The wavefunction of the electron in the hydrogen atom is $ k* exp(-r/a)$ (k is the normalization constant), but it does not take n into account, whereas the solution of Schrödinger's equation ($H(wavefunction)=E*wavefunction$) says the energy of it is $E=E(0)/n^2$, but the variable in the wavefunction is $r$, and $n$ isn't there. If I apply the Hamiltonian operator on the wavefunction, no $n$ will appear, I don't understand!
Your equation k*exp(-r/a) is the wavefunction(n=1,I=0,m=0), so n=1 = ground state. So while n does not appear explicitly in the equation, it’s really there and it’s equal to 1 in this case. The equation should really be written H x wavefunction(n) = En x wavefunction(n), En = E(0)/n^2.