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Problem understanding operator algebra

  1. Oct 27, 2013 #1
    "It is left as a problem for the reader to show that if [itex][S,T][/itex] commutes with S and T, then [itex][e^{tT}, S] = -t[S,T]e^{tT}[/itex]

    I'm not sure if i'm missing something here, but i dont even see how it is possible to arrive at this answer.

    I get:
    [tex][e^{tT}, S] = e^{tT}S - Se^{tT}[/tex]

    Then using the fact that [itex][S,T][/itex] commutes with S and T this gives:

    [itex]SST-STS = STS-TSS[/itex]
    and
    [itex]TST-TTS = STT-TST[/itex]

    and see no way to go further.
    One major thing is I don't even see how the factor of -t just appears in the identity?
    [itex][e^{tT}, S] = -t[S,T]e^{tT}[/itex]
     
  2. jcsd
  3. Oct 27, 2013 #2
    The typical way to do something like this would be to use a Taylor expansion of the function of the operator, in this case [itex]e^{tT}[/itex]. What you were trying to show is an example of a more general result.
     
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