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!

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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?



Similar Discussions: Linearly indep. annihilation operator
  1. Heat engine operations (Replies: 0)

Loading...