What is the Relationship Between Quantum and Classical Path Integrals?

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SUMMARY

The relationship between quantum mechanical path integrals and classical mechanics path integrals is fundamentally analogous. The Schrödinger equation is derived from a quantum version of classical Hamiltonian mechanics, while the path integral is based on a quantum version of Lagrangian mechanics. The classical limit of the path integral corresponds to the traditional Lagrangian path, achieved by taking Planck's constant, \hbar, to zero. This indicates that the classical path is the stationary path of the path integral, reinforcing the expected relationships between quantum and classical frameworks.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically path integrals.
  • Familiarity with classical mechanics, particularly Hamiltonian and Lagrangian formulations.
  • Knowledge of Planck's constant and its significance in quantum theory.
  • Basic grasp of the concept of action in physics.
NEXT STEPS
  • Explore the derivation of the Schrödinger equation from Hamiltonian mechanics.
  • Study the formalism of path integrals in quantum mechanics.
  • Investigate the implications of taking the limit of Planck's constant in quantum mechanics.
  • Learn about the concept of action and its role in both classical and quantum mechanics.
USEFUL FOR

Physicists, students of quantum mechanics, and anyone interested in the foundational relationships between quantum and classical physics will benefit from this discussion.

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What is the relationship between the quantum mechanical path integral and
classical mechanics path integral?
 
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They are basically just analogues of each other. The Schroedinger equation is based around a quantum version of classical Hamiltonian mechanics. By the same token, the path integral is built around a quantum version of Lagrangian mechanics. Another thing to consider is that the classical limit of the path integral is the traditional Lagrangian path. This can be found by taking Planck's constant, \hbar, to the limit of zero. Or, in another way of looking at it, the classical path is the stationary path of the path integral.

There are a lot of these little relationships that you can make between the two but most of these (like the classical limit) are obviously expected for the quantum path integral to make sense. And there is a bunch of formalism behind this. Can't recall a good reference that explains this though...
 
Can the relationship between the quantum mechanical path integral and
classical mechanics be stated as this?
A path integral involves an exponential of the action S.
 

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