QM I - Decomposition of countable basic states into coherent states

[BasslineSanta]

Homework Statement

Consider a quantum system with a countable number of basic states \left|n\right\rangle.
Calculate the decomposition into a basis of coherent states \left|λ \right\rangle all obeying \hat{a} \left|λ \right\rangle = λ \left|λ \right\rangle

Homework Equations

\hat{a} is the lowering operator:
\hat{a} \left|n\right\rangle = √n \left|n-1\right\rangle

The Attempt at a Solution

Because \left|λ\right\rangle form a basis, i can equate \left|n\right\rangle = Ʃλ_{n}\left|λ\right\rangle.
Applying the lowering operator n-times to both sides of the equation, i get: √n! \left|0\right\rangle = λ^{n} Ʃλ_{n} \left|λ\right\rangle
By equality of two vectors i can say that √n! = λ^{n} and that \left|0\right\rangle = Ʃλ_{n} \left|λ\right\rangle.

Now i got kinda stuck. I thought if i get to the \left|0\right\rangle vector, i can just keep applying the raising operator to get any state \left|n\right\rangle written in my new vectors \left|λ\right\rangle. But i realized i do not know how the raising operator acts on them. Neither do i know if i chose the right approach, but it feels like the only thing i could have done, considering the information given.

I would really appreciate some help.
Thanks a lot in advance!

 

[Oxvillian]

I'm a little confused here as to whether you're trying to express a coherent state as a sum of the |n>'s, or one of the |n>'s as a sum of coherent states

The former is easy enough, but the latter is trickier - the answer won't be unique, because the coherent state basis is an overcomplete basis. Probably you'll have to use a coherent state resolution of unity, which involves integrating over the complex λ plane.