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Minimising for Lambda; Differentiation?

  1. Mar 30, 2010 #1
    Hi, so first post on this forum so I hope I'm doing everything good-as-gold!

    I've been twisting my head around a derivation of the uncertainty principle and I'm a little stumped on something that I feel I really should know. Hope someone would like to help.

    Certain things are defined at the start,
    [tex]\hat{Q}'=\hat{Q}-\left\langle\hat{Q}\right\rangle[/tex]
    [tex]\hat{R}'=\hat{R}-\left\langle\hat{R}\right\rangle[/tex]
    [tex]\phi(x)=(\hat{Q}'+\textit{i}\lambda\hat{R}')\psi(x)[/tex]
    [tex]I(\lambda)=\intdx\phi^{*}(x)\phi(x)\geq0[/tex]

    I've followed the manipulation of [tex]I(\lambda)[/tex] all the way to the following line;
    [tex]I(\lambda)=(\Delta Q)^{2}+\lambda^{2}(\Delta R)^{2}+\textit{i}\lambda\left\langle\left[\hat{Q},\hat{R}\right]\right\rangle\geq0[/tex]

    That's all fine. In the notes he then says "Minimise for [tex]\lambda[/tex]" following it with;
    [tex]2 \lambda(\Delta R)^{2}+\textit{i}\left\langle\left[\hat{Q},\hat{R}\right]\right\rangle=0[/tex]

    He then subs it back into [tex]I(\lambda)[/tex]
    It looks like he's differentiated wrt lambda. But I'm not entirely sure what he means when he says minimise for lambda.
    Also, I haven't seen anywhere where 'I' has been defined. Not sure what it is!
    Has anyone got any ideas?

    MUCH appreciated!
     
  2. jcsd
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