Commutator Properties: [A,B]C+B[A,C]=[A,B](C+B)?

In summary, the commutator property states that [A,BC] = [A,B]C+B[A,C]. If B=C, then the equation becomes [A,B]C+B[A,C]=[A,B]B+B[A,B]. However, because the operators do not commute, you cannot switch orders.
  • #1
Kyle Nemeth
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Given the property,

[A,BC] = [A,B]C+B[A,C],

is it true that, if B=C, then

[A,B]C+B[A,C]=[A,B]C+B[A,B]=[A,B](C+B)?

I apologize if I have posted in the wrong forum.
 
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  • #2
Kyle Nemeth said:
Given the property,

[A,BC] = [A,B]C+B[A,C],

is it true that, if B=C, then

[A,B]C+B[A,C]=[A,B]C+B[A,B]=[A,B](C+B)?

I apologize if I have posted in the wrong forum.
The commutator (in the assumed context above) is defined as ##[A,B]=AB-BA##.
Now you have ##[A,B]C+B[A,C]=[A,B]B+B[A,B]=AB^2-B^2A## on the left and a factor ##2## on the right.
The (presumably) operators (or linear mappings) ##A,B,C## do not commutate, so you cannot switch orders.
 
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  • #3
Thank you for your help. I understand now.
 

FAQ: Commutator Properties: [A,B]C+B[A,C]=[A,B](C+B)?

1. What is a commutator in mathematics?

A commutator in mathematics is an operation that measures how much two operations do not commute with each other. In other words, it is a measure of how much the order in which two operations are performed affects the outcome.

2. How is the commutator represented mathematically?

The commutator of two operators A and B is represented as [A,B]. It is calculated by taking the product of A and B and subtracting the product of B and A, [A,B]=AB-BA.

3. What are the properties of the commutator?

One of the main properties of the commutator is that it is anti-commutative, meaning that [A,B]=-[B,A]. It also follows the Jacobi identity, [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0. Additionally, the commutator obeys the distributive property, [A,B+C]=[A,B]+[A,C].

4. How does the commutator property [A,B]C+B[A,C]=[A,B](C+B) relate to the distributive property?

The commutator property [A,B]C+B[A,C]=[A,B](C+B) is a special case of the distributive property where C is a constant. It states that a commutator operation applied to a constant is equal to the constant multiplied by the commutator of the operators.

5. What are some real-world applications of commutator properties?

Commutator properties have various applications in fields such as quantum mechanics, computer science, and engineering. In quantum mechanics, they are used to describe the behavior of particles and operators. In computer science, they are used in algorithms for efficient matrix computations. In engineering, they are used in control systems to analyze the response of systems to different inputs.

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