Just a little continuation of Jakohn's write up for those laypersons who would like to further understand the equation. H is your Hamiltonian operator and describes the energy of the thing in question. It can be summarized as the sum of

potential energies(V) and kinetic energies. The Hamiltonian in one dimension is described from two

operators hbar/

i*d/dx, and x. The first operator, although actually a group(

numbers are operators) corresponds to

momentum†(p) and the second operator corresponds to position. For a simple harmonic oscillator potential energy as a function of position is -kx

^{2}/2 (remember that in order find potential enrgy you

integrate force with repect to

displacement) and

kinetic energy as a function of momentum is always the same, p

^{2}/2m, where m is mass. Remember E(

energy) is always equal to hv (where v is frequency). Then you find &Psi as a function of x through manipulation of ordinary second order differential equations.

That all is a bit simplistic cause in the real world we generally like to consider things in three dimensions plus time. So then the time independent schrödinger equation becomes becomes H(hbar/i*∂/∂x, hbar/i*∂/∂y, hbar/i*∂/∂z, x, y, z)ψ=Eψ. Thus for a simple harmonic oscillator I think it'd go something like this -hbar ^{2}/2m*(∂^{2}ψ/∂x^{2}+∂^{2}ψ/∂y^{2}+∂^{2}ψ/∂z^{2})+k/(x^{2}+y^{2}+z^{2})=Eψ.

Even though Matrices are more complicated I like them more, and so I'm personally more fond of Heisenberg's Matrix Mechanics.

~~† this is not momentum per se but rather a quantum equivalent which behaves similarly. This was one of the original pitfalls of Schrödinger's equation, as how can a wave have momentum or mass?~~
Actually turns out this is an incorrect interpretation as pointed out by unperson Momentum does meat obey the canonical definition. But rather it is not mass times velocity in this sense. Also apparently electromagnetic waves did have some momentum even in classical theory.

Also thanks to Suvrat for pointing out some minor errors as per potential energy in three dimensions of the harmonic oscillator. That was bad integration on my part sorry.