Are Fock States Eigenstates for the Operators a^{\dagger} and a?

[Ylle]

Homework Statement

I have to show that the Fock states are eigenstates for the operators ${{\hat{a}}^{\dagger }}$ and/or ${\hat{a}}$

And I'm not totally sure how to show this.

Homework Equations

?

The Attempt at a Solution

I know that if I use the operators on a random Fock state i get:

$$a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle$$
$$a|n\rangle=\sqrt{n}|n-1\rangle$$

So what's next ?


Regards

 

[nickjer]

Are you sure you are typing the question correctly? Because as you just showed, Fock states are not eigenstates of the creation and annihilation operators. Maybe you are getting it confused with coherent states.

 

[Ylle]

Ahhh...
It said: IS the Fock states eigenstates for the operators, doh...

So when I use the operators on the Fock state, they are not eigenstates because of the |n-1> and |n+1> at the end, which should have been just |n> in each case, or am I way off ?

 

[nickjer]

Yes, an eigenstate of an operator won't change under an operation on the operator. So you would just get back |n> with a coefficient called the eigenvalue.

 

[Ylle]

Well, thank you :)
Didn't know it was that easy.