# Ground state of hamonic oscilator

1. Mar 24, 2015

### naima

When i take a coherent state $|\alpha>$ if $\alpha -> 0$ then the limit is the Fock state for n = 0. so $|n = 0> = |\alpha = 0>$
The problem is that they seem to have different Wigner functions:
Where is the error?
Thanks.

Edit sorry, in the link the W function is for a (n = 1) Fock state. So no more problem.

Last edited: Mar 24, 2015
2. Mar 24, 2015

### vanhees71

I don't understand your question. Among other definitions, you can define a coherent state of a given mode as an eigenvector of the corresponding annihilation operator
$$\hat{a} |\alpha \rangle=\alpha |\alpha \rangle,$$
where $\alpha \in \mathbb{C}$. For $\alpha=0$ that's the definition of the "vacuum" (absence of the considered mode). Thus also the vacuum is a special coherent state.

3. Mar 24, 2015

### naima

I had in mind the fact that for an integer $\alpha$ the Fock state $|\alpha>$ is just one of the terms of the serie of the coherenet state $|\alpha>$.
I did not saw that for 0 the serie has only one term.