Let us consider a particular microscopic system in a particular state, which we
label : *e.g.*, a photon with a particular energy, momentum, and polarization.
We can represent this state as a particular vector, which we also
label , residing in some vector space, where the other elements of the space
represent all of the other possible states of the system. Such a space
is called a *ket space* (after Dirac). The state vector is
conventionally written

(1) |

(2) |

Suppose that we want to construct a state whose plane of polarization makes
an arbitrary angle with the -direction. We can do this
via a suitably weighted superposition of states and . By analogy
with classical physics, we require of state , and
of state . This new state is represented by

(4) |

(5) |

We have seen, in Eq. (3),
that any plane polarized state of a photon can be represented
as a linear superposition of two orthogonal polarization states
in which the weights are real numbers. Suppose that
we want to construct a circularly polarized photon state. Well, we know from
classical physics that a circularly polarized wave is a superposition of two
waves of equal amplitude, plane polarized in orthogonal directions,
which are in *phase quadrature*. This suggests that a circularly polarized photon
is the superposition of a photon polarized in the -direction (state )
and a photon polarized in the -direction (state ), with equal weights given
to the two states, but with the proviso that state is out of phase
with state . By analogy with classical physics, we can use *complex numbers*
to simultaneously represent the weighting and relative phase in a
linear superposition. Thus, a circularly polarized photon is represented by

(6) |

(7) |

Suppose that the ket is expressible linearly in terms of the kets
and , so that

(8) |

The dimensionality of a conventional vector space is defined as the number
of independent vectors contained in the space. Likewise, the dimensionality
of a ket space is equivalent to the number of independent ket vectors it contains.
Thus, the ket space which represents the possible polarization
states of a photon propagating in the -direction is two-dimensional
(the two independent vectors correspond to photons plane polarized in the
- and -directions, respectively). Some microscopic
systems have a finite number of independent states (*e.g.*, the spin states
of an electron in a magnetic field). If there are independent states,
then the possible states of the
system are represented as an -dimensional ket space. Some microscopic
systems have a denumerably infinite number of independent states (*e.g.*,
a particle in an infinitely deep, one-dimensional potential well).
The possible states of such a system are represented as a ket space whose
dimensions are denumerably infinite. Such a space can be treated in more or
less the same manner as a finite-dimensional space. Unfortunately, some
microscopic systems have a nondenumerably infinite number of independent states
(*e.g.*, a free particle). The possible states of such a system are represented
as a ket space whose dimensions are nondenumerably infinite. This type of
space requires a slightly different treatment to spaces of finite, or
denumerably infinite, dimensions.

In conclusion, the states of a general microscopic system can be represented
as a complex vector space of (possibly) infinite dimensions. Such a space
is termed a *Hilbert space* by mathematicians.