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One way to begin to understand wave-particle duality is to think of a particle as a wave packet, constructed from a superposition of waves. Consider the wave given by the equation y(x, t) = A cos(kx − ωt) or for one instant in time (picking t = 0 for convenience), y(x) = A cos(kx). Restart.

1. Look at this wave. Where it is located?
2. Keep k₂ the same and choose k₁ = 8.0 rad/m.  Now, there is a localized packet. What is the uncertainty in x (measure the distance from one zero amplitude to the next)? What is the uncertainty in k (Δk = |k₁ − k₂|). What is the uncertainty in x and the uncertainty in k for part (a)?
3. Pick several more values of the wave number, k₁.  As the uncertainty in k (Δk) increases, what happens to the uncertainty in x (Δx)?

In general, as the uncertainty in the x position decreases, the uncertainty in the k value increases so that ΔxΔk ~ 1.  Note that this is just an approximate relationship. In fact,   ΔxΔk ≥ 1/2.

Instead of simply adding two waves together with different k values (and possibly different amplitudes), at a time t = 0, we can add a group of waves together. The wave given by the sum (or in the case of a continuous distribution of k values, an integral) of all the superimposed waves is

y(x) = ∫ A(k) cos(kx) dk,                     (5.10)

where the amplitude of each individual wave added together can depend on the k value. Consider the simplest case where the amplitude is equal to 1 over a range of k values and is zero otherwise.

4. Integrate the above equation and show that (again this is at t = 0): y(x) = (2/x) sin(Δkx/2) cos(k₀x), where the amplitude is 1 for k₀ − Δk/2 < k < k₀ + Δk/2.
5. Use the "Wave Packet: set values" button above to see this wave packet (here, k₀ is the average of k₁ and k₂ and Δk = |k₁ − k₂|) and measure Δx for different Δk values.

Sections 8.4-8.7 discuss constructing quantum-mechanical wave packets and the time evolution of such wave packets.