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Schrodinger Equation
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The graphical interpretation of the Schrodinger equation is not difficult either. The equation says that inside a well (where E>U) the curvature of Y(x) is toward the axis (like a sine wave), and outside the well the curvature is away from the axis (like an exponential). Usually a function that keeps curving away from the axis will end up going to infinity. In the Schrodinger equation, the curvature is proportional to the magnitude of the function, so Y(x) flattens out as it approaches the axis, and for an appropriate value of E the approach to the axis will be asymptotic.
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It is evident that these thoughts recurred to Erwin Schrodinger when he devised his wave equation leading to discovery of wave mechanics. He found the reality of physics in wave motions and he ... based this reality on an underlying unity of mind. Schrodinger was well versed in the techniques of Bhagavat Gita and he knew that he was a "Jnanayogi.". His intellect showed him the way, and throughout his life he expressed in graceful essays his belief in Vedanta but he remained what the Indians called a Mahavit, a person who knows that theory but has failed to achieve a practical realization of it in his own life. He knew from Chandogya Upanishad "I am Mahavit, a knower of the world and not an Atmavit, a knower of the atman"
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This is not a derivationof the schrodinger equation. There is no derivation of the equation for nonzero potential. At the same time its form is made very plausible by the above considerations. This is the Schrodinger equation for one dimensional motion of a single particle only. It can be extended to three dimensions and to an arbitrary number of particles, so that for example
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There have been, at times, heroic attempts to find tractable approximations to solve the Schrodinger equation for larger and larger molecules. We’ve ... realized that the Schrodinger equation is actually an insufficient principle to do chemistry because in the end, the Schrodinger equation is a single electron equation. Reality is made up of more than 1 electron. However, there does not exist an authoritative multi-electron theory, there are several competing multi-electron theories, from quantum field theory, fractional electron solids, Wilson effective field theories, to the Walter-Kohn approximations.
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The semilinear Schrodinger equation (NLS) is displayed on the right-hand box, where the exponent p is fixed and is larger than one. One can ... replace the nonlinearity by a more general nonlinearity
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When studying the dynamics of a quantum particle interacting with a quantum fluid, the relevant equation to consider is a multiconfiguration Schrodinger equation. The wavefunction represents a probability distribution in configuration space, while the fluid lives only in real space. This gives rise to strange terms where one must divide by the integral over some variables before computing the nonlinearity.
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Schrodinger Equation