Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Part I From classical to wave mechanics
- Part II Weyl quantization and algebraic methods
- Part III Selected topics
- 14 From classical to quantum statistical mechanics
- 15 Lagrangian and phase-space formulations
- 16 Dirac equation and no-interaction theorem
- References
- Index
15 - Lagrangian and phase-space formulations
Published online by Cambridge University Press: 14 January 2010
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Part I From classical to wave mechanics
- Part II Weyl quantization and algebraic methods
- Part III Selected topics
- 14 From classical to quantum statistical mechanics
- 15 Lagrangian and phase-space formulations
- 16 Dirac equation and no-interaction theorem
- References
- Index
Summary
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location xi at time ti and the final location xf at time tf. Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle.
Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space.
The Schwinger formulation of quantum dynamics
In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
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- From Classical to Quantum MechanicsAn Introduction to the Formalism, Foundations and Applications, pp. 526 - 549Publisher: Cambridge University PressPrint publication year: 2004