# Do L² and r² Commute in Quantum Mechanics?

• dreamspy
In summary, the conversation is discussing whether [L^2,r^2] = 0 is true and what variables L^2 depends on. The conversation also mentions the commutation relationships between different operators, including [x_i,p_j] = i\hbar\delta_{i,j}. One person suggests using spherical coordinates for further clarification.
dreamspy
Hi

The subject says it all. I'm wondering if $$[L^2,r^2] = 0$$ is true?

regards
Frímannn

What do you think ? On what variables does $L^2$ depend ?

bigubau said:
What do you think ? On what variables does $L^2$ depend ?

Well we have that:

$$\underline{\hat L }^2 = \hat L_1^2+\hat L_2^2+\hat L_3^2$$

$$\hat L_1 = \hat x_2 \hat p_3 - \hat x_3 \hat p_2$$

$$\hat L_2 = \hat x_3 \hat p_3 - \hat x_1 \hat p_3$$

$$\hat L_3 = \hat x_1 \hat p_2 - \hat x_2 \hat p_1$$

$$\underline{r }^2 = \hat x_1^2+\hat x_2^2+\hat x_3^2$$

and

$$[x_i,p_j] = i\hbar\delta_{i,j}$$

$$[L_i,p_j] = 0, i = j$$

$$[L_i,p_j] \ne 0, i \ne j$$

$$[L_i,x_j] = 0, i = j$$

$$[L_i,x_j] \ne 0, i \ne j$$

So it seems to me that they don't commute. But I'we been told otherwize so I'm trying to figure it out :)

Last edited:
Are you sure about that? I get $$[x_1,L_2] = i\hbar x_3$$

Sorry that was a mistake. That should have been a $$\ne$$. I fixed the original post.

I'll give you a further hint: use spherical coordinates.

## 1. What does it mean for two operators to commute?

Two operators commute if their order of operations does not affect the final result. In other words, if the operators can be applied in any order and still give the same result, then they commute.

## 2. What is L^2 and r^2 in this context?

L^2 and r^2 are operators commonly used in quantum mechanics. L^2 represents the squared angular momentum operator, while r^2 represents the squared radial distance operator.

## 3. Do L^2 and r^2 commute in all cases?

No, L^2 and r^2 do not always commute. They only commute in certain cases, such as when the potential energy is independent of the angular momentum.

## 4. What is the significance of L^2 and r^2 commuting?

If L^2 and r^2 commute, it means that they have a set of common eigenvectors. This allows for simpler calculations and makes it easier to determine the energy levels of a quantum system.

## 5. How can we determine if L^2 and r^2 commute in a specific case?

To determine if L^2 and r^2 commute in a specific case, we can use the commutator of the two operators. If the commutator is equal to zero, then the operators commute. If the commutator is non-zero, then the operators do not commute.

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