Pensieve header: Examples for the UCLA FDA talk.

Startup

SetDirectory["C:\\drorbn\\AcademicPensieve\\Talks\\UCLA-191101"];

hm

{(OverHat[x])=(({{0, 1, 0}, {0, 0, 0}, {0, 0, 0}})),(OverHat[y])=(({{0, 0, 0}, {0, 0, ℏ}, {0, 0, 0}})),(OverHat[c])=(({{0, 0, 1}, {0, 0, 0}, {0, 0, 0}}))};
{(OverHat[x]).(OverHat[y])-(OverHat[y]).(OverHat[x])==ℏ (OverHat[c]), (OverHat[x]).(OverHat[c])==(OverHat[c]).(OverHat[x]),(OverHat[y]).(OverHat[c])==(OverHat[c]).(OverHat[y])}

Λ=-ℏ (Subscript[η, i])(Subscript[ξ, j])(Subscript[c, k])+((Subscript[ξ, i])+(Subscript[ξ, j]))(Subscript[x, k])+((Subscript[η, i])+(Subscript[η, j]))(Subscript[y, k]);
Simplify@With[{𝔼=MatrixExp},
𝔼[(OverHat[x])(Subscript[ξ, i])].𝔼[(OverHat[y])(Subscript[η, i])].𝔼[(OverHat[x])(Subscript[ξ, j])].𝔼[(OverHat[y])(Subscript[η, j])]==𝔼[(OverHat[x])(∂_(Subscript[x, k]))Λ].𝔼[(OverHat[y])(∂_(Subscript[y, k]))Λ].𝔼[(OverHat[c])(∂_(Subscript[c, k]))Λ]]

sl2

Λ0=HoldForm[((Subscript[η, i])+(((E^(-(Subscript[α, i])-ϵ (Subscript[β, i]))) (Subscript[η, j]))/(1+ϵ (Subscript[η, j]) (Subscript[ξ, i]))))(Subscript[y, k])+((Subscript[β, i])+(Subscript[β, j])+((Log[1+ϵ (Subscript[η, j]) (Subscript[ξ, i])])/ϵ))(Subscript[b, k])+((Subscript[α, i])+(Subscript[α, j])+Log[1+ϵ (Subscript[η, j]) (Subscript[ξ, i])])(Subscript[a, k])+((((E^(-(Subscript[α, j])-ϵ (Subscript[β, j]))) (Subscript[ξ, i]))/(1+ϵ (Subscript[η, j]) (Subscript[ξ, i])))+(Subscript[ξ, j]))(Subscript[x, k])];
TeXForm[Λ0]
Λ=ReleaseHold[Λ0]

{(OverHat[y])=(({{0, 0}, {ϵ, 0}})),(OverHat[b])=(({{0, 0}, {0, -ϵ}})),(OverHat[a])=(({{1, 0}, {0, 0}})),(OverHat[x])=(({{0, 1}, {0, 0}}))};
{(OverHat[a]).(OverHat[x])-(OverHat[x]).(OverHat[a])==(OverHat[x]), (OverHat[a]).(OverHat[y])-(OverHat[y]).(OverHat[a])==-(OverHat[y]),(OverHat[b]).(OverHat[y])-(OverHat[y]).(OverHat[b])==-ϵ (OverHat[y]), (OverHat[b]).(OverHat[x])-(OverHat[x]).(OverHat[b])==ϵ (OverHat[x]),(OverHat[x]).(OverHat[y])-(OverHat[y]).(OverHat[x])==(OverHat[b])+ϵ (OverHat[a])}

Simplify@With[{𝔼=MatrixExp},
𝔼[(Subscript[η, i]) (OverHat[y])].𝔼[(Subscript[β, i]) (OverHat[b])].𝔼[(Subscript[α, i]) (OverHat[a])].𝔼[(Subscript[ξ, i]) (OverHat[x])].𝔼[(Subscript[η, j]) (OverHat[y])].𝔼[(Subscript[β, j]) (OverHat[b])].𝔼[(Subscript[α, j]) (OverHat[a])].𝔼[(Subscript[ξ, j]) (OverHat[x])]==𝔼[(OverHat[y])(∂_(Subscript[y, k]))Λ].𝔼[(OverHat[b])(∂_(Subscript[b, k]))Λ].𝔼[(OverHat[a])(∂_(Subscript[a, k]))Λ].𝔼[(OverHat[x])(∂_(Subscript[x, k]))Λ]]

Series[Λ,{ϵ,0,2}]