Equivalence between path integral formulation and matrix formulation

[wenty]

Does anyone know where to find the "direct" (not by prove they are both equal to Schrödinger formualtion )proof?

 

[dextercioby]

There's no direct proof.Both matrix mechanics and Schrödinger's wave mechanics are particular representations of Dirac's abstract formulation.So the equivalence is between formulations:path integral (R.P.Feynman) and traditional (vectors and operators,P.A.M.Dirac).

There's a famous book which deals with this issue:

Feynman and Hibbs,"Quantum Mechanics and Path Integrals",McGraw & Hill,1965.

The essential of that book is found as a chapter/subchapter in many QM texts,outta which i'd like to mention Sakurai and Cohen-Tannoudji.

Daniel.

 

[R0man]

The equivalence between the path integral formulation and the matrix formulation is a fundamental concept in quantum mechanics. It states that both formulations lead to the same physical predictions and are essentially two different ways of expressing the same underlying mathematical structure.

The path integral formulation, first introduced by Richard Feynman, is based on the idea of summing over all possible paths that a particle can take to go from one point to another. This is represented by an integral over all possible paths in space and time. On the other hand, the matrix formulation, developed by Werner Heisenberg, represents quantum states as vectors in a complex vector space and operators as matrices. The time evolution of a quantum state is then given by matrix multiplication.

To show the equivalence between these two formulations, one can directly derive the Schrödinger equation from both the path integral and matrix formulations. However, to find a "direct" proof that does not involve showing their equivalence to the Schrödinger equation, one can look at the underlying mathematical structure of both formulations.

Both the path integral and matrix formulations are based on the principle of superposition, which states that the total wavefunction of a system is the sum of all individual wavefunctions. In the path integral formulation, this is represented by summing over all possible paths, while in the matrix formulation, it is represented by matrix multiplication.

Furthermore, both formulations also involve the concept of time evolution, which is represented by the time evolution operator in the matrix formulation and the action integral in the path integral formulation. These operators are related to each other through the Hamiltonian of the system.

Therefore, one can see that the path integral and matrix formulations are essentially two different ways of representing the same underlying mathematical structure. This provides a "direct" proof of their equivalence without having to show their relationship to the Schrödinger equation.

In conclusion, the equivalence between the path integral and matrix formulations is a fundamental concept in quantum mechanics, and their underlying mathematical structures are closely related. While a direct proof may not be readily available, understanding the fundamental principles and concepts behind both formulations can provide a deeper insight into their equivalence.