Coherent states: Orthonormal set? Overcomplete basis?

[LagrangeEuler]

For two different coherent states
[tex]\langle \alpha|\beta \rangle=e^{-\frac{|\alpha|^2+|\beta|^2}{2}}e^{\alpha^* \beta}[/tex]

In wikipedia is stated
https://en.wikipedia.org/wiki/Coherent_state"Thus, if the oscillator is in the quantum state | α ⟩ {\displaystyle |\alpha \rangle } |\alpha \rangle it is also with nonzero probability in the other quantum state | β ⟩ {\displaystyle |\beta \rangle } |\beta \rangle (but the farther apart the states are situated in phase space, the lower the probability is). However, since they obey a closure relation, any state can be decomposed on the set of coherent states. They hence form an overcomplete basis, in which one can diagonally decompose any state. This is the premise for the Sudarshan-Glauber P representation. "

Could you please explain me what is OVERCOMPLETE BASIS?Also, when some authors write ## \langle \alpha|\beta \rangle \neq \delta(\alpha-\beta)## is ##\delta(\alpha-\beta)## Dirac ##\delta## or some type of continuous Kronecker?

 

[hilbert2]

It's a Dirac ##\delta##. An overcomplete basis means that it has more vectors than needed to span the required subspace of the Hilbert space where the vectors like ##\left|\alpha\right.\rangle## exist. If it's said to be an overcomplete basis of the whole ##\mathcal{H}##, then it means that it's a basis and it should contain vectors that are not all orthogonal with every other basis vector (therefore not a linearly independent set).

You can compare this to the concept of an overdetermined linear system of equations:

https://en.wikipedia.org/wiki/Overdetermined_system

Edit: And it's quite obvious that the set of all generalized Gaussians

##\displaystyle\psi (x) = Ae^{-kx^2 + bx + c}##,

with ##b## and ##c## complex numbers, is an overcomplete basis because it depends on continuum parameters while all normalizable states of the 1D harmonic oscillator can be spanned by a discrete basis. An example of that kind of discrete basis is the set of all eigenstates of the Hamiltonian

##\displaystyle\hat{H} = \frac{1}{2m}\hat{p}^2 + \frac{1}{2}m\omega^2 \hat{x}^2##.