Spreading of the wave packet

Spreading of a wave packet
We can terminate the Taylor series expansion
β(k) = β(k0) + (k - k0)dβ(k)/dk|k0 + (k - k0)2d2β(k)/dk2|k0  + ...
after the first term if all higher order terms are much smaller than 2π.  However, as t increases, the higher order terms increase steadily with t.  Therefore the second order term must be included for times greater than some t1.
We have
d2β(k)/dk2|k0 = d2α(k)/dk2|k0 + d2ω(k)/d2k|k0t,   d2ω(k)/d2k =  ħ/m.
Therefore d2β(k)/dk2|k0 depends on t, and for large enough times t its magnitude increases with time.
The wave packet is the of the form
Ψ(x,t) ∝ ∫|g(k)| exp(i((k - k0)(x - x0 - vgt) - (k - k0)2x1) dk,   x1 = -d2β(k)/dk2|k0.
x1 depends on t, and for large enough times t its magnitude increases with time.
Ψ(x,t) ∝ ∫|g(k)| exp(i2π(k - k0)/τ) dk,  with  τ = 2π/((x - x0 - vgt) - (k - k0)x1).
Now |x - x0 - vt |>> Δk-1 is no longer sufficient for |τ| << Δk. 
We need |(x - (k - k0)x1) - x0 - vt | >> Δk-1 for the integrand to oscillate rapidly and the contribution to the integral to be zero. 
The width of the wave packet increases with time.

Example:
Let |g(k)| = 1 for k0 < k < (k0 + π).  Then
Ψ(x,t) ∝ ∫k0k0+π exp(i(k - k0)(x - x0 - vgt) - (k - k0)2x1) dk
= ∫k0k0+π cos((k - k0)(x - x0 - vgt) - (k - k0)2x1) dk + i∫k0k0+π sin((k - k0)(x - x0 - vgt) - (k - k0)2x1) dk.
P(x,t) = |ψ(x, 0)|2 = [∫k0k0+π cos((k - k0)(x - x0 - vgt) - (k - k0)2x1) dk]2
+ [∫k0k0+π sin((k - k0)(x - x0 - vgt) - (k - k0)2x1) dk]2.
Let x - x0 - vgt = a, x1 = b.
Compare P(x,t) at t = 0 when b = 0 and at some later time when b ≠ 0.

Problem:
In one dimension, at t = 0 the normalized wave function of a free particle of mass m in k-space is
Φ(k,0) = N exp(-k2/(2b2)).
(a)  Find the normalization constant N.  Find the expectation value <p> = ħ<k>.
(b)  Find the FWHM in of |Φ(k,0)|2 in k-space.
(c)  Find the corresponding wave packet Ψ(x,0) in coordinate space.  Find <x>.
(d)  Find the FWHM in of |Ψ(x,0)|2 in coordinate space.
(e)  Find the FWHM in of |Ψ(x,t)|2 an some later time t.  Does it change with time?
(f)  Find the FWHM in of |Φ(k,t)|.  Does it change with time?

Hint:  ∫-∞+∞exp(-a2(x + c)2)dx = √π/a

Solution:


Summary
The wave function of a free particle is a wave packet.  The center of this wave packet moves with constant velocity vg, called the group velocity.  The width of the wave packet increases with time, the wave packet spreads, we gradually loose position information.