Commutators of angular momentum

In summary, the three components of angular momentum (L_x, L_y, and L_z) commute with nabla^2 and r^2 = x^2 + y^2 = z^2. This can be shown by using the commutator equation [A, B] = AB - BA and rewriting nabla^2 as a differential operator. When applied to a differentiable function f(x,y,z), the differentiation rules can simplify the equation.
  • #1
SunGod87
30
0

Homework Statement


Show the three components of angular momentum: L_x, L_y and L_z commute with nabla^2 and r^2 = x^2 + y^2 = z^2

Homework Equations


[A, B] = AB - BA
For example:
[itex]
[L_x, \nabla^2] = L_x \nabla^2 - \nabla^2 L_x
[/itex]

The Attempt at a Solution


[itex]
L_x \nabla^2 = -i\hbar(y\frac{\partial}{\partial z} \nabla^2 - z \frac{\partial}{\partial y}\nabla^2)
[/itex]

[itex]
\nabla^2 L_x = -i\hbar\nabla^2(y\frac{\partial}{\partial z} - z \frac{\partial}{\partial y})
[/itex]

How can I simplify these?
 
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  • #2


If you have no better ideas, then why not rewrite [itex]\nabla^2[/itex], just like you did with [itex]L_x[/itex]?
 
  • #3


after rewriting [itex]\nabla^2[/itex] as Hurkyl suggested, you should apply the commutator (it is also an operator) to a differentiable function [itex]f(x,y,z)[/itex] and see whether the differentiation rules (e.g.differentiation of a product) make the equation look simplier.
 

Related to Commutators of angular momentum

What is a commutator of angular momentum?

A commutator of angular momentum is a mathematical operation that describes the relationship between two different components of angular momentum. It is used to determine the uncertainty in the measurement of angular momentum in quantum mechanics.

What is the significance of commutators of angular momentum in quantum mechanics?

Commutators of angular momentum play a crucial role in quantum mechanics as they help us understand the fundamental properties of particles, such as spin and orbital angular momentum. They also help us determine the possible states and energy levels of a system.

How do you calculate the commutator of angular momentum?

The commutator of angular momentum can be calculated by taking the difference between the product of the two angular momentum operators and the product of their corresponding Hermitian conjugates. This can be written as [A, B] = AB - BA.

What is the physical interpretation of a non-zero commutator of angular momentum?

A non-zero commutator of angular momentum indicates that the two components of angular momentum do not commute, meaning they cannot be measured simultaneously with complete precision. This is known as the Heisenberg uncertainty principle.

How do commutators of angular momentum affect the behavior of particles?

The commutator of angular momentum affects the behavior of particles by determining the possible values of their angular momentum and the probabilities of measuring these values. It also plays a role in the conservation of angular momentum, as any changes in one component will affect the other component through the commutator relationship.

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