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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.83 The Schrödinger Equation

Terence Tao

In mathematical physics, the Schrödinger equation (and the closely related Heisenberg equation) are the most fundamental equations in nonrelativistic quantum mechanics, playing the same role as Hamilton’s laws of motion (and the closely related Poisson equation) in nonrelativistic classical mechanics. (In relativistic quantum mechanics, the equations of quantum field theory take over the role of Heisenberg’s equation, while Schrödinger’s equation does not have a natural direct analogue.) In pure mathematics, the Schrödinger equation, together with its variants, is one of the basic equations studied in the field of PARTIAL DIFFERENTIAL EQUATIONS [IV.12], and has applications to geometry, to spectral ...

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