Quantum mechanics, way of writing the eigenstates

[Chronos000]

Homework Statement

I'm having some trouble understanding exactly how the eigenstates of this matrix are being presented.

A= ( 0 -i
i 0 ) <- matrix

I can find the eigenvalues to be +/- 1 which gives the eigenvectors to be x=-iy or x=iy.

The eigenvectors are then being presented as:

|A+> = (|1> +i|2> )/root2

|A-> = (|1> -i|2> )/root2

I thought to get this you take the action 2 of A away from action 1 and the resolve it.(for A-). But I can't seem to get this.

I really just need some clarification on this as I have a further complicated matrix to do

 

[grey_earl]

What they give you as |1> is simply the vector (1, 0), and |2> is the vector (0,1), so |A+> = (1,i)/sqrt(2) which satisfies x = -i y, as you got. Similar for |A->.

 

[Chronos000]

thanks for your reply. I think I may be blind here- but I don't see how A=(1,i)/sqrt2 satisfies x=-iy. - if x=1 and y equals 1 from the vectors you spoke of

 

[JaWiB]

|A+> = (1,i) = |1>+i|2>

i.e., x=1 and y=i, which satisfies x=-i*y

similarly for the |A-> eigenvector.

 

[paranormal]

.

I would first commend the individual for working through the eigenvalues and eigenvectors of the matrix. Quantum mechanics can be a complex subject, and it's great to see someone taking the time to understand it.

In terms of the presentation of the eigenstates, it is important to note that in quantum mechanics, we often use the bra-ket notation to represent vectors and operators. The "A+" and "A-" in this context are not actions, but rather the eigenstates of the matrix A. The notation |1> and |2> represent the eigenvectors of the matrix, with 1 and 2 being the corresponding eigenvalues.

To obtain the eigenvectors presented, you can solve the matrix equation (A-λI)|x> = 0, where λ is the eigenvalue and |x> is the eigenvector. This will give you the two eigenvectors presented in the problem.

I would also suggest looking into the concept of Hermitian matrices in quantum mechanics, as they play a significant role in the eigenvalue and eigenvector calculations.

I hope this helps clarify the presentation of the eigenstates in this problem. Keep up the good work in your studies of quantum mechanics!