# Elementary question:[A,B] = [A-<A>, B-<B>]

In summary, the conversation discussed the steps needed to show that [A,B] = [A-<A>, B-<B>], with <A> being a real or complex number and <A> interpreted as <A>I where I is the identity operator. The forum's policy on homework and textbook-style questions prevented a complete answer, but it was suggested to use the formulas [A+B,C]=[A,C]+[B,C] and [A,B+C]=[A,B]+[A,C] to deal with [A+B,C+D]. It was also mentioned that operators that commute with everything can be used to further simplify the solution. The conversation ended with gratitude for the knowledgeable individuals on the forum.
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I know this should be obvious, and I should be ashamed of asking it, but could someone fill in the steps to show that [A,B] = [A-<A>, B-<B>]? Thanks from a non-physicist.

<A> is a real number (or if A isn't self-adjoint, a complex number), so A-<A> must be interpreted as A-<A>I, where I is the identity operator, which commutes with everything.

The forum's policy on homework and textbook-style questions prevents me from giving you the complete answer, but I think you will find it easy to show that [A+B,C]=[A,C]+[B,C], and [A,B+C]=[A,B]+[A,C] for all A,B,C. Then you can use these formulas to deal with [A+B,C+D]. When you have done that, let B and D be operators that commute with everything ([B,X]=[D,X] for all X) and see what you get.

Edit: Hehe. I cleverly made sure my reply would be the first by posting the first paragraph as soon as I was done with it and then adding the second one in an edit.

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Well, A-<A> means A-<A>1 , where the '1' is the unit operator on the vector space these operators act on. The unit operator commutes with every other operator, including itself, that's why you have the equality of the 2 commutators.

Thanks, first-past-the-post Fredrik, and also to bigubau. That should do it.

Fredrik: Just as a side note, this was not a homework question: I wish it were, because I miss academia. But you're right, it's textbook style. Alas, I am limited in the number of textbooks I have as reference. Hence my double gratitude for this Forum and knowledgeable people like you on it.

## 2. What does the notation and represent in "Elementary question:[A,B] = [A-, B-]"?

The notation and represents placeholders for values or expressions that will be substituted into the equation. These values or expressions can represent any numerical or mathematical quantity.

## 3. How is "Elementary question:[A,B] = [A-, B-]" solved?

The equation is solved by substituting the values of and into their respective places in the equation. This could involve simplifying expressions or performing mathematical operations to find the numerical values of and .

## 4. What is the purpose of using "Elementary question:[A,B] = [A-, B-]" in mathematics?

The purpose of using this notation is to represent a basic question or problem involving two elements in a concise and standardized way. It allows for a clear and organized presentation of the problem, making it easier to solve and understand.

## 5. Can "Elementary question:[A,B] = [A-, B-]" be used for more than two elements?

Yes, this notation can be expanded to represent problems or questions involving more than two elements. For example, "Elementary question:[A,B,C] = [A-, B-, C-]" could represent a problem involving three elements, and so on.

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