Proof of exp(a)exp(b) = exp(a+b)exp(1/2 [a,b])

[asantas93]

Homework Statement

a, b are operators that commute with their commutator

(1) Show that f(x) = e^xae^xb satisfies df/dx = (a + b + x[a,b])f

(2) use (1) to show that e^ae^b = e^a+be^(1/2)[a,b]

Homework Equations

[a,[b,a]] = [b,[b,a]] = 0

The Attempt at a Solution

I tried a Taylor expansion but just got a mess.

 

[Oxvillian]

Did you get part (1)?

If so, the strategy is to slip an x into the exponent of the object in (2) and then differentiate it with respect to x. You should find that the object in (2) satisfies the same differential equation, with the same boundary condition, as the object in (1). It's therefore the same object. Then at the end you just set x=1 to get the answer.

Btw this is an example of a "BCH" relation (Baker-Campbell-Hausdorff) and if you search around for that you should find loads of stuff.

 

[dextercioby]

This is not an exercise about Taylor series, it's simply a neat trick to get a useful relation between exponentials of particular bounded operators.

 

[asantas93]

Thanks, knowing the name of this sort of equation helped me find a proof.

 

[paranormal]

I would approach this problem by first understanding the definitions and properties of the operators a and b. I would also look at the properties of the exponential operator, exp(x), which is defined as the infinite sum of x^n/n! and has the property that d/dx(exp(x)) = exp(x).

To solve part (1), I would start with the definition of f(x) = exp(xa)exp(xb) and use the product rule to calculate df/dx. Using the definition of the exponential operator and the commutativity of a and b, I would then manipulate the expression to get (a + b + x[a,b])f. This shows that f(x) satisfies the desired differential equation.

For part (2), I would use the result from part (1) to write eaeb = exp(a)exp(b) = exp(a+b)(1/2)[a,b]. This can be simplified using the definition of the exponential operator and the commutativity of a and b to get eaeb = ea+be(1/2)[a,b]. This proves the desired result.

In conclusion, the proof relies on understanding the definitions and properties of the operators and using them to manipulate the expressions to get the desired results. It is also important to use the properties of the exponential operator to simplify the expressions.