Why there is no constant wave-function of an electron, while all the electrons are same? The mass and charge of all electrons are same, so why don't we have a single wavefunction for electrons.
 A: Electrons are indeed all the same in the sense that they are all excitations of the same electron quantum field. However they can have different momenta and energy. The momentum is obtained from the wavefunction using the momentum operator:
$$ \hat p = -i\hbar \frac{\partial}{\partial x} $$
And the energy from the Hamiltonian operator:
$$ \hat H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V $$
And for these to return the differing energies and momenta of different electrons clearly those electrons have to have different wavefunctions.
The wavefunction is generally used to describe single electrons, or groups of a few interacting electrons, and those electrons will have differing properties like the momentum and energy already mentioned, and also potential energy, spin orientation and probably others I haven't thought of. All these different quantities have to be reflected in different wavefunctions.
However all electrons are ultimately described by a single mathematical object called the electron quantum field. In this sense you are correct that there is a single object that describes all electrons - it's just not the wavefunction.
