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"/>
Bounded operators on a Hilbert space H → https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq3665.tif"/> have many of the properties of operators defined on V → N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq3666.tif"/> . For example, Theorem 13.1(1) and Corollary 13.1(1) apply, e.g., for bounded operators we have ( 18.1 ) 〈 ϕ → | A ^ ϕ → 〉 = 〈 ϕ → | B ^ ϕ → 〉 ∀ ϕ → ∈ H → ⇒ A ^ = B ^ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203702413/2f242b9b-104e-4a03-b478-d3f6577cf44e/content/eq3667.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"/> .
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"/>
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"/>
Using the Planck’s formula for P n , show that The average number of photons is given by ( 4.1 ) 〈 n 〉 = 1 e x − 1 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/eqn4_1.tif"/> where x = ℏ ω k B T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/ineqn4_1.tif"/> , and k B is the Boltzmann constant. Show that for large temperatures ( T ≫ 1), the average energy is proportional to temperature, i.e., 〈 E 〉 = k B T . Calculate 〈 n 2 〉 and show that the ratio ( 4.2 ) α = 〈 n 2 〉 − 〈 n 〉 〈 n 〉 2 = 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429092053/5e12ab0e-ba42-4caa-9bdf-6325e9117d3e/content/eqn4_2.tif"/>