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

Lagrangian Densities and Principle of Least Action in Nonrelativistic Quantum Mechanics