Question

I have the following Hamiltonian for a triangular tight binding model:

H = e^(iϕ) (|2⟩ ⟨1| + |3⟩⟨ 2| + |1⟩⟨ 3| ) + e^(−iϕ) ( |1 ⟩⟨ 2| + |2 ⟩⟨ 3| + |3 ⟩⟨ 1| )
where the states |1⟩ , |2⟩ , |3⟩ are the basis states and ϕ ∈ R.
I wanted to find the eigenvalues and eigenstates of this Hamiltonian. The matrix turns out to be :
H = [0 e^(−iϕ) e^(iϕ) e^(iϕ) 0 e^(−iϕ) e^(−iϕ) e^(iϕ) 0].
I got λ1 = λ2 = λ3 = 2cos(ϕ). Now, one of the eigenvectors is
v1 = 1√3[1 1 1].
How do I find the other eigenvectors?
Also, how do I find the time-dependent solution |ψ(t)⟩ of the Schrödinger equation with the given Hamiltonian?

Answer 1

To find the other eigenvectors, use the eigenvalue equation (H - λI)v = 0 and apply the Gram-Schmidt process. The time-dependent solution of the Schrödinger equation is given by |ψ(t)⟩ = e^(-iEt/ħ) |ψ(0)⟩.

Step-by-step explanation:

To find the other eigenvectors, we solve the eigenvalue equation (H - λI)v = 0, where H is the Hamiltonian matrix, λ is the eigenvalue, I is the identity matrix, and v is the eigenvector. For the given Hamiltonian matrix:

H = [0 e^(−iϕ) e^(iϕ) e^(iϕ) 0 e^(−iϕ) e^(−iϕ) e^(iϕ) 0],

• For λ1 = λ2 = λ3 = 2cos(ϕ), the eigenvector v1 = [1/sqrt(3), 1/sqrt(3), 1/sqrt(3)].
• To find the other eigenvectors, we can use the Gram-Schmidt process. Let's denote the second eigenvector as v2 and the third eigenvector as v3.
• By applying the Gram-Schmidt process, we can find the orthogonal vectors v2 and v3 that are linearly independent to v1.

To find the time-dependent solution |ψ(t)⟩ of the Schrödinger equation, we can use the fact that the time evolution of a state is given by |ψ(t)⟩ = e^(-iEt/ħ) |ψ(0)⟩, where E is the eigenvalue and ħ is the reduced Planck's constant.