It is instructive to note that the principal advantage of linear operators is that their action on any vector is defined purely by their action on a set of basis vectors. Let us consider the linear operator F ̂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqni2_1.jpg"/> transforming the vector a via its components a k : 2.1 F ̂ a i = ∑ k F i k a k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqn2_1.jpg"/>
Let  , A ^ , B ^ , C ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn10_1.tif"/> Ĉ be arbitrary linear operators. Prove that [ A ^ B ^ , C ^ ] = [ A ^ , C ^ ] B ^ + A ^ [ B ^ , C ^ ] , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn10_2.tif"/> [ A ^ , [ B ^ , C ^ ] ] + [ B ^ , [ C ^ , A ^ ] ] + [ C ^ , [ A ^ , B ^ ] ] = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn10_3.tif"/>
Q24(1) Consider a three-dimensional Hilbert space ℋ → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1209.tif"/> with a preferred decomposition as the direct sum of a complete orthogonal family of one-dimensional subspaces S → ( − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1210.tif"/> , S → ( 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1211.tif"/> and S → ( + ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1212.tif"/> , i.e., ℋ → = ℋ → ⊕ = S → ( − ) ⊕ S → ( 0 ) ⊕ S → ( + ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0459.tif"/> Let η → ( − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1213.tif"/> , η → ( 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1214.tif"/> and η → ( + ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1215.tif"/> be unit vectors in S → ( − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1216.tif"/> , S → ( 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1217.tif"/> and S → ( + ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1218.tif"/> , respectively. 1 A vector in ℋ → ⊕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1219.tif"/> is of the form η → ⊕ = c − η → ( − ) ⊕ c 0 η → ( 0 ) ⊕ c + η → ( + ) = c − η → ( − ) ⊕ + c 0 η → ( 0 ) ⊕ + c + η → ( + ) ⊕ , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0460.tif"/> where C − , c 0 , c + ∈ ℂ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1220.tif"/> .
Calculate the square A → ⋅ S → 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn13_1.tif"/> of the scalar product of an arbitrary vector A → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn13_2.tif"/> and of the spin vector S → = S x i → + S y j → + S Z k → . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn13_3.tif"/>
The transition amplitudes 4.47 and 4.82 of chapter 4 may be generalized for a problem with s degrees of freedom that may be described by a set of operators q ̂ = q ̂ α α = 1 s , p ̂ = p ̂ α α = 1 s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqni6_1.jpg"/> : 6.1 Κ q ″ , t ″ ; q ′ , t ′ = ∫ ∏ α = 1 s D p α D q α exp i ℏ ∫ t ′ t ″ ∑ α = 1 s p α q . α − Η p q d t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003145554/875ff22b-09e9-4373-a808-2051f13516c8/content/eqn6_1.jpg"/>
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"/>
