Commutator Relation: Exploring A, B and \DeltaA, \DeltaB

In summary, the conversation discusses the expressions of \DeltaA and \DeltaB, and how they are related to the operators A and B. It is mentioned that \DeltaA and \DeltaB can be found by subtracting <A> and <B> from A and B respectively. It is also noted that <A> and <B> are scalars, not operators, and therefore commute with each other and all other operators. The conversation concludes with the statement that the answer has been found.
  • #1
Seanskahn
29
0
hi

I found this in textbook:

[A,B] = [[tex]\Delta[/tex]A, [tex]\Delta[/tex]B]

Experimenting witht he expressions of [tex]\Delta[/tex]A and [tex]\Delta[/tex]B, I find
[[tex]\Delta[/tex]A, [tex]\Delta[/tex]B] = [A,B] - [A, <B>] - [<A>, B] + [<A>,<B>]

A, and B are two hermitean operators, and [tex]\Delta[/tex]A = A - <A> etc, so <A> and <B> do not commute in general.

What am I missing?
 
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  • #2
[tex]\langle A\rangle,\langle B\rangle[/tex] are real numbers (times the identity operator). They commute with everything.
 
  • #3
ok, found it

For those who might want to know the answer, <A>, <B> are scalers, not operators, so they commute with each other, and the operators, A and B
 
  • #4
Good. So you found it exactly at the same time when I posted my comment.
 

FAQ: Commutator Relation: Exploring A, B and \DeltaA, \DeltaB

What is a commutator relation?

A commutator relation is an important concept in quantum mechanics that describes the relationship between two physical observables, A and B. It is represented by the expression [A, B] = AB - BA, where the brackets denote the commutator. In simpler terms, it tells us how the measurements of A and B affect each other.

How do A and B affect each other in a commutator relation?

In a commutator relation, the order in which A and B are measured matters. If the commutator [A, B] is zero, then A and B are said to commute, meaning their measurements do not affect each other. On the other hand, if the commutator is non-zero, then A and B do not commute, and their measurements do affect each other.

What is the physical significance of a non-zero commutator?

A non-zero commutator indicates that the observables A and B do not have simultaneous eigenstates, meaning they cannot be measured simultaneously with definite values. This is a fundamental principle in quantum mechanics, known as the uncertainty principle.

What are \DeltaA and \DeltaB in the commutator relation?

\DeltaA and \DeltaB represent the uncertainties or fluctuations in the measurements of A and B, respectively. They are related to the non-commutativity of A and B, and their values can be calculated using the commutator relation.

How is the commutator relation used in quantum mechanics?

The commutator relation is used to understand the behavior of physical observables in quantum systems. It allows us to calculate the uncertainties in measurements and make predictions about the outcomes of measurements. It is also an essential tool in the development of quantum algorithms and quantum technologies.

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