Baker–Campbell–Hausdorff equation

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In summary, the conversation is about the Baker-Campbell-Hausdorff identity in quantum mechanics and whether it still holds when trying to write it in multiplication form. The person is unsure if the identity is still valid after adding and subtracting a commutator term. They are also seeking further knowledge about the commutator [X,Y] and whether it is equivalent to another operator or a number.
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Franky4
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I have a question about Baker–Campbell–Hausdorff identity in QM: exp(X)*exp(Y) = exp(X + Y + [X,Y]/2)
I have exp(X + Y) where X and Y don't commute and want to try to write in multiplication form. Tried adding and subtracting commutator term: exp(X + Y + [X,Y]/2 - [X,Y]/2), but now I am not sure whether identity still holds: exp(X + Y + [X,Y]/2 - [X,Y]/2) =? exp(X + Y)*exp(- [X,Y]/2) .
 
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Do you have further knowledge about [X,Y], i.e. whether it equals to another operator or just a number instead?
 

What is the Baker–Campbell–Hausdorff equation?

The Baker–Campbell–Hausdorff (BCH) equation is a mathematical formula used in the field of Lie algebras to find the commutator of two exponentials. It is used to simplify the calculation of certain operations involving these exponentials, such as matrix exponentials and differential equations.

Who discovered the Baker–Campbell–Hausdorff equation?

The Baker–Campbell–Hausdorff equation was first discovered by British mathematician Ernest William Hobson in 1891. However, it was later independently rediscovered and popularized by mathematicians Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff, after whom it is named.

What is the significance of the Baker–Campbell–Hausdorff equation in physics?

The Baker–Campbell–Hausdorff equation has various applications in physics, particularly in quantum mechanics and statistical mechanics. It is used to calculate the time evolution of quantum mechanical systems and to derive equations of motion in statistical mechanics.

What are some other names for the Baker–Campbell–Hausdorff equation?

The Baker–Campbell–Hausdorff equation is also known as the Campbell–Baker–Hausdorff formula, the Campbell–Hausdorff theorem, and the BCH formula. It is also sometimes referred to as the Lie theorem or the Lie product formula.

What are some practical applications of the Baker–Campbell–Hausdorff equation?

The Baker–Campbell–Hausdorff equation has a wide range of applications in areas such as quantum mechanics, statistical mechanics, and differential geometry. It is used in the calculation of quantum mechanical systems, the study of symmetries in physics, and the analysis of nonlinear dynamical systems. It also has applications in engineering, chemistry, and other fields.

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