In investigating the behavior of electrons in atoms, physicists came across the following conundrum:
Electrons seemed to be captured in orbits around the atoms under the influence of electrostatic forces, much the way the planets are in orbit around the sun under the influence of gravity. The force law for gravity and electrostatic attraction are in fact identical, and the electrons seem to move in orbits which seem to have definite energies.
On the other hand we know that the electron is a charged particle, and we also know that accelerating charged matter gives off radiation, according to Maxwell's equations, loses energy and eventually stops.
The only conclusion that anyone could think of was that the electrons were somehow in orbits for which their charge density remains constant over time.
But how can this be if the electron is envisioned as charged particle confined to a single point? The only plausible stationary point would be the origin, but electrons where spread out over quite large areas. (on the order of 10-8cm in diameter)
Physicists responded by changing the way they envision physical systems.
Originally they thought of the electron as having measurable variables like position and momentum that are numbers with the problem of describing their motion being that of discovering their orbits.
They came instead to the notion that the states of the system should be described as vectors, and the measurable variables be described as linear transformations (think of matrices) that act on these vectors.
The value of a variable describable by matrix M in state with row vector <s| and column vector |s> in this description is <s|M|s>.
The dynamical development of the system in this formulation is derived from the basic equation of motion: the time derivative of a state vector is proportional to its Hamiltonian matrix applied to that vector.
Another interesting feature of quantum mechanics is that using the basis in which position x (a linear transformation or operator or matrix) is represented by a number which means that the state is an eigenvector of the position operator, the corresponding momentum variable is represented as a multiple of (you can also define a basis in which the opposite is true). These transformations (or operators or matrices) obviously do not commute.
What does it mean to measure a variable and find it has value z?
It means to project the state of the system from what it was onto the eigenvector (or if there are more than one "eigenspace") of your variable with eigenvalue z.
What does it mean when operators representing two variables like x and do not commute?
It means that they can have no common eigenvector and hence cannot be simultaneously measured. (If simultaneously measured the system would be characterized by an eigenvector of both. But there is no common eigenvector of x and .)
What sort of a vector space is used here?
It is what is called a Hilbert space; it is infinite in dimension, and the numbers that describe components in each direction are complex numbers.
The states of an electron in an atom are stationary states that persist for some time. They are eigenstates or eigenvectors of the electron's Hamiltonian.
They actually do evolve in time, but not in an easily observable way; only the angle of their state in the complex plane changes and that does not change such things as their charge density.
This discussion cannot pretend to teach you anything worthwhile about quantum mechanics, except the fact that it can be considered as the application of the concepts of this course, each and every one of them, run amok.
This scheme has been able to describe atomic spectra with amazing precision. And it explains lots more besides.