Normalized solution (mathematics): Difference between revisions

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{{Duplication|dupe=Normalized solutions (mathematics)}}

[[File:Quantum particle probability density.png|thumb|upright=1.35|The probability density distribution of a quantum particle in three-dimensional space. The points in the image represent the probability of finding the particle at those locations, with darker colors indicating higher probabilities. To simplify and clarify the visualization, low-probability regions have been filtered out. In fact, the total probability 1 means that the particle exists everywhere in the entire space.]] {{Quantum mechanics}}

In mathematics, a '''normalized solution''' to an [[Ordinary differential equation|ordinary]] or [[partial differential equation]] is a solution with prescribed norm, that is, a solution which satisfies a condition like <math>\int_{\mathbb{R}^N} |u(x)|^2 \, dx = 1.</math> In this article, the normalized solution is introduced by using the [[nonlinear Schrödinger equation]]. The nonlinear [[Schrödinger equation]] (NLSE) is a fundamental equation in [[quantum mechanics]] and other various fields of physics, describing the evolution of complex [[wave functions]]. In Quantum Physics, normalization means that the total probability of finding a quantum particle anywhere in the universe is unity.<ref>{{Cite journal |last1=Berestycki |first1=H. |last2=Lions |first2=P.-L. |date=1983 |title=Nonlinear scalar field equations. I. Existence of a ground state |journal=Arch. Rational Mech. Anal. |volume=82 |issue=4 |pages=313–345 |mr=0695535}}</ref>