Commutator Relation: What is [\hat{A}\hat{B}, \hat{C}\hat{D}] Equal to?

  • Context: Graduate 
  • Thread starter Thread starter M. next
  • Start date Start date
  • Tags Tags
    Commutator Relation
Click For Summary

Discussion Overview

The discussion centers around the commutator relation [\hat{A}\hat{B}, \hat{C}\hat{D}] in quantum mechanics, specifically focusing on how to distribute and simplify the expression. The scope includes theoretical exploration and mathematical reasoning related to commutators.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant asks for the value of the commutator [\hat{A}\hat{B}, \hat{C}\hat{D}] and how to distribute the terms inside.
  • Another participant proposes that the expression can be simplified to (ab)(cd) - (cd)(ab).
  • Several participants reference the rule [A,BC]=B[A,C]+[A,C]B, suggesting it as a method to approach the problem.
  • There is a clarification regarding the correct formulation of the commutator rule, with one participant confirming the expression [A,BC]=[A,B]C+B[A,C].

Areas of Agreement / Disagreement

Participants appear to agree on the use of the commutator rule, but there is some confusion regarding its correct application and formulation. The discussion remains unresolved regarding the final expression for the commutator.

Contextual Notes

There are potential limitations in the assumptions made about the operators involved and the specific conditions under which the commutator is evaluated. The discussion does not clarify these aspects.

M. next
Messages
380
Reaction score
0
What is the commutator [[itex]\hat{A}[/itex][itex]\hat{B}[/itex], [itex]\hat{C}[/itex][itex]\hat{D}[/itex]] equal to? How to distribute what's inside?
 
Physics news on Phys.org
=(ab)(cd)-(cd)(ab)
 
Use the rule [A,BC]=B[A,C]+[A,C]B. It's very easy to prove, so I suggest that you do that as an exercise.
 
Fredrik said:
Use the rule [A,BC]=B[A,C]+[A,C]B. It's very easy to prove, so I suggest that you do that as an exercise.

Do you mean [A,BC]=[A,B]C+B[A,C]?
 
DocZaius said:
Do you mean [A,BC]=[A,B]C+B[A,C]?
Ah, yes of course. Thanks.
 

Similar threads

Graduate Problem while simulating spin polarized interacting SSH model
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Can anyone give me the details of creating a cat state in Circuit QED?
  • · Replies 16 ·
Replies
16
Views
4K
Undergrad Use of the Beam Splitter Operator
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad What does ## \psi^{*}\hat{H}\psi ## mean?
  • · Replies 9 ·
Replies
9
Views
4K
Graduate Maxwell field commutation relations
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Dot product of two vector operators in unusual coordinates
  • · Replies 14 ·
Replies
14
Views
3K
Graduate What is the cause of the Mexican hat potential of the Higgs field?
  • · Replies 30 ·
2
Replies
30
Views
4K
Graduate Proof for commutator $[\hat{H},\hat{a}] = - \hbar \omega \hat{a}
  • · Replies 8 ·
Replies
8
Views
4K
Undergrad Polarization and the beam splitter
  • · Replies 7 ·
Replies
7
Views
2K
Graduate What happens when you commute Sx and Sz in spin operators?
  • · Replies 2 ·
Replies
2
Views
2K