Final answer:
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.