# Express Eigenvectors in terms of Eigenkets.

• jhosamelly
In summary, the problem at hand involves finding the eigenvectors of the operator J_y and expressing them in terms of the eigenkets of J^2 and J_z. This can be done by solving the corresponding eigenvalue equations and using linear combinations of the 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 $J_z$ .

## 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?

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!

## 1. What are eigenvectors and eigenkets?

Eigenvectors and eigenkets are both mathematical concepts used in linear algebra to represent the behavior of a linear transformation on a vector space. An eigenvector is a vector that, when multiplied by a transformation, remains in the same direction but may be scaled by a scalar factor. An eigenket is the quantum mechanical counterpart of an eigenvector, representing the state of a quantum system in a particular basis.

## 2. How are eigenvectors and eigenkets related?

Eigenvectors and eigenkets are essentially the same concept, just in different mathematical contexts. They represent the same idea of a vector or state that is unchanged by a transformation or measurement, respectively.

## 3. Why is it useful to express eigenvectors in terms of eigenkets?

Expressing eigenvectors in terms of eigenkets allows for a more general and abstract representation of the concept. It also allows for the application of quantum mechanical principles to linear algebra, making it applicable in a wider range of contexts.

## 4. How do you express eigenvectors in terms of eigenkets?

To express an eigenvector in terms of eigenkets, we use the spectral theorem, which states that any linear transformation can be represented by a set of eigenvectors and eigenvalues. We can then write the eigenvector as a linear combination of eigenkets, with the eigenvalues as coefficients.

## 5. Can you provide an example of expressing eigenvectors in terms of eigenkets?

Sure, let's say we have a linear transformation represented by the matrix A = [1 0; 0 2]. The eigenvalues of this matrix are 1 and 2, and the corresponding eigenvectors are [1 0] and [0 1]. We can express these eigenvectors in terms of eigenkets as |1> and |2>, respectively. So, the eigenvector [1 0] can be written as |1> = |1> and the eigenvector [0 1] can be written as |2> = |2>. This allows us to easily apply quantum mechanical principles to this linear transformation.

Replies
9
Views
2K
Replies
9
Views
2K
Replies
1
Views
1K
Replies
18
Views
2K
Replies
1
Views
1K
Replies
1
Views
1K
Replies
5
Views
2K
Replies
1
Views
2K
Replies
2
Views
1K
Replies
20
Views
2K