# Derivative of an operator

1. May 31, 2012

### sunrah

1. The problem statement, all variables and given/known data
calculate $\frac{d}{dt}e^{\hat{A}t}$ where $\hat{A} \neq \hat{A}(t)$ in other words operator A doesn't depend explicitly on t.

2. Relevant equations

3. The attempt at a solution

$\frac{d}{dt}e^{\hat{A}t} = (\frac{d}{dt}(\hat{A})t + \hat{A})e^{\hat{A}t} = (\sum^{n}_{i=0}\frac{d\hat{A}}{dx_{i}}\frac{dx_{i}}{dt}t + \hat{A})e^{\hat{A}t}$

if the xi ≠ xi(t) we get $\hat{A}e^{\hat{A}t}$

but is this correct I know how to define the derivative of an operator if it is explicitly dependent on the variable of differentiation but not in this case.

Last edited: May 31, 2012
2. May 31, 2012

### conquest

First off are you sure this isn't just a partial differntiation in which case there is no problem. Otherwise this looks quite allright.

3. Jun 1, 2012

### dimension10

Yup, there's nothing wrong with your solution.

4. Jun 3, 2012

### dextercioby

It makes a world of difference if the operator in the exponent is bounded or not. Either way, there's a strict definition of such a derivative in terms of limits which can be found in almost all books on functional analysis.