How to Find Ψ(x,t) for a Given Hamiltonian Matrix and Initial State?

[Mlisjak]

Homework Statement

I have the matrix form of the Hamiltonian:

H = ( 1 2-i
2+i 3)

If in the t=0, system is in the state a = (1 0)^T, what is Ψ(x,t)?

Homework Equations

Eigenvalue equation

The Attempt at a Solution

So, I have diagonalized given matrix and got the eigenvalues: 2+√6 and 2-√6. I am suspecting that these are not good, since I can't get eigenvectors I can use. When trying to calculate eigenvectors, I get:
a = 1 and b=(-1+√6)/(2+i). This is the one I got when I used 2+√6 but after that I didn't even try with the other eigenvalue since it will be similar.
I don't know what to do with those and don't know how to normalize them. Also, even if I knew how to get correct eigenvectors, I am not sure how to proceed and get Ψ(x,t). [/B]

 

[DrClaude]

I don't see how you got that value for b (I don't get the same signs), so check your math. If what bothers you is the (2+i) in the denominator, simply multiply by (2-i)/(2-i).

 

[Mlisjak]

Do you get the same eigenvalues? I put it in the form:
( 1 2-i
2+i 3 ) * (a b)^t = (2+√6) (a b)^T

I'm sorry, I don't know to write it properly.

Then I got:
a+(2-i)b = (2+√6)a
That's where the expression comes from. And by using the second equation i got a=a in which cases we always put 1 in our class. Is that wrong?

 

[DrClaude]

a+(2-i)b = (2+√6)a
That's where the expression comes from. And by using the second equation i got a=a in which cases we always put 1 in our class.

That gives
$$
b = \frac{1 + \sqrt{6}}{2 - i}
$$
which is not what you wrote above.

Try multiplying with $(2 + i)/(2 + i)$.

 

[Mlisjak]

Ok, I got the vectors but don't know how to proceed to get Ψ (x,t)

 

[DrClaude]

What have you learned about time evolution?

 

[Mlisjak]

To be precise, I don't understand how to get right coefficients to write Ψ(x,0) as a linear combination of the vectors I got. When i get that, I believe that I just have to add time dependence e^-iEt/†

 

[DrClaude]

Scalar product. You have found the eigenvectors φ₁ and φ₂ and want to write Ψ(t=0) = c₁ φ₁ + c₂ φ₂, you find the coefficients using
$$
c_n = \phi_n^\dagger \Psi
$$

 

[Mlisjak]

Is that equal to finding the norm of the eigenvectors? For example, I have three eigenvectors: v1 = v2 = (1 0 0)^† and v3=1/√2 (0 -i 1)^†.

Would I write Ψ_o= 2*(1 0 0)^†+1/√2 (0 -i 1)^†?

 

[Mlisjak]

I have used different example, since I get too complicated vectors in the exercise I posted originally