Abstract
Quantum mechanics is a linear theory, based on the superposition principle, which associates with each quantum system a wavefunction that satisfies the Schrödinger equation. In certain cases, constituents of the multi-particle system, for example electrons or atoms, can be described by the single-particle linear Schrödinger equation, which incorporates the presence of the other quantum particles in the effective mass and/or the periodic potential V. Such situations were discussed in Chap. 2, which focused on ballistic electrons, and in Chap. 6, which was dedicated to particle optics. However, this simplifying assumption is not valid for all quantum systems, and there are situations when the presence of other quantum particles in the system must be treated by introducing nonlinear terms into the expression for the potential energy felt by a constituent of the quantum system. In particular, these situations include the cases where many-body interactions cannot be neglected. Quantum nonlinear equations, especially the nonlinear Schrödinger equation (NLS), have many interesting analogies in nonlinear optics or gravity. These analogies are the subject of this chapter.
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Dragoman, D., Dragoman, M. (2004). Quantum/Classical Nonlinear Phenomena. In: Quantum-Classical Analogies. The Frontiers Collection. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09647-5_7
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DOI: https://doi.org/10.1007/978-3-662-09647-5_7
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