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Representation of linear operator using series ?

  1. May 19, 2013 #1
    representation of linear operator using "series"?

    I was looking into the progression of quantum states with respect to time. From what I understood the progression of a state ## \left|\psi(t)\right> ## is given by:
    $$ \left|\psi(t)\right> = U(t)\left|\psi(0)\right> $$
    I'm not sure if that's right. But it's okay if I haven't got that yet.. That was just to give some context to my actual question.

    What is this representation of the linear operator ## U(t) ## at ## t = \epsilon ##, where ## \epsilon ## is an infinitesimal?
    $$ U(\epsilon) = I - i\epsilon H $$
    Where ## i ## is the imaginary unit, ## I ## is the identity matrix and I think ## H ## is the hamiltonian.
    It also apparently has more terms of order ## \epsilon^2 ## and so on. What "series" is this? Is it some first order approximation of ## U(t) ##? What should I look into to understand where those terms are coming from?
     
  2. jcsd
  3. May 19, 2013 #2

    micromass

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  4. May 19, 2013 #3
    It's right.

    Those are the first two terms of the Taylor series expansion for [itex]e^{-i\mathbf{H}\epsilon}[/itex]. It just so happens that [itex]U(t) = e^{-i\mathbf{H}t/\hbar}[/itex]. This can be related to the idea that [itex]E = \hbar \omega \rightarrow \omega = E/\hbar[/itex]. A state [itex]\mid n\rangle[/itex] with definite energy [itex]E_n[/itex] has the following time evolution.
    [itex]\mid n(t)\rangle = e^{-i\omega t}\mid n(0)\rangle = e^{-iE_n t/\hbar}\mid n(0)\rangle [/itex]

    A function of an operator like [itex]e^{-i\mathbf{H}t/\hbar}[/itex] has the same eigenvectors as the operator, but the eigenvalues are replaced with the corresponding function of the eigenvalues.

    Here are some relevant video lectures:
    https://www.youtube.com/watch?v=cVbB6wFNqYc&list=PL9LRV0x7N1NCpd-ZerPxiTzm97y9VGPLp
    https://www.youtube.com/watch?v=31XrxGMRwtw&list=PL9LRV0x7N1NCpd-ZerPxiTzm97y9VGPLp
     
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