How to Estimate Uncertainty in a Proton's Energy in an Atomic Nucleus

Question:

What is the uncertainty in the proton's energy in an atomic nucleus, given the uncertainty in the position of a proton as the radius of the nucleus?

Answer:

The uncertainty in the proton's energy in the nucleus cannot be numerically estimated without auxiliary information or assumptions because the Heisenberg Uncertainty Principle offers a lower limit not a precise value. The calculation requires the uncertainty in momentum which can then be used to compute the uncertainty in energy assuming non-relativistic speeds.

Explanation:

To estimate the uncertainty in the proton's energy in an atomic nucleus, we must use the principles of Quantum Mechanics, more specifically, the Heisenberg Uncertainty Principle. This principle denotes there is a limit to the precision with which pairs of physical properties of a particle, such as position and momentum (and consequently, energy if non-relativistic approximation is valid), can be simultaneously known.

Applying this principle, if we consider the uncertainty in the position of a proton (Δx) to be the radius of a nucleus (5.0 x 10¹⁵ m as given in the question), the associated uncertainty in the momentum must then be calculated. Once this is estimated, one can use the kinetic energy expression E= p²/2m, where m is the mass of the proton, to compute the uncertainty in energy (ΔE). Note that this computation will assume non-relativistic speeds.

While the Heisenberg Uncertainty principle provides a minimum bound on the uncertainties, it doesn't give an exact value on its own. Therefore, without knowing more about the specific case or using specific quantum mechanical models (such as a particle in a box), a numerical estimate isn't feasible.