# Using Heisenberg's Uncertainty Principle to find the kinetic energy of an atomic particle

I'm working on a problem set for my modern physics course and a couple of the problems have asked me to use Heisenberg's uncertainty principle, given atomic nuclear radius (or uncertainty of an electron's position), to find the uncertainty of the particle's momentum. This is of course not a problem. It then however asks me to use that value to estimate the kinetic energy of the particle. I'm confused as to why the uncertainty in momentum can be used to find that, rather than some experimentally determined average momentum. It seems to me what I'm finding is the uncertainty in the kinetic energy, but the problem seems to be suggesting it is the actual value, unless of course I'm misunderstanding it.

The problem I'm referring to states:

"The nucleus of a gold atom has a radius of 7.0 fm. Estimate the kinetic energy of a proton or neutron confined to a gold nucleus."

The problem prior asked me to do the same thing for an electron with a given size of space it was confined to.

Relativity was quite confusing already, but this confusion only seems to be increasing. This is slightly unrelated, but I've never experienced conceptual misunderstanding to this degree in any other physics course I've taken, and if anybody who happens to see this has any advice it would be greatly appreciated!

• Have you had a course in quantum mechanics? – Oct 12, 2022 at 4:51
• @annav I haven't, although I am scheduled to next semester. In this course as well as my intro chemistry courses, however, I've experienced some very rudimentary, basic quantum theory (such as quantum numbers). Oct 12, 2022 at 5:06
• The point is that the Heisenber uncertainty principle was proposed at the necessity of having quanutm mechanics was developing, and is an envelope that can be derived from the theory that formed afterwards."'Im confused..........rather than some experimentally determined average momentum. " In quantum states one cannot measure momenta per particle , one can only predict an accumulation of measurements . The quantum mechanical wavefunction can predict the distribution. The HUP is a way to use the quantum knolwledge in getting numbers as you are asked. Oct 12, 2022 at 5:55
• I should qualify that you cannot predict the track even of free particles after the interaction (as your question involves bound particles within the nucleus). Only accumulatins of measurents can be predicted at the quantum mechanical level. this recent answer of mine might help you understand what "accumulation is" physics.stackexchange.com/questions/731578/… Oct 12, 2022 at 6:25

If we assume that a nucleon is in its ground state, it should have as small an energy as physically possible. The uncertainty principle provides a fundamental limit on how small that energy can be as $$\Delta p$$, the uncertainty in the particle's momentum. Therefore, we can use the $$\Delta p$$ which satisfies the uncertainty principle limit for a given $$\Delta x$$ (i.e. $$\Delta p = \frac{\hbar}{2 \Delta x}$$) to calculate an estimate for the minimum energy of the nucleon.