The mutual dependence of the uncertainty in position and the uncertainty in momentum was first stated by Werner Heisenberg. His statement is known as Heisenberg's Uncertainty Principle. It says that the uncertainty in position and the uncertainty in momentum are closely related. If one decreases, the other increases by the same factor. Mathematically, the Uncertainty Principle says:
(Uncertainty in position) x (Uncertainty in momentum) = constant
Or, if you like symbols for uncertainty in position (Dx) and uncertainty in momentum (Dp).
Dx × D p = constant
The constant is Planck's constant (h) divided by 2p:
This principle states that we can never know both the exact position and the exact momentum of an electron at the same time. The best we could do is that the uncertainties are related by this equation. We would get these results only with perfect measuring instruments. We can always do worse. For this reason, Heisenberg's Uncertainty Principle is usually stated as an inequality:
(Uncertainty in position) x (Uncertainty in momentum) is at least h/2p
For example, suppose we establish the position of an electron precisely to within a tenth of a nanometer (0.0000000001 m). Using Dx = 10-10 m, what would be the minimum uncertainty in the electron's momentum (Dp)? (The value of Planck's constant is h = 6.63 × 10-34 J×s)
With this uncertainty in momentum, what would be the corresponding uncertainty in the speed of the electron? (The electron's mass is 9.11 × 10 -31 kg.) Recall that: momentum = mass × velocity.
A reasonable speed for an electron might be around 10 6 m/s. Is the uncertainty in speed that you calculated significant when compared to this speed? Explain.
***************** To begin let's use the Wave Packet Explorer program to create wave functions with different uncertainties in momentum.
Use Wave Function Explorer to produce wave functions as shown in Figure1.
(a) (b) (c) (d) (e)
Figure 1: Wave functions with different uncertainties in position.
Compare the result of all groups. Which one needed to include the largest number of momenta?
Which one needed to include the smallest number of momenta?
How is the uncertainty in position related to the uncertainty in momentum?
This exercise indicates that as one of the uncertainties increases the other decreases. This conclusion is built into the wave nature of matter. It does not depend on our measurement instruments. (We have not discussed measurement here - only creating wave functions.)