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Homework Help: Linearly indep. annihilation operator

  1. Nov 15, 2008 #1
    1. The problem statement, all variables and given/known data
    If [tex]\varphi[/tex] and [tex]\widetilde{ \varphi }[/tex] are linearly independent and
    [tex]\hat{N}[/tex][tex]\varphi[/tex]=n[tex]\varphi[/tex] and
    [tex]\hat{N}[/tex][tex]\widetilde{\varphi}[/tex]=n[tex]\widetilde{\varphi}[/tex], with [tex]n\geq 1[/tex]
    [tex]\ \text{Prove that } \hat{a}\varphi_{n} \text{ and } \hat{a}\widetilde{\varphi}_{n} \text{ are also linearly independent.}[/tex]

    2. Relevant equations
    [tex]\ \text{I have to use the fact that if } \hat{N}\varphi=\nu\varphi \text{ then } \nu\geq0 \text{ and } \nu=0 \text { iff } \hat{a}\varphi=0
    \hat{N}=\hat{a^{*}}\hat{a}
    \hat{a} \text { is the annihilation operator.}[/tex]

    3. The attempt at a solution
    If i suppose they aren't linearly independent then i can show that this means that
    [tex]\hat{a}[/tex][tex]\varphi[/tex][tex]_{n}[/tex] and the same with tildas are linearly dependent. But does this show that the converse is true?
     
  2. jcsd
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