Delta p = infinity does not mean momentum is infinite
Understanding Infinite Uncertainty in Momentum
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.
Leave a Reply