Commutator question. [A,B] =.lambda proof

[Jreyes613]

Homework Statement

Hello!
I'm having troubles with this proof.
given two operators A &B , such that [A,B] = λ where λ is complex,and μ is also complex, show that

exp{μ(A+B)} = exp{μA}exp{μB}exp{(-μ^2λ)/2}

Homework Equations

[A,B] = λ.
[A,B] = AB-BA = λ

The Attempt at a Solution

I have tried expanding the right side of the equation into three different exponentials by subbing in λ=AB-BA.
I get exp{μ(2A+2B-μAB+μBA}
but i seem to have a factor of two extra...
I'm not even sure if i am on the right track, so if anyone could give me a starting point it would be greatly appreciated!

 

[MisterX]

It seems like maybe you tried adding the exponents on the right hand side. That's not allowed. If $e^Ae^B$ generally is not equal to $e^{A+B}$. That would contradict what you are trying to show. Anyway, I'm not sure the best way to solve this problem, but what you are trying to show is.


$$\sum_{n=0}^\infty \frac{\mu^nA^n}{n!} \sum_{m=0}^\infty \frac{\mu^mB^m}{m!} \sum_{a=0}^\infty \frac{\left(-1\right)^a\mu^{2a}\lambda^a}{2^a a!}$$
$$=\sum_{b=0}^\infty \frac{\mu^b (A+B)^b}{b!}$$

You could try seeing if you can figure out where at the terms come from in the first expression.
$(A+B)^0= 1$
$(A+B)^1= A + B$
$(A+B)^2= A^2 + 2AB - [A,B]+ B^2 = A^2 + 2AB + B^2 - \lambda$
$(A+B)^3= (A+B)\left( A^2 + 2AB + B^2 - \lambda\right)$

However I'm really not sure exactly how to get this result.

 

[hilbert2]

That's the Baker-Campbell-Hausdorff formula. Search with Google and you will probably find help in the proof.

 

[Jreyes613]

Thank you guys.
Yes , my TA said that you can use the Baker -Campbell-Hausdorff formula.

I will be able to solve it now, thanks!

 

[paranormal]

Hi there,

It looks like you are on the right track! The extra factor of two is likely coming from the expansion of the exponential function itself. Remember that exp(x+y) = exp(x)exp(y), so when you expand out the right side of the equation, you will end up with an extra factor of exp(μ^2λ/2) because you are multiplying three exponential terms together.

To solve this, you can use the fact that exp(x+y) = exp(x)exp(y) and also the commutator relation [A,B] = AB-BA = λ. From there, you can manipulate the terms to get the desired result. I would suggest starting with the right side of the equation and manipulating it until you get something that looks like the left side.

I hope this helps and good luck with your proof!