Free wave packet

In this section, we are going to describe some important features about wave packets, which is the way to explain the particle wave duality in Quantum Mechanics.

Moving free particles are some times represented by special wave functions called wave packets. Most of the time, the momentum of a particle is not know with complete certainty, that is, it doesn't have a definite wavelength. Their wave functions are rather a sum of waves which cover a spectrum of wave numbers. The easiest way to represent this is through a Gaussian wave packet, which is nothing more than a sum of plane waves with amplitudes that vary according to a Gaussian distribution. If the wave number of maximum amplitude corresponds to k_0, then the normalized wave function of such particle at t=0 is

$$\psi(x,0) =\left( \frac{2}{\pi a^2} \right)^{1/4}e^{ik_0x}e^{-x^2/a^2}$$

The time evolution of such packet can be found by multiplying each plane wave by e^{-iwt}. After integration, the probability density of the particle in code units is given by the following expression:

$$|\psi(x,t)|^2 = \sqrt{\frac{2}{\pi a^2}}\frac{1}{\sqrt{1+\frac{4t^2}{a^4}}}\exp \left\{ -\frac{2(x-k_0t)^2}{a^2+\frac{4t^2}{a^2}} \right\}$$

What we do is evolve the initial conditions and illustrate the dynamics of the wave packet and the results appear in the animation below. What happens is that the Gaussian density widens and moves to the right while time passes.

The parameters used for this particular animation, were the following:

Numerical Domain
xmin = -1
xmax =  3
Nx = 2000
CFL = 0.125
Nt = 40000

Initial Conditions
k_0 = 100
a = 0.1

The animation also illustrates the difference between the phase and group velocities. The blue arrow is moving at the same speed as the maximum of the wave packet, while the red arrow is moving at the same speed as one of the maximums of the red graph (the real part of the wave function). We can see that the group velocity (blue arrow) is twice as fast as the phase velocity (red arrow). All of this comes from the dispersion relation of Schrödinger's equation, which says that:

$$\hbar \omega = \frac{\hbar^2 k^2}{2m}$$

The group velocity is:

$$v_g = \frac{d\omega}{dk} = \frac{\hbar k}{m},$$

which for a particle with definite momentum, concides with the velocity of the particle. On the other hand, the phase velocity is given by the following formula:

$$v_p = \frac{\omega}{k} = \frac{\hbar k}{2m} = \frac{v_g}{2},$$

which is exactly half the group velocity. So, in the following examples, we are going to mainly care about the group velocity, which is the speed at which the Gaussian moves and consides with the velocity of the classical analog.