(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Linearly indep. annihilation operator

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