Influence of applied voltage to an electron of a metal I would like to ask what would happen to the potential well of an electron being trapped in a metal? If I apply a voltage trying to accelerate the electron out of the potential well. Would It make the potential well shallower? Or would it rather shift the fermi level to a lower fermi energy level? 
 A: First of all I think we are thinking about the free electrons in the metal rather than the ones which are bound to individual atoms.
Generally to apply a voltage we attach connections to each end of the metal and apply a potential difference. One way of thinking about what happens when we apply a voltage is that the fermi level gets slightly tipped from one end of the metal to the other so that it is slightly raised at one end and lowered at the other and electrons flow because of this. 
Another way to think about this is using the free electron model where electrons fill the 3D space of the metal. This has to be solved quantum mechanically and is like the particle in a box calculation except, of course, because electrons are femions they each have to have separate levels and fill up to the fermi level. Now we often think of levels in the particle in the box as having different energies, but in the 3D box the electrons are confined to we can also think of the electrons as having different wave vectors $k$ ($E$ is proportional to $k^2$. The wave vector $k$ has direction and points anywhere in 3D. If we fill up to the fermi level we will fill up all states up to some maximum $k$ value. We can plot these states in 3D phase space where the axes are $k_x$, $k_y$ and $k_z$ and we would get a filled sphere centred on the origin - we can think about the electrons in this sphere travelling in all different directions with a net current flow of zero because the sphere is centred on the origin.... 
.... Now applying a voltage to the metal will slightly shift this sphere in $k$ space so that it is no longer centred on the origin. This will meant that there is a net flow of electrons.  
If you want to think about this without using metal contacts then UV photons can knock an electron out of the metal by the photoelectric effect. 
Alternatively in Field Emission Microscopy a high voltage near a sharp metal tip 'pulls' electrons out of the metal - this raises the fermi level at the tip to the vacuum level and electrons can escape.
