Schroedinger's Equation is:

-((h/2π)2/2m)(∂2Ψ(x,t)/∂x2)+V(x,t)Ψ(x,t) = j(h/2π)(∂Ψ(x,t)/∂t)

A description of each variable:

The essence of Schroedinger's Equation is that it sums the energy for the particle, ala the classical physics equation E = K + V, where E is the total energy, K is the kinetic energy, and V is the potential energy.

In Schroedinger's Equation E is expressed in the right-hand side of the equation, V is expressed in V(x,t)Ψ(x,t), and K is expressed in -((h/2π)2/2m)(∂2Ψ(x,t)/∂x2)

This is the time-dependent Schroedinger Equation. There also exists a time-independent Schroedinger Equation, which easily derived from the time-dependent version via the technique of separation of variables (using Ψ(x,t)=ψ(x)φ(t)), and which can be used if the potential energy of the particle does not vary with time. It is:

-((h/2π)2/2m)(∂2ψ(x)/∂x2)+V(x)ψ(x) = Eψ(x)

Which can be rewritten, by factoring out ψ from the left-hand side, and using the Hamiltonian operator (H):

Hψ = Eψ

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 -kx2/2 (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, p2/2m, 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*(∂2ψ/∂x2+∂2ψ/∂y2+∂2ψ/∂z2)+k/(x2+y2+z2)=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.

This is fundamental equation of Quantum mechanics. It comes in two versions: the time-independent version, and the more general but less useful time-dependent version.

Theoretically, everything in our everyday world follows this equation (the time dependent one). This includes all of chemistry, biology, anthropology, geology, economics, etc. While this equation is probabilistic, on most large scales it quickly becomes effectively deterministic at it's behavior merges with classical mechanics. So if one could solve my Schrodinger equation and perhaps the equation for my local area they would probably be able to predict a great deal of my future.

Unfortunately, or perhaps fortunately, while this may be true, it is impossible. To try and manipulate this equation with anything much more complicated then a few atoms, or perhaps an infinite lattice of identical atoms, and you have a nightmare of a problem. You run into the same many body problem people have when trying to solve any system with too many degrees of freedom plus a big, ugly algebra mess.

Hence, our non-deterministic Romantic world is preserved. We don't have to worry about mad scientists using Quantum mechanics to play God.

Note: The only things that violates this equation are things approaching the speed of light and perhaps things at extreme energies/times/masses/temperatures. Not something I see every day.

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