Express Eigenvectors in terms of Eigenkets.

[jhosamelly]

Homework Statement

One of the problems in our test is this.

Express the eigenvectors of $J_y$ in terms of the eigenkets of $J^2$ and $J_z$ .

Homework Equations

The Attempt at a Solution

I know the matrix of $J_y$ and the operators or eigenkets for $J_y$ ,$J^2$ and $J_z$. I just don't seem to understand the question. Can someone please explain it to me please? What should I do?

 

[mark726]

Hello there,

Firstly, let's define what eigenvectors and eigenkets are. Eigenvectors are special vectors that do not change direction when a linear transformation is applied to them. In quantum mechanics, these eigenvectors are represented by eigenkets. They are the "building blocks" of quantum states and can be used to describe the state of a quantum system.

Now, in terms of the problem at hand, we are looking at the eigenvectors of the operator J_y. This means that when we apply the operator J_y to these eigenvectors, we get a scalar multiple of the original eigenvector. In other words, the eigenvectors of J_y are the vectors that remain unchanged when J_y is applied to them.

To express these eigenvectors in terms of the eigenkets of J^2 and J_z, we need to use the properties of these operators. The operator J^2 represents the total angular momentum squared, while J_z represents the z-component of the angular momentum. Therefore, we can use the eigenkets of J^2 and J_z to construct the eigenvectors of J_y.

To do this, we can start by finding the eigenkets of J^2 and J_z. These can be found by solving the corresponding eigenvalue equations. Once we have the eigenkets, we can use them to construct the eigenvectors of J_y by taking linear combinations of the eigenkets.

I hope this helps to clarify the problem for you. Let me know if you have any further questions or need more assistance. Best of luck with your test!