Lagrangian Densities and Principle of Least Action in Nonrelativistic Quantum Mechanics

Details Lagrangian Densities and Principle of Least Action in Nonrelativistic Quantum Mechanics Lagrangian Densities and Principle of Least Action in Nonrelativistic Quantum Mechanics

2007-12-10

arxiv.org/abs/0712.1608v1

Physics

Quantum Physics

17 pages, no figures

Scientific paper

The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schroedinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton's equations also give the Schroedinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order derivatives. For a time-independent system the Principle of Least Action reduces to the energy variational principle. For time-dependent systems the Principle of Least Action gives time-dependent approximations. Using a Hartree product trial wave function for a time-dependent many-boson system, we apply the Principle of Least Action to obtain the Gross-Pitaevskii equation that describes a Bose-Einstein condensate.

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