QM:Finding the probabilities of a Hamiltonian measurement

[~Sam~]

Homework Statement

This problem is from Zetelli 3.21
http://imgur.com/wYTNVwz
http://imgur.com/wYTNVwz

Homework Equations

Just the standard probability via product between the eigenfunction and the wavefunction

The Attempt at a Solution

I've found the eigenvectors for the Hamiltonian (-i,5,3) (-i,-2,3) (3i,0,1)

I can orthonormalize them, as well as normalize the initial state |y(0)> however I run into a problem when actually calculating the probabilities. I only get complex values and can't seem to get rid of them, especially since the normaliztion factor for |y(0)> is 1/sqrt(-1-16i). Can anyone help?

 

[blue_leaf77]

I can orthonormalize them

They are already orthogonal, you only need to normalize them.

especially since the normaliztion factor for |y(0)> is 1/sqrt(-1-16i).

Then you must do the normalization wrong. Normalization constant must be real. Can you show us how you normalized $\psi(0)$?

 

[~Sam~]

They are already orthogonal, you only need to normalize them.

Then you must do the normalization wrong. Normalization constant must be real. Can you show us how you normalized $\psi(0)$?

Ahh yes I realize now, I was trying to normalize then in terms of linear algebra instead of quantum mechanics. It looks like it's actually 1/sqrt(59). Normalizing the rest of the vectors and calculating the probabilities using bra-ket squared gives me ~59.322% ~23.73% ~16.95% which sum up to 100% so that looks right? Thank you

 

[blue_leaf77]

I was trying to normalize then in terms of linear algebra instead of quantum mechanics.

Both ways should coincide as linear algebra is one of the bases on which quantum mechanics was established.

 

[~Sam~]

Both ways should coincide as linear algebra is one of the bases on which quantum mechanics was established.

Seems like the way I was doing it with linear algebra is wrong for complex numbers, just taking the dot product sums and square rooting. Any ways do you have any insight in helping in the part C of finding it how it evolves in time? I know I just need to write out the initial state as a linear combination of the eigenvectors, but the first number is 4-i, and there is no real numbers in the first entry in either three of the eigenvectors, they are just i, -i, and 3i. Same for the other entries

 

[blue_leaf77]

I know I just need to write out the initial state as a linear combination of the eigenvectors

Right.
What you have to do is that you have to write $\psi(0) = c_1 u_1 + c_2 u_2 + c_3 u_3$. You already have $\psi(0)$, $u_1$, $u_2$, and $u_3$. What are yet to calculate are the expansion coefficients. How do you normally find the expansion coefficients given the basis your vector is expanded to is orthonormal?

 

[~Sam~]

Right.
What you have to do is that you have to write $\psi(0) = c_1 u_1 + c_2 u_2 + c_3 u_3$. You already have $\psi(0)$, $u_1$, $u_2$, and $u_3$. What are yet to calculate are the expansion coefficients. How do you normally find the expansion coefficients given the basis your vector is expanded to is orthonormal?

Yeah just by using the formula for the C_n and taking the bra-ket of the eigenvectors and the initial wavefunction. I've finished the rest of the problem. Thank you!