Is the Functional Integral in Quantum Mechanics Solvable?

[eljose]

I would like to know if the functional integral:

$$D[\phi]e^{iS[\phi]/\hbar$$ (1)

where S is the classical action of a system of a Lagrangian with a potential in the form:

$$V=\sum_{n}\delta(x-n)$$ n=0,1,2,3,4,5,6,...

The Schroedinguer equation if the sum is finite can be transformed into a solvable "integral equation"..but i would like to know if the functional is exactly integrable,...by the way i would like to know how Feynman obtained the Schroedinguer equation by calculating the infinite integral (1) for S the action:

$$S=\int_{a}^{b}dt\alpha{(dx/dt)^{2}}$$

then this integral in (1) would be a Gaussian and could be calculated exactly but how you derive Schroedinguer equation?..thanks.

 

[eljose79]

I am happy to provide some information about the functional integral and its relationship to the Schrödinger equation.

First of all, the functional integral (1) is a mathematical tool used in quantum field theory to calculate the probability of a particle or system of particles to move from one state to another. It is a generalization of the path integral in classical mechanics, and it allows us to calculate the probability amplitude for all possible paths of the system.

In regards to your question about the functional integral being exactly integrable, the answer is not straightforward. It depends on the specific system and potential being studied. In some cases, the integral can be evaluated exactly, but in many cases, it is not possible to find an analytical solution and numerical methods must be used.

Regarding Feynman's derivation of the Schrödinger equation, it is based on his path integral formulation of quantum mechanics. In this formulation, the probability amplitude for a particle to go from one point to another is given by a sum over all possible paths that the particle could take. Feynman showed that in the limit of small time steps, this sum can be approximated by a functional integral, and by applying the principle of least action, the Schrödinger equation can be derived.

In summary, the functional integral is a powerful tool in quantum field theory, and its exact solvability depends on the specific system being studied. Feynman's path integral formulation provides a connection between the functional integral and the Schrödinger equation, which is a fundamental equation in quantum mechanics. I hope this information helps to answer your questions.

 

[denian]

I can provide a response to your query about the solvability of the functional integral in quantum mechanics.

Firstly, let me clarify that the functional integral in quantum mechanics is a mathematical tool used to calculate the probability amplitude of a particle moving from one point to another in space and time. It is based on the path integral formulation of quantum mechanics, which was developed by Richard Feynman.

Now, to answer your question, the functional integral in quantum mechanics is not exactly solvable in most cases. This is because it involves an integration over an infinite number of paths, making it a highly complex mathematical problem. However, there are some special cases where the functional integral can be solved exactly, such as for free particles or particles in a harmonic potential.

In regards to your specific query about the functional integral for a Lagrangian with a potential in the form of a sum of delta functions, it is possible to transform this into a solvable "integral equation" if the sum is finite. However, for an infinite sum, it is not exactly solvable.

Regarding your question about how Feynman obtained the Schrödinger equation, he derived it by using his path integral formulation of quantum mechanics. He showed that the probability amplitude for a particle to move from one point to another can be expressed as a sum over all possible paths that the particle can take. By using this approach and applying the principle of least action, he was able to derive the Schrödinger equation.

In conclusion, while the functional integral in quantum mechanics is not exactly solvable in most cases, it is still a valuable tool for understanding and predicting the behavior of quantum systems. Feynman's path integral formulation has greatly influenced our understanding of quantum mechanics and continues to be a fundamental concept in the field.