Phys 711 — Assignment 1

Q1(a)

ψ={α,β,γ}

Unprotect[Conjugate];

Conjugate/:MakeBoxes[Conjugate[x_],StandardForm]:=TemplateBox[{MakeBoxes[x,StandardForm]},"Conjugate",DisplayFunction->(SuperscriptBox[#1,"*"]&)];

Protect[Conjugate];

$Post=#/. {z_ Conjugate[z_]:>Abs[z]^2,Conjugate[z_] z_:>Abs[z]^2}&;

Transpose[ψ\[Conjugate]]

(Subscript[Norm, ψ])=Transpose[ψ\[Conjugate]].ψ

(OverTilde[ψ])=ψ/SqrtN

Assuming[α∈Complexes&&β∈Complexes&&γ∈Complexes&&SqrtN ∈Reals,Simplify[Transpose[(OverTilde[ψ])]\[Conjugate].(OverTilde[ψ])]]

Assuming[α∈Complexes&&β∈Complexes&&γ∈Complexes&&SqrtN ∈Reals,Simplify[Transpose[(OverTilde[ψ])]\[Conjugate].(OverTilde[ψ])]]/.SqrtN->Sqrt[(Subscript[Norm, ψ])]

Q1(b)

Pb={{0,0,0},{0,1,0},{0,0,0}}

Assuming[α∈Complexes&&β∈Complexes&&γ∈Complexes&&SqrtN ∈Reals,Simplify[Transpose[(OverTilde[ψ])]\[Conjugate].Pb.(OverTilde[ψ])]]/.SqrtN->Sqrt[(Subscript[Norm, ψ])]

Q1(e)

T={{0,0,1},{1,0,0},{0,1,0}}
{Evals,Evecs}=Eigensystem[T]
Evecs=Normalize/@Evecs

Evals

Transpose[Evecs]//MatrixForm

Q1(f)

V=Transpose[Evecs]
V//MatrixForm
Expand[Transpose[Conjugate[V]].T.V]//MatrixForm

Q1(g)

Inverse[T]==Transpose[Conjugate[T]]
T.T.T==IdentityMatrix[3]

Q2(a)

h=ArrayReshape[Range[16],{4,4}]
H=h+Transpose[h]
H//MatrixForm

Table[5(i+j)-8,{i,1,4},{j,1,4}]//MatrixForm

Q2(b)

H.H==Table[Sum[H[[i,k]]*H[[k,j]],{k,1,4}],{i,1,4},{j,1,4}]

Q2(c)

{Evals,Evecs}=Eigensystem[H]

Evals[[3]]
Evals[[4]]

Evecs[[3]]
Evecs[[4]]

H.Evecs[[3]]

H.Evecs[[4]]

Q2(d)

(Subscript[ϕ, a])=Evecs[[3]]
(Subscript[ϕ, b])=Evecs[[4]]

Nmat = Table[If[i==j,i,0],{i,1,4},{j,1,4}]

((Subscript[ϕ, a]).Nmat.(Subscript[ϕ, a]))/((Subscript[ϕ, a]).(Subscript[ϕ, a]))
((Subscript[ϕ, b]).Nmat.(Subscript[ϕ, b]))/((Subscript[ϕ, b]).(Subscript[ϕ, b]))