Wave packets and the Fourier transform

Fourier transform

Problem:

Find the Fourier transform of the δ function δ(x-  x0),

Solution:

Problem:

The wave function of a particle at t = 0 is 
Ψ(x) = 1/L½,  |x| < L/2,  Ψ(x) = 0 otherwise. 
At t = 0, what possible values of the momentum of the particle can be found, and with what probability?

Solution:

Problem:

Consider a free particle which is described at t = 0 by the normalized Gaussian wave function ψ(x,0) = Nexp(ik0x)exp(-ax2).
(a)  Normalize the wave function.
(b)  Find the probability density |ψ(x,0)|2 of the particle. 
(c)  Find its Fourier transform Φ(k,0) of the wave function and the probability density
|Φ(k,0)|2 in k-space.

Solution:


Evolution of a wave packet

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) = Nexp(-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:

Problem:

In one dimension, at t = 0 the normalized wave function of a free particle in momentum space is
Φ(p,0) = Nexp(-(p – p0)2/(2b2ħ2))exp(-ipx0/ħ).
(a)  Find the normalization constant N.  Find the expectation value <p>.
(b)  Find the FWHM in of |Φ(p,0)|2 in momentum space.
(c)  Find the corresponding wave packet Ψ(x,0) in coordinate space.  Find the expectation value <x>.
(d)  Find the FWHM in of |Ψ(x,0)|2 in coordinate space.

Solution:

Problem:

The wave packet for a quantum mechanical particle of mass m in one dimension is described by

Ψ(x,t) = [1/(2π)½] lim(R-->∞) ∫-R+R dk Φ(k) exp(i(kx - ω(k)t)),

where  Φ(k) = N exp(-(k – k0)2/(4(Δk)2)) is a “strongly peaked” distribution around k = k0 with ΔkΔx = ½ at t = 0.
(a)  Show that  ψ(x, 0) =  [Δx(2π)] exp(ik0x)exp(-x2/(4Δx2)) (evaluate the integral),
and find the probability density |ψ(x, 0)|2 of the particle.  Use ΔkΔx = ½  to eliminate Δk.  Show your work!
(b)  For a free particle E = ħω = ħ2k2/(2m).  Show that for a free particle
|ψ(x, t)|2 = exp[-½ (x - ħk0t/m)2/Δx(t)2]/[ (2π)½ Δx(t)],
with Δx(t) = (Δx2 + ħ2t2/(4m2Δx2)½.
(c)  Determine the group velocity of the wave packet.
(d)  Evaluate the time it takes for the wave-packet to double in spatial extent, specifically if the particle is an electron and Δx ~10 nm, at t = 0.

Useful integrals: 
-∞+∞exp(-αx2)dx = √(π/α) 
-∞+∞exp(-αx2 + iβx))dx = (π/α)½exp(-β2/(4α))
Note:  You can complete some parts of the problem by using the given results without evaluating the integrals.

Solution: