Imagine a particle moving around in the three-dimensional space that we see around us. In classical mechanics, the state is described in terms of a six-dimensional space. In fact, there are two ways to do this. The first way is the picture of Lagrangian mechanics, in which the six coordinates are the position of the particle along the three axes of the physical space and the speed in which it’s moving in each of those directions. In Hamiltonian mechanics, the six coordinates are the three position coordinates and the momenta of the particle in the three directions. Both of these pictures are equivalent to each other although their mathematical formulations are pretty different, and they’re both equivalent to the more familiar formulations of Newtonian mechanics. In both cases, the particle’s state is described by a point in a manifold, which is just a mathematical space we can cover with coordinates. The laws of physics then determine the way the particle’s point moves around in the manifold.

Each “observable” is a function on the manifold. For example, the energy observable is a number associated with each point, so that if we know the particle’s position and velocity (or momentum) then we know its energy. The key point is that a given particle or system of particles (in which case the dimension of the manifold in question is bigger) has a definite value for all observables. (In statistical mechanics, our ignorance blurs our view of the manifold - we aren’t quite sure what the state of the particle is so we aren’t quite sure of the values of each observable.)

Quantum mechanics is both simpler and more complicated than classical mechanics. It’s simpler in two respects. Firstly, the states don’t live in general manifolds but in vector spaces, which are nice rigid Euclidean manifolds that are much simpler to deal with mathematically. Also, we don’t have to worry about both the position and the velocity/momentum of the particle. It’s more complicated, though, because in general the vector spaces have an infinite number of dimensions. There’s one dimension for each possible location of the particle. A state of the particle is now a point in this vector space, which is just the same as a vector in the vector space. Roughly speaking, the amount by which this vector points in the direction corresponding to a location determines the probability that the particle will be found in that location. For example, if the particle’s state points along the same direction as the “here” vector then the particle will be definitely found here. If it points at equal angles to “here” and “there” then it has equal probabilities to be found here or there. Just as in classical mechanics, time development is just a rule for how the state moves around inside the state space as time advances.

In quantum mechanics, observables aren’t functions on the state space, but rather sets of directions in that space. Just as the probability for the particle to be found at various locations is determined by the angles between its state and the vectors corresponding to those locations, so the probability for the particle to have various values of other properties are determind by the angle between its state and the vectors corresponding to the relevant observable. Curiously, however, not all the directions corresponding to the different observables coincide with each other. This means that, unlike in classical mechanics, a particle with a definite value for one observable won’t necessarily have definite values for all the other observables. This is the ultimate origin of the various uncertainty principles.

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