# Equivalence between path integral formulation and matrix formulation

• wenty
In summary, the conversation discusses the search for a direct proof of the equivalence between matrix mechanics and Schrödinger's wave mechanics. It is mentioned that both are representations of Dirac's abstract formulation, and the equivalence is between path integral and traditional formulations. A recommended resource for further reading on this topic is the book "Quantum Mechanics and Path Integrals" by Feynman and Hibbs. This concept is also covered in many other quantum mechanics textbooks, such as those by Sakurai and Cohen-Tannoudji.
wenty
Does anyone know where to find the "direct" (not by prove they are both equal to Schrodinger formualtion )proof?

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.

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 Schrodinger equation from both the path integral and matrix formulations. However, to find a "direct" proof that does not involve showing their equivalence to the Schrodinger 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 Schrodinger 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.

## 1. What is the path integral formulation and the matrix formulation?

The path integral formulation and the matrix formulation are two different mathematical approaches used to describe the behavior of quantum systems. The path integral formulation is based on the concept of a particle taking all possible paths simultaneously, while the matrix formulation uses matrices to represent the operators and states of a quantum system.

## 2. How are these two formulations related?

The path integral formulation and the matrix formulation are equivalent, meaning they can both be used to describe the same quantum system. The path integral can be converted into a matrix expression through a process called discretization, where the continuous path is divided into a finite number of steps.

## 3. What are the advantages of the path integral formulation?

The path integral formulation allows for a more intuitive understanding of quantum mechanics, as it is based on the concept of a particle taking all possible paths. It also provides a way to deal with non-perturbative effects, which are difficult to handle in the matrix formulation.

## 4. What are the advantages of the matrix formulation?

The matrix formulation is more mathematically rigorous and can handle a wider range of quantum systems, including those with an infinite number of states. It is also more computationally efficient, making it easier to apply to complex systems.

## 5. Which formulation is used more frequently in research?

Both the path integral and matrix formulation are used extensively in research, as they each have their own advantages and are applicable to different types of systems. However, the matrix formulation is more commonly used in theoretical physics, while the path integral formulation is often used in condensed matter physics and quantum field theory.

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