Chapter 4: Q7CQ (page 133)

Question: If a particle’s position uncertainty is zero, what can be said of its momentum uncertainty? If a particle’s position uncertainty is infinite, what can be said of its momentum uncertainty?

Short Answer

Expert verified

Answer:

For , $∆ x = 0, ∆ p$ has to be $\infty$ , however, for $∆ x = \infty$ , then $∆ p$ could carry any value including zero.

Step by step solution

Uncertainty relation

Mathematically, the uncertainty relation can be stated as $Δ x Δ p ⩾ \frac{h}{2}$ .

Explanation

Because this is the only method to preserve the uncertainty relation, which must be maintained for whatever measurements we take, if $∆ x$ is zero, we know that must $∆ p$ be $\infty$ . Physically, these degrees of uncertainty correspond to a particle whose real momentum cannot be established but whose position is completely determined (probability density at a single point).

Regarding the infinite uncertainty in location $∆ x = \infty$ , we are aware that any value of , up to and including zero, will result in the satisfaction of the uncertainty relation.

This condition is analogous to an infinitely wide wave that is capable of carrying any value of momentum.

Thus, $∆ x = 0$ $∆ p$ for $\infty$ , has to be , however, for $∆ x = \infty$ , then $∆ p$ could carry any value including zero