Wolfram Alpha:

```Wavepackets
-----------

Wavefunctions of the form ψ(x) = Aexp(ipx/h) are referred to as FREE PARTICLE
wavefunctions and are constant throughout space.  As such they are somewhat
artificial (but nonetheless very useful) and do not represent a localized
particle.  To represent a localized particle it is necessary to construct a
WAVEPACKET by superpositioning basis vectors (plane waves).

Consider the Gaussian distribution:

P(x) = (1/√2πσx2)exp(-[x - xo]2/2σx2)

Construct a wavepacket of the form:

ψ(x) = (1/2πα)1/4exp(-x2/4α)exp(ikox)

where (1/2πα)1/4 is the normalization factor and 4α = 2σx2.

This looks like:

|ψ(x)> = ΣkAk|exp(ikox)>

Where,

A(k) = (1/√2π)∫ψ(x)exp(-ikx)dx

= (1/√2π)∫(1/2πα)1/4exp(-x2/4α)exp(ikox)exp(-ikx)dx

= (2α/π)1/4exp(-α[k - ko]2)

So we can write out original wavefunction in an equivalent form as:

ψ(x) = Σk(2α/π)1/4exp(-α[k - ko]2)exp(ikx)

For a free electron E = h2k2/2m = hω

∴ ω = hk2/2m

Phase velocity:

vp = fλ = ω/k since k = 2π/λ

vp = ω/k = hk/2m

Group velocity:

vg = ∂ω/∂k = hk/m```