Earlier in 1925 than Schrödinger, Werner Heisenberg was working on a new description of matter. His ruminations led him to assert a new principle that has become a hallmark of quantum theory - it is even immortalized on bumper stickers ("Heisenberg MAY have slept here.") This is the well-known Heisenberg Uncertainty Principle. His work was grounded in Poisson's description of mechanics, rather than Hamilton's. As indicated, a central concept involves a quantity known as a Poisson Bracket, which has a simple definition and is most interesting when operators are being studied.

Consider the following situation. Two useful properties to measure of any object could be the objects position and its momentum. Restrict our discussion to motion of a particle of mass m moving in a single direction, say the x direction. The value for x(t) is its position and the value of p_x(t) is its momentum, both at a given moment in time t.
P.W. Atkins, in his book "Molecular Quantum Mechanics" provides a cute little development of the uncertainty principle using these ideas for the particle undergoing simple harmonic motion. This development follows here:

The basic equation of classical physics is Newton's Second Law:

For simple harmonic motion, the definition is that the force F is proportional to the displacement, or that F = -k x(t), k being the proportionality, or force constant. With this definition for F, Newton's Second Law can be solved for x(t) giving

This expresses the displacement as a function of time. By definition of momentum we find that for this particle

With these two equations for the position and momentum of this oscillating particle, we can explore some properties of the system. Referring back to Poisson's mechanics, an important relation between two properties is that defined by the Poisson Bracket:

With the two expressions given above, we can find a result for this expression. Obviously we simply take the product xp and then the product px and take the difference.

It is obvious, yea even trivial, to note that for these two expressions, px-xp must be 0. But to quote Atkins "Trivial it may be, but true it is not." This is the remarkable contribution from Heisenberg. He suggested that rather than being identically zero, such a quantity must be extremely small, but non-zero. This is where Planck's Constant is first introduced into his work. The Uncertainty Principle amounts to requiring that x and p_x must satisfy the following relation.

This is the most fundamental equation of quantum mechanics. The entire theory flows from it. Look here for a brief look at Heisenberg's Matrix Mechanics.