Vector identities in quantum mechanics

In summary, to prove the expression [L^2,[L^2,\hat{r}]] = 2\hbar^2 {L^2,r} in quantum mechanics, we can use commutator identities to simplify the expression and show that it is equal to zero, proving the desired result.
  • #1
RabbiSnail
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The overall problem is to prove that [L^2,[L^2,\hat{r}]]=2\hbar^2 {L^2,r}

I feel I am very close to solving this problem but I need a quantum version of the vector identity ax(bxc). Because the relevant vectors are operators that don't commute, there is a problem.

Does anybody know of a source of vector identities that don't assume commutation?

Thanks
 
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  • #2
for your post! This is a great question and a common problem in quantum mechanics. One way to approach this problem is to use the formalism of quantum commutators.

In quantum mechanics, the commutator of two operators A and B is defined as [A,B] = AB-BA. This is similar to the cross product of two vectors in classical mechanics. Using this, we can rewrite the expression [L^2,[L^2,\hat{r}]] as [L^2, L^2\hat{r} - \hat{r}L^2]. Now, using the commutator identity [AB,C] = A[B,C] + [A,C]B, we can expand this expression to [L^2,L^2]\hat{r} - \hat{r}[L^2,L^2]. Since [L^2,L^2] = 0 (since L^2 is a scalar operator), we are left with -\hat{r}[L^2,L^2].

Next, we can use another commutator identity, [A,BC] = [A,B]C + B[A,C], to expand this further to -\hat{r}(L^2L^2 - L^2L^2) = 0. This means that the commutator [L^2,[L^2,\hat{r}]] is equal to zero, which proves the desired result.

In general, when dealing with non-commuting operators, it is useful to use the commutator identity and the definition of the commutator to simplify expressions. You can find more information about these identities in any standard quantum mechanics textbook or online resource.

I hope this helps and good luck with your research!
 

1. What is a vector identity in quantum mechanics?

A vector identity in quantum mechanics refers to a mathematical equation or relationship that involves vectors, which are mathematical quantities that have both magnitude and direction. In quantum mechanics, these vector identities are used to describe the behavior and properties of particles on a quantum level.

2. Why are vector identities important in quantum mechanics?

Vector identities are important in quantum mechanics because they allow us to describe and understand the behavior of particles on a quantum level. They provide a mathematical framework for analyzing and predicting the behavior of particles in a quantum system, and are essential for solving problems and making predictions in quantum mechanics.

3. How are vector identities used in quantum mechanics?

In quantum mechanics, vector identities are used to describe the properties and behavior of particles, such as their position, momentum, and spin. They are also used in mathematical equations, such as the Schrödinger equation, which describes the time evolution of quantum systems. Vector identities are also used in calculations and theoretical models to make predictions about the behavior of particles in a quantum system.

4. What are some common vector identities used in quantum mechanics?

Some common vector identities used in quantum mechanics include the commutator and anticommutator relationships, which describe the relationship between two operators, as well as the angular momentum and spin operators, which describe the rotational properties of particles. Other important vector identities include the momentum operator and the position operator, which are used to describe the momentum and position of particles in a quantum system.

5. How do vector identities relate to other concepts in quantum mechanics?

Vector identities are closely related to other concepts in quantum mechanics, such as operators, wave functions, and quantum states. They are used to describe the properties and behavior of these concepts, and are essential for understanding the fundamental principles of quantum mechanics. Vector identities also play a crucial role in the development of quantum algorithms and technologies, such as quantum computers and quantum communication systems.

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