Why eigenvalues of L_x^2 and L_z^2 identical?

[fandango92]

Homework Statement

Calculate the eigenvalues of the $L_x^2$ matrix.
Calculate the eigenvalues of the $L_z^2$ matrix.
Compare these and comment on the result.

Homework Equations

$L_x=\frac{1}{2}(L_+ + L_- )$

The Attempt at a Solution

I have derived eigenvalues for each: $0$ and $\hbar^2$ for both $L_x^2$ and $L_z^2$. But why are they identical? I'm finding it difficult in qualitatively explaining why the eigenvalues are the same for both.

 

[Oxvillian]

Because the God of Physics does not care which direction you call the $x$-direction and which direction you call the $z$-direction.

 

[fandango92]

Sorry I forgot to mention this is for $l=1$.

Okay, but I used $L_z$ eigenvalues of $m\hbar$, where $m=-1,0,1$ in this case, and used $L_x=\frac{1}{2}(L_+ + L_- )$. I have called the z component the one in which is certain, so how can the x component squared in this case have the same eigenvalues as the z component squared?

 

[vela]

The operators will have the same eigenvalues for the reason Oxvillian said, but that's not saying anything about the state a particle is in. If a particle is in an eigenstate of $\hat{L}_z$, it's not in an eigenstate of $\hat{L}_x$.

 

[paranormal]

As a scientist, it is important to understand the underlying principles and mathematical concepts behind any results we obtain. In this case, the similarity in eigenvalues for L_x^2 and L_z^2 can be explained by the commutation relationship between the two operators.

In quantum mechanics, operators representing physical observables, such as the angular momentum operators L_x and L_z, do not necessarily commute with each other. This means that their order of operations matters and they may not yield the same result if applied in different orders.

However, the operators L_x^2 and L_z^2 do commute with each other, meaning that their order of operations does not affect the outcome. This is due to the fact that L_x and L_z are both components of the total angular momentum operator L, and L_x^2 and L_z^2 can be thought of as measuring the square of the total angular momentum along the x and z axes, respectively.

Since L_x and L_z represent different components of the same physical quantity, it makes sense that their squares would have the same eigenvalues. This is similar to how the x and y components of a vector have the same magnitude, but different directions.

In summary, the identical eigenvalues of L_x^2 and L_z^2 can be attributed to the commutation relationship between the operators and the underlying physical concept of total angular momentum.