Each state as obtained as a solution of the Schrödinger equation of the hydrogen atom is
characterised by a unique combination of the quantum numbers *n*, *l*, and *m*.
We have seen that each state can be populated by up to two electrons. Even if we reduce the
energy of the system to near absolute zero temperature, not all of the electrons converge on
the *n*=1 state but remain in higher states because the *n*=1 state can't take more than
two electrons. This has lead to *Pauli's exclusion principle*: No two electrons can
share the same state (*i.e.* have the same quantum numbers) at the same time. Therefore,
we need another quantum number, *spin*, to distinguish the two electrons sharing the same
hydrogen, *i.e.* (*n,l,m*) state.

Here is an overview of the quantum numbers:

The *main quantum number*, *n*, determines the energy of the state (exactly in the
case of hydrogen-like atoms; with minor corrections involving the other quantum numbers for multi-electron
atoms). It also determines the size of the spherical envelope of the wave function: The higher *n*,
the further the probability cloud stretches out into space. Possible values of *n* are positive
integer numbers.

The *angular momentum quantum number*, *l*, governs the ellipticity of the probability cloud
and the number of planar nodes going through the nucleus. In the Fig., red and blue areas indicate regions
with opposite sign of the wave function and the black dot represents the position of the nucleus.
For a given *n*, the possible values are integers from 0 up to *n*-1.

The *magnetic quantum number*, *m*, is related to the magnetic moment of any non-spherical
electron (probability) distribution. Therefore, the different *m* states relate to differently
orientated orbitals. Which one is which depends on the preferential axis, *i.e.* the orientation of
an external field, if any. All (*n,l*) states with different *m* look the same. In the Fig.,
the third one is the same shape as the others but orientated perpendicular to the screen. The values of
*m* for a given (*n,l*) range from -*l* to +*l*.

The *spin (quantum number)*, *s*, is similar to *l* in that it refers to an angular
momentum component. It can be visualised as the rotation of the electron, thought of as a particle,
around its own axis rather than around the nucleus. This is, however, a bit of a crutch because the
electron's behaviour has both aspects of particle (such as scattering) and wave (such as diffraction),
and if carried through quantitatively doesn't give accurate results. The possible values of the spin
quantum number are -½ and +½ for electrons.

Classical objects in great numbers can be treated by classical statistics, also known as *Boltzmann
statistics*. This is the basis of properties such as temperature and pressure, which have little
meaning for an individual submicroscopic particle but are properties of an ensemble of such particles.
See 2nd year Thermal Physics for a discussion of Boltzmann
statistics.

Quantum-mechanical particles obey a different sort of statistics which takes into account the discrete, quantised nature of physical properties on very small length scales. When the size of the objects making up the ensemble increases, the quantum statistics approach the classical limit, which is the familiar Boltzmann statistics. The classical description is not wrong, but it is not accurate enough to deal with the smallest of objects.

There are two different quantum statistics (the gory mathematical detail will be left for
3rd year Condensed Matter Physics). The *Fermi-Dirac
statistics* applies to particles with half-integer spin (such as electrons). It is only for these
*Fermions* that the exclusion principle applies. *Bosons*, particles with integer spin,
have their own quantum statistics, *Bose-Einstein statistics*.
Bosons have some quite counter-intuitive properties
at extremely low temperatures which follow from the fact that all of them can occupy the same lowest-energy
state. The helium isotope ^{3}He, for example, is a superfluid, *i.e.* a fluid that flows
without friction. This surprising behaviour is due to the condensation of the ^{3}He atoms in
the lowest energy state, a process known as Bose-Einstein condensation.

The components of angular momentum are quantised. This applies both to orbital angular momentum
(quantum number *l*) and to spin (quantum number *s*).

The uncertainty principle applies to all products of observables whose units combine to [J s] =
[kg m^{2} s^{-1}], *i.e.*
the dimension of Planck's constant. The best-known uncertainty principle is the one linking the
uncertainties of position, *x*, and momentum, *p*. We have also used the energy-time
uncertainty principle when discussing energy bands.

Angular momentum is a vector defined by the cross product
. Judging by its units,
[kg][m]×[m s^{-1}]=[kg m^{2} s^{-1}], the complementary uncertainty
should be dimensionless. It is in fact the uncertainty of the angle of rotation, *φ*.

The uncertainty of the angle must be limited to 2π - the angular momentum has to point *somewhere*
on a circular orbit. Therefore, the uncertainty of the component of the angular momentum perpendicular
to the direction of motion must be of the order of Planck's constant. We may therefore conclude that
values of *L _{z}* are quantised in multiples of

There are 2*l*+1 possible orientations of the angular momentum,
,
and two possible orientations of the spin,
, of an electron. The Fig. shows the situation for
the spin: The spin aligns at an angle with the z axis (which could be a preferential axis generated by
applying a magnetic field or by a chemical bond). There are two possible orientations (these are
often called *parallel* or *up* and *anti-parallel* or *down* although
that's exactly what they are *not*!). The angle with the z axis is given by the need to distribute
the 2*l*+1 states uniformly over the whole angular range from the +z to the -z direction. For example,
in the case of an *l*=1 angular momentum state, there are three possible orientations, one of which
is perpendicular to the z axis, and the other two are 45^{o} of the positive and negative z axis,
respectively. In no case is the angular momentum vector exactly (anti-)parallel to the z axis. If it was,
we would know both the angular momentum *and* the angle precisely, which would defy the uncertainty
principle.