# [Physics] Is the vacuum state a coherent state

coherent-statesharmonic-oscillatoropticsquantum mechanics

I'm asking because I got introduced to the state $|0\rangle$ as a fock-state. Nevertheless:
$$\hat{a} |0\rangle = 0 |0 \rangle$$
It is an eigenstate of $\hat{a}$ with eigenvalue $0$, and it can be obtained the same way any other coherent states are obtained via the displacement operator with parameter 0:
$$\hat{D}(\alpha=0)|0 \rangle = e^{0 \hat{a}^\dagger – 0 \hat{a}}|0\rangle = |0\rangle$$

Would one consider the vacuum state a coherent state?

The coherent state $$\vert \alpha\rangle$$ is just a vacuum state $$\vert 0\rangle$$ translated in $$x$$ and $$p$$ space so $$\alpha=x_0+ip_0$$. Thus the vacuum state is a coherent state that has not been displaced, i.e. $$x_0=p_0=0$$.
In fact, a nice way to see this is in the Wigner function formalism. The vacuum state is just a Gaussian sitting at the centre of $$(x,p)$$ space whereas a coherent state is the same state displaced to another point. This is illustrated in the figures below, taken from this site: on the left is the Wigner function of the vacuum state, and on the right that of a coherent state.
A short movie illustrating the time evolution of the Wigner function of a coherent state can be found on the coherent state wikipage; it shows the Wigner function does not deform and remains non-negative at all times Of course since the vacuum state is an eigenstate of the Hamiltonian and lies at the centre of $$(x,p)$$, its Wigner function would actually remain there at all times.