# Linearly indep. annihilation operator

1. Nov 15, 2008

### yakattack

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

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

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