When a system is in a state where the measurement of an observable
may take on various values (as in the final location in a
single state diffraction experiment) the state of the system somehow
``contains,'' in a special quantum mechanical sense, the pure states
corresponding to the alternative values of the observable. We say
that the state of the system is a superposition of those pure
states of the observable which represent the possible outcomes. As we
can imagine a quantum state composed of any combination of pure
states, the superposition of any two states leads to a new valid
quantum state. We express the physical concept of the composition of
an arbitrary state in terms of pure states symbolically as
where the 
represent pure states of the observable
. Note that if the allowed values of 
are
continuous, then we would write an integral instead
Because the measurement of an observable must always yield some value
and a system in any state 
may be measured, it is a
general principle of quantum mechanics that all states may be written
as superpositions of the pure states of any valid physical observable.