Exploring the Connection: Quantum & Classical Path Integrals

The quantum mechanical path integral involves a similar exponential, but with the quantum action, which is the classical action plus Planck's constant times the change in action over time.In summary, the quantum mechanical path integral and classical mechanics path integral are analogues of each other, with the former being based on quantum Hamiltonian mechanics and the latter being based on Lagrangian mechanics. The classical limit of the path integral is the traditional Lagrangian path, and there are other relationships between the two such as the quantum action being the classical action plus Planck's constant times the change in action over time.
  • #1
sourena
13
0
What is the relationship between the quantum mechanical path integral and
classical mechanics path integral?
 
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  • #2
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...
 
  • #3
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.
 

Related to Exploring the Connection: Quantum & Classical Path Integrals

1. What is the difference between quantum and classical path integrals?

Quantum and classical path integrals are mathematical tools used to describe the behavior of particles in different systems. The main difference between them is that classical path integrals consider the particle to be a point-like object with a definite trajectory, while quantum path integrals take into account the wave-like nature of particles and their probabilistic behavior.

2. How are path integrals used in quantum mechanics?

In quantum mechanics, path integrals are used to calculate the probability of a particle moving from one point to another in a given system. They allow us to take into account all possible paths that the particle could take, rather than just considering a single classical trajectory. Path integrals are also used to calculate the wave function of a system, which describes the probability of finding a particle at a given position and time.

3. What are some real-world applications of path integrals?

Path integrals have a wide range of applications in various fields, such as physics, chemistry, and engineering. They are used in quantum field theory to describe the behavior of particles at the subatomic level. In chemistry, path integrals are used to study the behavior of molecules and chemical reactions. They are also used in finance to model the behavior of stock prices and in computer graphics for simulating the movement of fluids.

4. Can classical and quantum path integrals be combined?

Yes, classical and quantum path integrals can be combined in some cases to describe systems that exhibit both classical and quantum behaviors. This is known as semi-classical path integration, where the classical and quantum descriptions are weighted according to their relative contributions to the overall behavior of the system.

5. How do path integrals relate to the uncertainty principle?

The uncertainty principle, a fundamental principle in quantum mechanics, states that it is impossible to simultaneously know the exact position and momentum of a particle. Path integrals provide a way to calculate the probability of a particle's position and momentum, taking into account all possible paths it could take. This allows us to understand the probabilistic nature of particles and their behavior, as described by the uncertainty principle.

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