## Building a Localized Single-Particle Wave Packet

We now have a wave function for a free particle with a definite momentum

where the wave number is defined by and the angular frequency satisfies . It is not localized since everywhere.

We would like a state which is localized and normalized to one particle.

To make a wave packet which is localized in space, we must add components of different wave number. Recall that we can use a Fourier Series to compose any function when we limit the range to . We do not want to limit our states in , so we will take the limit that . In that limit, every wave number is allowed so the sum turns into an integral. The result is the very closely related Fourier Transform

with coefficients which are computable,

The normalizations of and are the same (with this symmetric form) and both can represent probability amplitudes.

We understand as a wave packet made up of definite momentum terms . The coefficient of each term is . The probability for a particle to be found in a region around some value of is . The probability for a particle to have wave number in region around some value of is . (Remember that so the momentum distribution is very closely related. We work with for a while for economy of notation.)

Jim Branson 2013-04-22