On Hamiltonian Formulations of the Schrödinger System

2002/06/01 by L. Gergely, László Á. Gergely · 1 voice · 14 citations
Engineering · Mathematics · Physics and Astronomy · #Applied mathematics #Classical mechanics #Covariant Hamiltonian field theory #Dirac equation #Electromagnetic Simulation and Numerical Methods #Hamiltonian (control theory) #Hamiltonian system #Lagrangian #Mathematical physics #Mathematics #Numerical methods for differential equations #Phase space #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #Schrödinger equation #gr-qc #hep-th #quant-ph

paper · pdf · doi:10.1006/aphy.2002.6262

published in Annals of Physics 298(2), 394-402 (Elsevier BV)

openalex publication_date 2002/06/01 · arxiv created 2003/01/09 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We review and compare different variational formulations for the Schrödinger field. Some of them rely on the addition of a conveniently chosen total time derivative to the hermitic Lagrangian. Alternatively, the Dirac-Bergmann algorithm yields the Schrödinger equation first as a consistency condition in the full phase space, second as canonical equation in the reduced phase space. The two methods lead to the same (reduced) Hamiltonian. As a third possibility, the Faddeev-Jackiw method is shown to be a shortcut of the Dirac method. By implementing the quantization scheme for systems with second class constraints, inconsistencies of previous treatments are eliminated.

Citations

• Generalized Hamiltonian Dynamics
• Elementary Quantum Field Theory
• Space-Time Approach to Non-Relativistic Quantum Mechanics
• The Development of the Space-Time View of Quantum Electrodynamics
• Quantisierung als Eigenwertproblem

Discussions

• The Lagrangian density of the Schrödinger's Eqn can be expressed as $$ \mathscr{L} = \frac{i\hbar}{2}(\psi^*\dot{\psi}-\dot{\psi}^*\psi)-\frac{\hbar^2}{2m}\nabla\psi\cdot\nabla\psi^* -V(x)\psi^*\psi \ [stackexchange, 1 points]