Robertson uncertainty relation for the angular momentum components

In summary, the conversation discusses the Robertson uncertainty relation for the components of orbital angular momentum in the quantum domain. It is important to consider the triviality problem, where a commutator of zero leads to a product of standard deviations of zero, meaning there is no information about one of the observables. However, there is no angle operator that satisfies the necessary relation for applying the Robertson uncertainty relation in this context. The speaker is seeking clarification or an example on this matter.
  • #1
Yan Campo
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TL;DR Summary
I would like any explanation about Robertson the uncertainty relation for the angular momentum components and compatibility between the components
I'm studying orbital angular momentum in the quantum domain, and I've come up with the Robertson uncertainty relation for the components of orbital angular momentum. Therefore, I read that it is necessary to pay attention to the triviality problem, because in the case where the commutator is zero, the product of the standard deviations is zero, so the variance is also zero. This means that we don't have information about one of the observables and, therefore, we don't know the incompatibility between the two, I think. But, I can't see any kind of problem in using the Robertson uncertainty relation in the orbital angular momentum components. Can anyone explain to me, or give me an example about this? I really want to understand.
 
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  • #2
I am afraid there is no angle operator such that
[tex][\hat{\theta},\hat{L}]=i\hbar[/tex]
to which we apply Roberson uncertainty relation.
 
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Likes vanhees71
  • #3
Yan, the Robertson uncertainty principle is regarding two operator have a common complete set of eigenfunctions, i.e., in such basis both operators are diagonal. This is usually expressed, for example, as

$$\Delta A\Delta B \geq \frac{1}{2}\left | \int \psi^{*}[A,B]\psi d\tau\right |$$

But, in the case of angular momentum components, it does not mean that some of the eigenfunctions of ##L_{z}## cannot also be simultaneous eigenfunctions of ##L_{x}## and ##L_{y}##. See the case of ##Y_{0}^{0}(\theta,\phi)## spherical harmonic. In such case, it is allowed to have ##\Delta L_{x} = 0##, ##\Delta L_{y} = 0## and ##\Delta L_{z} = 0##.
 

FAQ: Robertson uncertainty relation for the angular momentum components

What is the Robertson uncertainty relation for the angular momentum components?

The Robertson uncertainty relation is a generalized form of the Heisenberg uncertainty principle. For angular momentum components, it states that the product of the uncertainties in two non-commuting components of angular momentum (e.g., Lx and Ly) is bounded by the expectation value of their commutator. Mathematically, it is expressed as ΔLx * ΔLy ≥ (1/2) ||, where ΔLx and ΔLy are the standard deviations of the angular momentum components Lx and Ly, respectively, and is the expectation value of the third component.

How does the Robertson uncertainty relation differ from the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle typically refers to the uncertainty relation between position and momentum. The Robertson uncertainty relation generalizes this concept to any pair of non-commuting observables, including the components of angular momentum. While Heisenberg's principle is usually stated in terms of standard deviations, Robertson's relation incorporates the expectation value of the commutator of the two observables, providing a more general and rigorous framework.

Why is the Robertson uncertainty relation important for angular momentum components?

The Robertson uncertainty relation is crucial for understanding the intrinsic limitations in measuring angular momentum components simultaneously. Since angular momentum components do not commute, their uncertainties are interrelated. This relation helps in quantifying these uncertainties and is fundamental in quantum mechanics, affecting how we interpret measurements and the behavior of quantum systems involving angular momentum, such as electrons in atoms.

Can the Robertson uncertainty relation be violated?

No, the Robertson uncertainty relation cannot be violated. It is a fundamental principle derived from the mathematical structure of quantum mechanics. Any apparent violation would indicate a misunderstanding or miscalculation in the application of the theory. It is a cornerstone of quantum mechanics that ensures the consistency and coherence of the theory.

How is the commutator of angular momentum components related to the Robertson uncertainty relation?

The commutator of two angular momentum components, such as [Lx, Ly], is directly related to the third component, Lz, and is given by [Lx, Ly] = iħLz. This commutation relation is key to deriving the Robertson uncertainty relation for angular momentum components. The expectation value of this commutator appears in the inequality that defines the uncertainty relation, linking the uncertainties of Lx and Ly to the expectation value of Lz, thereby providing a quantitative measure of the intrinsic quantum uncertainties.

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