1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Derivative of an operator

  1. May 31, 2012 #1
    1. The problem statement, all variables and given/known data
    calculate [itex]\frac{d}{dt}e^{\hat{A}t}[/itex] where [itex]\hat{A} \neq \hat{A}(t)[/itex] in other words operator A doesn't depend explicitly on t.

    2. Relevant equations

    3. The attempt at a solution

    [itex]\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} [/itex]

    if the xi ≠ xi(t) we get [itex]\hat{A}e^{\hat{A}t} [/itex]

    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. jcsd
  3. May 31, 2012 #2
    First off are you sure this isn't just a partial differntiation in which case there is no problem. Otherwise this looks quite allright.
  4. Jun 1, 2012 #3
    Yup, there's nothing wrong with your solution.
  5. Jun 3, 2012 #4


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook