# Commutator algebra

1. Oct 12, 2012

### helpcometk

1. The problem statement, all variables and given/known data
Analytic functions of operators (matrices) A are defined via their Taylor expansion about A=0 .Consider the function

g(x) = exp(xA)Bexp(-xA)

Compute : dng(x) /dxn |x=0
for integer n

and then show that :exp(A)Bexp(-A)= B+[A,B] +1/2 [A,[A,B]] +1/6[A,[A,[A,B]]]+ ...

2. Relevant equations

for the first part of the question i don't understand what x=0 means in the formalism and it is unclear how one should proceed ,becuase we are dealing with operators.

about the second part i would like to note that :

g(1) =g(0) +g'(0) +1/2 g''(0)+ ...
if A=0 e^0 =1 and we start with B in the expansion formula

3. The attempt at a solution

About the first part i cannot imagine how i could proceed ,but about the second question i think we must use somehow a taylor expansion to resemble the monsterous expression of the RHS. But what i can't see is how one should manipulate [A,B], how could this come from a taylor expansion of the exponential ?

2. Oct 12, 2012

### D H

Staff Emeritus
x is a scalar. Some hints:
• Matrix multiplication is transitive, but not necessarily commutative. That the latter is the case is the point of the commutator.
• The chain rule still works as expected with your dg(x)/dx. You will either need to show this or cite the relevant theorem from your text/lectures.
• For any square matrix A, exp(A) commutes with An for all nonnegative integers n. Once again, you will either need to show this or cite the relevant passage.
• What is $\frac{d}{dx} \exp(xA)$?

3. Oct 12, 2012

### helpcometk

If chain rule is correct ,then i would have that the derivative is equal to 0 ,right?
about the second part do i just take the taylor expansions of both exponentials and mmultiply them ?

4. Oct 12, 2012

### D H

Staff Emeritus
No! You apparently don't understand the notation.

This means you are to compute the value of the nth derivative of g with respect to x at x=0. It says nothing about the derivative being zero.

Ooops. My bad. Sorry about that. I meant product rule, not chain rule.

5. Oct 12, 2012

### helpcometk

I understand the notation ,but im very tired becuase i dont sleep overnight and i make a mistake ,
so we have :
dng(x) /dxn = -2xAg(x) ,do you agree with this ?

6. Oct 24, 2012

### D H

Staff Emeritus
Sorry for the very late response. I didn't see that you had added a post until just now.

No, I don't agree with that. Let's start with $dg/dx$. Applying the product rule (which is still valid because matrix multiplication is associative),
\begin{align} \frac {d\,g(x)}{dx} &= \frac d {dx} \bigl(\exp(xA)\,B\,\exp(-xA)\bigr) \\ &= \frac{d\exp(xA)}{dx}\,B\,\exp(-xA) + \exp(xA)\,B\,\frac{d\exp(-xA)}{dx} \end{align}
The key challenge is to evaluate $d\exp(xA)/dx$. Using the fact that $A$ and $\exp(xA)$ commute (show this!), you can reduce the derivative to something of the form
$$\frac {d\,g(x)}{dx} = \exp(xA)\,C\,\exp(-xA)$$
where $C$ is some matrix. The question itself suggests a form for this matrix: It has to somehow involve the commutator. (Hint: It is the commutator.) See if you can get to that. With this, getting $d^ng/dx^n$ should be a snap.

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