Section 5.10: Exploring the Uncertainty Principle

k1 = rad/m | k2 = rad/m

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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 kk = |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 kk) increases, what happens to the uncertainty in xx)?

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.

  1. 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.
  2. 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.