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Gideon
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i searched the forum, but nothing came up. My question, how do you prove that [A,B] = iC if A and B are hermitian operators? I understand how C is hermitian as well, but i can't figure out how to prove the equation.
Gideon said:i searched the forum, but nothing came up. My question, how do you prove that [A,B] = iC if A and B are hermitian operators? I understand how C is hermitian as well, but i can't figure out how to prove the equation.
dextercioby said:Here's how i'd do it
Let's consider the densely defined linear operator
[tex] \hat{O}=:\left[\hat{A},\hat{B}\right]_{-}=:\hat{A}\hat{B}-\hat{B}\hat{A} [/tex] (1)
,where [itex] \hat{A},\hat{B} [/itex] are densly defined linear operators on the Hilbert space [itex] \mathcal{H} [/itex].
The domain of this operator is
[tex]\mathcal{D}_{\hat{O}}=\mathcal{D}_{\hat{A}}\cap\mathcal{D}_{\hat{B}} [/tex]
...
To be continued.
Daniel.
HackaB said:WTF are you talking about? It appears that you have been reading too many math books, and that it has affected your brain. Or perhaps you are just trying to show off.
HackaB said:WTF are you talking about? It appears that you have been reading too many math books, and that it has affected your brain. Or perhaps you are just trying to show off.
HackaB said:OK, call it whatever you want. While you're here, how does that "proof that group multiplication is associative" go again?
The commutator of hermitian operators is a mathematical operation that measures how two operators, which represent physical quantities in quantum mechanics, interact with each other. It is defined as the difference between the product of the two operators and the product of the same two operators in the opposite order. In other words, it measures the non-commutativity of two operators.
The commutator of hermitian operators is important because it is directly related to the uncertainty principle in quantum mechanics. The uncertainty principle states that certain physical quantities, such as position and momentum, cannot be known simultaneously with absolute precision. The commutator of hermitian operators plays a crucial role in understanding and quantifying this uncertainty.
The commutator of hermitian operators is calculated by taking the product of the two operators and subtracting the product of the same two operators in the opposite order. This can be represented mathematically as [A,B] = AB - BA, where A and B are the two hermitian operators.
The commutator of hermitian operators has physical significance in quantum mechanics because it determines how two operators interact with each other. When the commutator of two operators is non-zero, it means that the two operators do not commute, which leads to an uncertainty in the measurement of these physical quantities. In contrast, when the commutator of two operators is zero, it means that the two operators commute, and the corresponding physical quantities can be known with certainty.
Yes, the commutator of hermitian operators can be used to determine the eigenvalues of an operator. If two operators, A and B, commute with each other, it means that they share a common set of eigenvectors. This allows us to simultaneously measure the two operators and determine their eigenvalues. However, if the commutator is non-zero, the operators do not share a common set of eigenvectors, and we cannot simultaneously measure the two operators with certainty.