When Δp = ∞, it means that the uncertainty in momentum is infinitely large, not that the actual momentum itself is infinite. This distinction is crucial in quantum mechanics.

Explanation:

1. Uncertainty Interpretation:

The Heisenberg uncertainty principle states:

If Δp is infinite, then Δx must be zero, meaning the particle’s position is known exactly. However, this does not mean the actual momentum p is infinite—it only means that the system does not have a well-defined momentum.

2. Example: Plane Wave States

Consider a plane wave described by:

This function extends infinitely in space, meaning Δx = 0. Since the Fourier transform of a delta function is a constant, the momentum is perfectly defined, with zero uncertainty (Δp = 0). The opposite case occurs when the wave function is a localized delta function:

Here, the position is perfectly known (Δx = 0), but the Fourier transform of δ(x) is a uniform distribution, meaning all momentum values are equally probable (Δp = ∞).

3. Physical Meaning:

A state with infinite uncertainty in momentum means that the particle’s momentum can take any value, but it does not mean that the particle necessarily has infinite momentum. Instead, it lacks a well-defined momentum value entirely.

This subtlety highlights the difference between knowing a precise quantity and having an undefined range of possible values.

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