Modern Physics: Chapter 4 – Wave Groups

In the last chapter, I discussed matter waves. Now, when one thinks about waves in general, one visualizes plane waves. Essentially something like this:

plane waves

Now, the mathematical function for a moving plane wave is of the form

  • f(x, t) = Acos(kx - wt)

where k = \frac{2 \pi}{\lambda} and w = 2 \pi f

You can use the following matlab script to generate a moving plane wave:


%Author: Muhammad A. Tirmazi
%Date: March 27, 2015
%Generating Moving Plane Waves

syms f x t;

A = 3;
k = 2;
w = 2;

f(x, t) = A*cos(k*x - w*t);

ix = 0:0.1:10;

for t = 0:0.1:60
    tic;
    iy = double(f(ix, t));
    plot(ix, iy);
    pause(0.1 - toc);
end

Since the intensity matter waves describe the probability of finding a particle at a specific point in space, and since particles are limited to specific regions of space (the probability of finding them can’t just be equal at every point in the universe, the earth for example is far far far far far far more likely to be found in its orbit in the solar system than in the andromeda galaxy), plane waves of infinite extend and the same amplitude throughout are an incorrect representation for a moving particle.

The matter wave associated with a localized moving particle is, in fact, a wave packet. This is what a wave-packet looks like:

wave groups

Wave packets are created by the super-position of waves with similar but not exactly equal wave-numbers and angular frequencies. In other words, the mathematical function for a simple wave-packet created by the super-position of two waves is something like:

f(x, t) = Acos(k_{1}x - w_{1}t) + Acos(k_{2}x + w_{2}t)

You can use the following matlab script to generate a moving wave-packet:

%Author: Muhammad A. Tirmazi
%Date: March 27, 2015
%Generating Moving Wave Groups

syms f g h x t;

A = 3;
k1 = 4;
w1 = 4;
k2 = 4.3;
w2 = 4.3;

f(x, t) = A*cos(k1*x - w1*t);
g(x, t) = A*cos(k2*x - w2*t);
h(x, t) = f(x, t) + g(x, t);

ix = 0:0.1:20;

for t = 0:0.1:60
 tic;
 iy = double(h(ix, t));
 plot(ix, iy);
 pause(0.1 - toc); 
end

The mathematical function describing a simple wave-packet can be simplified by using the identity

  • cos a = cos b = 2 \times cos \frac{1}{2}(a - b) \times cos \frac{1}{2}(a + b)

This results in a function of the form:

  • f(x, t) = 2Acos(\frac{\Delta x}{2}k - \frac{\Delta w}{2}t) \times cos(\frac{k_{1} + k_{2}}{2}x - \frac{w_{1} + w_{2}}{2}t)

where \Delta k = k_{2} - k_{1} and \Delta w = w_{2} - w_{1}.

However, we still have a problem with this function. A simple superposition of two slightly different waves generates wave-pulses that repeat at a constant interval. To represent a localized moving particle, we need a localized wave-packet. Essentially, we need something like this:

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