• Saxonphone
In summary, the conversation discusses the proof that |j1,j2,m1,m2> is not an eigenket of J2. This is demonstrated by writing out J^2 explicitly in terms of J1 and J2 and showing that the last term, which involves the raising and lowering operators, does not have eigenkets in the form of |j1,j2,m1,m2>. Therefore, |j1,j2,m1,m2> is not an eigenket of J^2.
Saxonphone

## Homework Statement

Say we are adding angular momentum: J = J1+J2.
Now, how can i prove that J2 |j1,j2,m1,m2>$$\neq$$ $$\lambda$$ |j1j,2,m1,m2>
a.k.a. |j1j2,m1,m2> is not a eigenket of J2?

## Homework Equations

J12|j1,j2,m1,m2>=j1|j1j,2,m1,m2>

J22|j1,j2,m1,m2>=j2|j1j,2,m1,m2>

J1z|j1,j2,m1,m2> = m1|j1,j2,m1,m2>

J2z|j1,j2,m1,m2> = m2|j1,j2,m1,m2>

J1=J1x+J1y+J1z

J2=J2x+J2y+J2z

Jx=J1x+J2x

Jy=J1y+J2y

Jz=J1z+J2z

Consider writing out $J^2$ explicitly in terms of J1 and J2:

$$J^2=(\vec{J_1}+\vec{J_2})^2=J_1^2+J_2^2+2\vec{J_1}\cdot\vec{J_2}$$

Obviously the first two terms have eigenkets of the form $|j_1,j_2,m_1,m_2>$, but what about that last term?

In fact that last term in $J^2$ can be written in terms of the z components of the angular momentum and the raising and lowering operators for angular momenta. And, $|j_1,j_2,m_1,m_2>$ are not eigenkets of the raising and lowering operators.

Thus, $|j_1,j_2,m_1,m_2>$, are not eigenkets of $J^2$.

Last edited:

## 1. What is an eigenket in terms of angular momentum?

An eigenket in terms of angular momentum is a state vector that represents the possible values of angular momentum that a system can have. It is a quantum mechanical concept that is used to describe the behavior of particles in a system.

## 2. How is the concept of eigenkets related to addiction of angular momentum?

The concept of eigenkets is closely related to the idea of conservation of angular momentum. Angular momentum is a conserved quantity, meaning that it cannot be created or destroyed. As a result, when a system undergoes a change, its eigenkets will also change in a predictable way to maintain the conservation of angular momentum.

## 3. Can eigenkets be observed in a physical system?

No, eigenkets themselves cannot be observed directly. They are a mathematical representation of the possible states of a system. However, the effects of eigenkets can be observed in experiments through the measurement of angular momentum values.

## 4. How do eigenkets affect the behavior of particles in a system?

Eigenkets play a crucial role in determining the behavior of particles in a system with respect to angular momentum. They determine the possible values of angular momentum that a particle can have and how these values will change when the system undergoes a change.

## 5. Can the concept of eigenkets be applied to other physical quantities?

Yes, the concept of eigenkets can be applied to other physical quantities that are conserved, such as energy and momentum. In these cases, the eigenkets represent the possible states of the system with respect to these quantities and how they will change under different conditions.

Replies
2
Views
3K
Replies
5
Views
3K
Replies
1
Views
2K
Replies
7
Views
3K
• Atomic and Condensed Matter
Replies
1
Views
2K
Replies
1
Views
2K
Replies
7
Views
2K
Replies
4
Views
3K