kind of force the Hamiltonian-Lagrangian formalism can deal with is one whose form is as that of the Lorentz force [295, 302] (see also Sec. 1-5, pp. 21-24 of [240]). For a simple pedagogic discussion of soluble models of quantum dumping in cavity QED, see [410]. For more on the polemic attempts to describe damping directly by a Lagrangian or Hamiltonian, see [45,76, 150,251,295,339,365,368,507,508,512,513,520,582] . The "Gardiner-Collett" Hamiltonian was known in the literature before Gardiner and Collett's two papers [109, 225] on their input-output theory. An analogous Hamiltonian was introduced in 1962 by Cohen et at. to explain experiments involving tunneling of electrons between superconducting films covered by a thin oxide layer [107]. This Hamiltonian was ultimately based [29] on some early work by Bardeen [28] on a many-body theory of tunneling. For this reason, in solid-state physics this Hamiltonian is known as the Bardeen Hamiltonian. The Bardeen Hamiltonian was deduced from first principles (for the solid-state case of tunneling of particles) by Prange [496], who pointed out that in general it gives rise to mode non orthogonality. Gardiner and Collett's input-outputformalism has been extended to bosonic matter fields to model the output coupling of atoms (as in an atom laser) from a Bose-Einstein conden16In 1948, Feynman derived Maxwell equations as a mathematical consequence from two apparently unconnected ingredients: Newton's second law and the quantum commutation relations for position and momentum! See Dyson's account ofFeynman's derivation [176].
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sate in a single-mode trap [296]. Recently, the input-output theory has been generalized to fermions [553], where the Pauli exclusion principle forces us to take into account all "cavity modes." In the quantum optics literature, the modes of open cavities are often referred to as quasimodes to distinguish them from the normal modes of closed (perfect) cavities. There is an impressive lack of consensus on how such modes should be defined. Most authors miss the essential point of the problem of using natural cavity modes to describe the quantized radiation field, which we discussed in Section 9.4, and the ideas of classical resonator theory (particularly Thomson's [594, 595]). For a review of some of these approaches, see [173]. Here I will just list the main papers: [30, 31,43, 78-80, 102, 103, 132-140, 169, 193, 194, 226-228, 247, 248, 258, 259, 294, 362, 374, 378-382, 390, 391, 400404,408,603,604,608], [618, 8], For a pedagogical account of the radiation condition from Sommerfeld himself, see Sec. 28 of [571]. For a treatment of the underlying mathematical issues, see [24, 651]. The possibility of representing the electromagnetic field by two scalar potentials alone was discovered by Sir Edmund Whittaker in 1903 [640]. Two other ways of representing the field by only two scalar potentials were developed later by Green and Wolf [250] and Bouwkamp and Casimir [70]. For a review, see Nisbet's paper [466] or Section 10 of Bremmer's article in the Handbuch der Physik 17 [72]. Problems 9.1 Use the Gardiner-Collett Hamiltonian, (9.1), to derive the Lindblad master equation for cavity damping, (6.68). Hint: Write down a general expression for the total density matrix in terms of number states of the cavity mode and the external modes. Then use (9.1) to obtain the master equation for this total density matrix and trace over the external modes to get a master equation for the reduced density matrix for the cavity mode.
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9.2 Calculate the first two integrals on the right-hand side of (9.9) by substituting (9.10) and (9.11) and using
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