## Section 5.10: Exploring the Uncertainty Principle

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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* − ω*t*) or for one instant in time (picking *t* = 0 for convenience), *y*(*x*) = *A* cos(*kx*). Restart.

- Look at this wave. Where it is
*located*? - Keep
*k*_{2}the same and choose*k*_{1}= 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*(Δ*k*= |*k*_{1 }−*k*_{2}|). What is the uncertainty in x and the uncertainty in*k*for part (a)? - Pick several more values of the wave number,
*k*_{1}. As the uncertainty in*k*(Δ*k*) increases, what happens to the uncertainty in*x*(Δ*x*)?

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

- Integrate the above equation and show that (again this is at
*t*= 0):*y*(*x*) = (2/*x*) sin(Δ*kx*/2) cos(*k*_{0}*x*), where the amplitude is 1 for*k*_{0 }− Δ*k*/2 <*k*<*k*_{0 }+ Δ*k*/2. - Use the "Wave Packet: set values" button above to see this wave packet (here,
*k*_{0}is the average of*k*_{1}and*k*_{2}and Δ*k*= |*k*_{1 }−*k*_{2}|) and measure Δ*x*for different Δ*k*values.

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