Eigenkets/addiction of angular momentum

[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>

J_1z|j₁,j₂,m₁,m₂> = m₁|j₁,j₂,m₁,m₂>

J_2z|j₁,j₂,m₁,m₂> = m₂|j₁,j₂,m₁,m₂>

J₁=J_1x+J_1y+J_1z

J₂=J_2x+J_2y+J_2z

J_x=J_1x+J_2x

J_y=J_1y+J_2y

J_z=J_1z+J_2z

 

[G01]

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 $J^2$.