The physical state of a quantum system is represented by a vector denoted
which is a column vector, whose components are probability amplitudes for different states in which the system might be found if a measurement were made on it.
A probability amplitude 
is a complex number, the square modulus of which
gives the corresponding probability 
The number of components of 
is equal to the number of possible states in which
the system might observed. The space that contains is called a
Hilbert space 
. The dimension of is also equal to the
number of states in which the system might be observed. It could be finite
or infinite (countable or not).
must be a unit vector. This means that the inner product:
In the above, if the vector , known as a Dirac ``ket'' vector,
is given by the column
then the vector 
, known as a Dirac ``bra'' vector, is given by
so that the inner product becomes
We can understand the meaning of this by noting that 
, the components
of the state vector, are probability amplitudes, and 
are
the corresponding probabilities. The above condition then implies that
the sum of all the probabilities of being in the various possible states
is 1, which we know must be true for probabilities.