Phys 711 — Assignment 2

Q1(a)

ClearAll[H,t,ϵ]

H={{ϵ,-t,0},{-t,2ϵ,-t},{0,-t,3ϵ}}

MatrixForm[H]

Eigenvalues[H]

Q1(b)

{Evals,Evecs}=Eigensystem[H];

Evals

Evecs

(Simplify[#/Sqrt[#[[1]]^2+#[[2]]^2+#[[3]]^2]])&/@ Evecs

V=Transpose[(Simplify[#/(#[[1]]^2+#[[2]]^2+#[[3]]^2)])&/@ Evecs];
MatrixForm[%]

Simplify[Inverse[V].V]
Simplify[V.Inverse[V]]

Inverse[V].H.V

N[Expand[Inverse[V].H.V]/.{t->1,ϵ->1}]//MatrixForm
{2ϵ,2ϵ-Sqrt[2t^2+ϵ^2],2ϵ+Sqrt[2t^2+ϵ^2]}/.{t->1,ϵ->1}
N[%]

N[Expand[Inverse[V].H.V]/.{t->2,ϵ->1/2}]//MatrixForm
{2ϵ,2ϵ-Sqrt[2t^2+ϵ^2],2ϵ+Sqrt[2t^2+ϵ^2]}/.{t->2,ϵ->1/2}
N[%]

Q1(c)

Tr[H]

Tr[H.H]

Simplify[Tr[H.H.H]]
Simplify[Tr[MatrixPower[H,3]]]

Q1(c)

TrHk=Simplify[Tr[MatrixPower[H,k]]]

Q1(d)

Simplify[Tr[MatrixExp[I H t/ℏ]]]

Simplify[Tr[MatrixExp[-β H]]]