What Are the Properties of the Commutator in the Dilation Operator?

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SUMMARY

The discussion focuses on the properties of the commutator in the context of the dilation operator defined as D = \vec{r} * \vec{p}. Participants compute the commutators [D, \vec{r}] and [D, \vec{p}], specifically examining [D, x] = [x, x p_x]. The key equation used is p = -i * ħ, which is fundamental in quantum mechanics. The conversation emphasizes the need to reference textbooks for understanding commutators and their properties.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically operators and commutators.
  • Familiarity with the dilation operator and its mathematical representation.
  • Knowledge of the momentum operator defined as p = -i * ħ.
  • Basic skills in manipulating algebraic expressions involving operators.
NEXT STEPS
  • Study the properties of commutators in quantum mechanics.
  • Learn about the mathematical formulation of the dilation operator.
  • Explore examples of operator algebra in quantum mechanics.
  • Review textbooks on quantum mechanics for deeper insights into operator theory.
USEFUL FOR

Students of quantum mechanics, physicists working with operator theory, and anyone interested in the mathematical foundations of quantum operators.

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Homework Statement



Concicer the dilation operator

[itex]D = \vec{r} * \vec{p}[/itex]

Compute [itex][D,\vec{r} ] and [D, \vec{p}][/itex]

Homework Equations



[itex]p = - i * hbar[/itex]


The Attempt at a Solution


I think the question is really if [itex][D, \vec{r}][/itex] commutes

I got this:
[itex]D = \vec{r} * \vec{p}[/itex]
[itex]D = x p_x + y p_y + z p_z[/itex]


for the x direction:

[itex][D, r] = [D, x] = [x, x p_x][/itex]

where do i take it from here?
 
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You have absolutely no idea ?? Pick up your book and find those brackets (called <commutator>) and the properties they have.
 

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