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sign (=) or the greater than sign (>) applies depends on the shapes of the probability distributions. The equal sign applies for a shape called a Gaussian after the great mathematician, Karl Friedrich Gauss (1777-1855). The curves shown in Figures 6.5 through 6.7 are Gaussians. A Gaussian is the standard “bell-shaped curve” that describes the distributions of things, such as test scores, for a properly designed test with a large enough number of people taking the test. Gaussian-shaped curves show up in many places in the physical sciences. The greater than sign applies to other shapes. For any shape, which is made up from a specific distribution of waves, it is possible to determine what the product ΔxΔp will be, but it will always be >h/4π unless the shape is a Gaussian.

To understand the nature of the Uncertainty Principle, it is sufficient to consider Gaussian shapes like those in Figure 6.7. Then, ΔxΔp = h/4π. The equation shows what it is possible to know simultaneously about the position and momentum of a particle. h/4π is a constant. Therefore, ΔxΔp equals a constant. So if the uncertainty in the momentum, Δp, is large, then the uncertainty in the position, Δx, must be small, so that the product is h/4π. On the other hand, if Δp is small, then Δx is large. The connection between Δp and Δx is illustrated in Figure 6.7. The uncertainty principle says that you can know something about the momentum of a particle and something about the position of a particle, but you can’t know both the position and the momentum exactly at the same time. This uncertainty in the simultaneous knowledge of the position and the momentum is in sharp contrast to classical mechanics. It is fundamental to classical mechanics theory, as illustrated in Figure 2.5, that the position and the momentum of a particle can be precisely known (measured) simultaneously. Quantum theory states that it is impossible to know both the position and the momentum precisely at the same time. We can know both within some uncertainties, Δx and Δp.

Examining the Uncertainty Principle relationship, ΔxΔp = h/4π, consider what happens as we make Δp smaller and smaller. As Δp becomes smaller and smaller, Δx grows. Dividing both sides of the equation by Δp gives Δx = As Δp becomes smaller, we are dividing by a smaller and smaller number, so Δx grows. As Δp gets closer and closer to zero, Δx gets closer and closer to infinity. In the limit that Δp goes to zero, Δx goes to infinity. This limit has an important meaning. If Δp is zero, the momentum is known precisely, but the position is totally unknown. With Δx = ∞, the particle can be found anywhere with equal probability. This result is in accord with the discussion surrounding Figure 6.1, which shows the wavefunction for a momentum eigenstate. When a particle is in a momentum eigenstate, it has a perfectly well-defined value of its momentum. However, its probability amplitude function, which describes the probability of finding the particle in some region of space, is spread out (delocalized) over all space. The probability of finding the particle anywhere is uniform; Δx = ∞. This is in contrast to the wave packets shown in Figure 6.7, where a superposition of momentum eigenstates produces a state in which there is no longer a perfectly well-defined momentum, but there is some knowledge of the position. We know the position and momentum within some ranges of uncertainty.

If we rearrange the uncertainty relation to give Δp = we see that in the limit that Δx goes to zero (perfect knowledge of the position), Δp goes to infinity. If we know the position perfectly, the momentum can have any value. A wave packet composed of all of the momentum eigenstates (Δp = ∞) has a perfectly well-defined value of the position. It is possible to know p precisely but with no knowledge of x; it is possible to know x precisely, but with no knowledge of p. This is called complementarity. You can know x or p but not both at the same time. In classical mechanics, you can know x and p. In quantum mechanics, you can know x or p. Generally for