The direct connection between the physical interpretation of
as a beam of particles and our formal theory comes through
the TDSE. The pure states of energy are stationary, and
the connection between them and the dynamical evolution pictured in
Figure 2 is not direct. Producing a wave packet
with an identifiable location in space requires taking
superpositions of the states .
Mathematically, a wave packet evolving under the TDSE is such a superposition of pure energy states times multiplied with the appropriate time-dependent phase factors,
where 
is sharply peaked near 
. The mathematical
form of the integral (4) naturally guarantees that the
resulting wave packet 
will be confined to a particular
region of space because the integral is essentially essentially a sum
of complex numbers with varying phases. For most values of x, these
phases vary rapidly, resulting in much cancellation and a small
absolute value of the integral.
To sketch the behavior of the integral, we write
as the product of its amplitude and a complex phase
where 
describes the phase of the packet.
Note is peaked about as in Figure 4.
Figure 4: Integrand of an integral to be analyzed using the method of stationary phase
With this separation we may rewrite (4) as the integral of the product of real amplitudes with complex phases,
Such integrals are best analyzed using the method of stationary phase as described in the next section.
