In quantum mechanics to each physical quantity in the real world correspond a mathematical function that is applied to the particle from the real world. To apply a function to a particle means in turn to compose that given function with the (quantum mechanics) wave function associated with the real world particle.
In particular the Schrodinger equation relates the energy function in quantum mechanics `hatH` to the real values of energy `E` that the particle can take. If the wave function that describes the state of the particle is `psi` one can write the Schrodinger equation in the form:
(remember `hatH` is a function applied to another function and `E` is a physical value). Therefore the numerical values `E` are the eigenvalues of the function `hatH` .
Since `hatH` represents the total energy, it is the sum of two functions each representing the kinetic energy and the potential energy:
`hatH =hat(E_k) +V(r) = (hat(P))^2/(2m) + V(r)`
where `hatP =ih/(2*pi)*grad` is the corresponding function for the linear momentum.
(Above `hat(E_k) =hatP^2/(2m)`
like in real world `E_k =(m*v^2)/2 =(m*v)^2/(2m) =p^2/(2m)` )
Thus the overall Schrodinger equation can be written as
`-h^2/(4*pi^2)*grad^2(psi) +V(r)*psi =E*psi`
Now, for Hydrogen atom and Hydrogen like atoms the potential function is
`V(r) = -Z*e^2/(4*pi*epsilon_0*R)` ,` `
or in other words the electric potential energy of the electron in the electric field produced by the `Z` protons in the nucleus.
Thus the Schrodinger equation for Hydrogen atom and Hydrogen like atoms is
`[-h^2/(2m)*grad^2 -(Z*e^2)/(4*pi*epsilon_0*R)]psi =E*psi`