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Matrix Element of a Physical Operator via Functional Integral

Lukong Cornelius Fai

Published Feb 19, 2021• 0 citations

Abstract

We limit ourselves for convenience to a system with one degree of freedom and having the Lagrangian of a particle in a potential field, as in 5.107 . For the transition amplitude of this system we have expression 5.97 . We have the following questions to ask: Is it possible on the basis of functional integration to obtain the formula for the evaluation of the matrix element of a physical operator of the type 4.82 ? How can we avoid the difficulty related to the normalization factor in the evaluation of these matrix elements? In order to answer these questions, we define the matrix element of a physical operator using the quantum mechanical definition. We limit ourselves to the quantum mechanical evaluation of the matrix element of the operator of a physical quantity F. We consider the Heisenberg localized state | q i , t i ⟩. Suppose the initial time moment t i corresponds to the initial state | q i , t i ⟩ and the final time moment t f corresponds to the final state | q f , t f ⟩. Consider the Heisenberg operator F ̂ t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline10_1.jpg"/> of a physical quantity Ϝ and t i ≤ t ≤ t f . By definition, considering the normalized state, the matrix element in the q representation of the operator F ̂ t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/inline10_2.jpg"/> at the given time moment t is given by the relation: 10.1 F if = q f , t f | F ̂ t | q i , t i q f , t f q i , t i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqn10_1.jpg"/>

DOI: 10.1201/9781003145554-10

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