Finding Eigenvalues and Wave Function in a Basis of Orthonormalized Vectors

[Lolek2322]

Homework Statement

Eigenvalues of the Hamiltonian with corresponding energies are:
Iv₁>=(I1>+I2>+I3>)/3^1/2 E₁=α + 2β
Iv₂>=(I1>-I3>) /2^1/2 E₂=α-β
Iv₃>= (2I2> - I1> I3>)/6^1/2 E₃=α-β

Write the matrix of the Hamiltonian in the basis of the orthonormalized vectors I1>, I2>, I3>

If in t=0, system is in the state I1>, what is the wave function in t?

Homework Equations

Hij = <ilHlj>

The Attempt at a Solution

Although I know that energy is the eigenvalue of the Hermitian operator, I am not sure how to incorporate that in this certain problem. I have used mentioned equation for previous problems, but I always had the form of the operator. With only eigenvectors and eigenvalues I am stuck and don't even know how to begin solving this.

 

[DrClaude]

There seems to be a part of the question that is missing. Can you write it out fully?

 

[Lolek2322]

There seems to be a part of the question that is missing. Can you write it out fully?

I appologize. I have written it now

 

[DrClaude]

One way to go about this is to start by writing the Hamiltonian in the |v1>, |v2>, |v3> basis, then applying the proper transformation operation to "rotate" the Hamiltonian to the |1>, |2>, |3> basis.

 

[Lolek2322]

But unfortunately I do not know how to do that

 

[kuruman]

The straightforward way to do it is
1. Find |1>, |2> and |3> as linear combinations of |v₁>, |v₂> and |v₃> and verify that they are orthonormal.
2. Note that H|v₁> = (α + 2β) |1> and get similar expressions for H operating on the other two v's.
3. Calculate things like < 1 | H | 2 > using the linear combinations from item 1 and substituting from item 2.