Hermitian operator matrix

1. Aug 30, 2016

Lolek2322

1. The problem statement, all variables and given/known data
Eigenvalues of the Hamiltonian with corresponding energies are:
Iv1>=(I1>+I2>+I3>)/31/2 E1=α + 2β
Iv2>=(I1>-I3>) /21/2 E2=α-β
Iv3>= (2I2> - I1> I3>)/61/2 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?

2. Relevant equations
Hij = <ilHlj>

3. 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.

Last edited: Aug 30, 2016
2. Aug 30, 2016

Staff: Mentor

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

3. Aug 30, 2016

Lolek2322

I appologize. I have written it now

4. Sep 1, 2016

Staff: Mentor

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.

5. Sep 2, 2016

Lolek2322

But unfortunately I do not know how to do that

6. Sep 8, 2016

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.