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/e05d5bc2-9f6b-44fc-95c4-8a9db92e5483/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/e05d5bc2-9f6b-44fc-95c4-8a9db92e5483/content/inline10_2.jpg"/> at the given time moment t is given by the relation: 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/e05d5bc2-9f6b-44fc-95c4-8a9db92e5483/content/eqn10_1.jpg"/>