1. Not finding help here? Sign up for a free 30min 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!

Associativity of operators in quantum mechanics

  1. Nov 24, 2012 #1
    1. The problem statement, all variables and given/known data
    What is the correct interpretation of
    [tex] < \frac{\partial {A}}{\partial t} >[/tex], where A is an operator?


    2. Relevant equations
    for a wave function [tex]\phi[/tex] and operator A,
    [tex]<A> = \int_{V}\phi^{*}(A\phi)dV[/tex]


    3. The attempt at a solution
    I thought it could mean
    [tex] < \frac{\partial {A}}{\partial t} > = \int_{V}\phi^{*}\frac{\partial}{\partial t}(A\phi)dV[/tex]
    but then again it might mean
    [tex] < \frac{\partial {A}}{\partial t} > = \int_{V}\phi^{*}(\frac{\partial A}{\partial t})(\phi)dV[/tex].

    I read an article saying that operators are associative. But, when I think about the operators [tex]t[/tex] and [tex]\frac{\partial}{\partial t}[/tex], then,

    [tex]\frac{\partial}{\partial t}\left(t\phi\right) = t\frac{\partial\phi}{\partial t} + \phi \neq \left(\frac{\partial t}{\partial t}\right)\phi = \phi[/tex]

    any thoughts?
     
  2. jcsd
  3. Nov 24, 2012 #2

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    The d/dt is not an operator in the normal sense. A(t) is interpreted as a set of operators which depend on a parameter, t. The derivative of this A(t) wrt the parameter is defined by means of a limiting procedure always in the presence of vectors in the domain of all A(t).

    [tex] \frac{d A(t)}{dt} \psi = \lim_{t\rightarrow 0} \frac{A(t)\psi - A(0)\psi}{t} [/tex]

    The expectation value is then simply [itex] \left\langle \phi, \frac{dA(t)}{dt}\phi\right\rangle [/itex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Associativity of operators in quantum mechanics
Loading...