# Product of exponential of operators

• cpsinkule
In summary, the "Product of exponential of operators" is a mathematical concept where two or more operators are multiplied together after having an exponential function applied to them. This allows for the calculation of complex functions and is commonly used in science, specifically in quantum mechanics. The product can be simplified in some cases, but it can only be applied to linear operators and may not always result in a well-defined operator, limiting its usefulness in certain situations.
cpsinkule
How does one show that
eAeB=eA+Be[A,B]/2
where A,B are operators and [ , ] is the commutator. The QM book I am using states it as a fact without proof, but I would like to see how it is proved. I've muddled around with the series expansion, but can't get farther than a few term by term products which lead nowhere.

## 1. What is the "Product of exponential of operators"?

The "Product of exponential of operators" is a mathematical concept in which two or more operators are multiplied together, with each operator having an exponential function applied to it. This results in a new operator that can be used to perform operations on a given function.

## 2. How is the product of exponential of operators calculated?

The product of exponential of operators is calculated by first applying the exponential function to each individual operator, and then multiplying the resulting operators together. The order in which the operators are multiplied can affect the final result.

## 3. What is the significance of the product of exponential of operators in science?

The product of exponential of operators is commonly used in quantum mechanics and other branches of science to represent the evolution of a system over time. It allows for the calculation of complex wave functions and the prediction of particle behavior.

## 4. Can the product of exponential of operators be simplified?

Yes, the product of exponential of operators can sometimes be simplified by using mathematical properties and identities. However, this is not always possible, and in many cases, the product must be calculated in its original form.

## 5. Are there any limitations to using the product of exponential of operators?

One limitation of using the product of exponential of operators is that it can only be applied to linear operators. Additionally, the product may not always result in a well-defined operator, which can limit its usefulness in certain applications.

Replies
3
Views
1K
Replies
0
Views
216
Replies
8
Views
2K
Replies
3
Views
2K
Replies
2
Views
697
Replies
1
Views
857
Replies
14
Views
2K
Replies
1
Views
1K
Replies
3
Views
962
Replies
2
Views
1K