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.
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 them.
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 sort of a vector space is used here?
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.