# How the Heisenberg principle holds for protons?

Does this principle holds for protons since their position is, in a manner of speaking, known and they are also assumed to be stationary at every popular physics book that I have read, including the school textbooks.

• Protons and neutrons move in orbitals, but introductory treatments don't usually mention that. Commented Jun 25, 2018 at 10:55
• I would say that the uncertainty in the position of the nucleus is negligible with respect to the size of the electrons orbits. Commented Jun 26, 2018 at 8:13

The Heisenberg uncertainty principle has quantum mechanical origin and in general holds for all bodies. Wave nature can be attributed to even macroscopic bodies courtesy of de Broglie's proposal of matter waves and hence an associated uncertainty in position.

Protons definitely have a wave character and there is a finite uncertainty in the position. I don't think your statement about protons being considered stationary is correct, you may be confusing protons with nuclei. In many body problems, the Hamiltonian of the nuclei and the electrons are considered separately as a first approximation (Born-Oppenheimer approximation). This is justified since the nuclei are much heavier than electrons and hence their motion is much more slower. As a further approximation, the nuclei are considered stationary. But it is important to understand that these approximations are just considered to simplify the many body problem for the electronic motion. The nuclei are not stationary and do not have fixed positions.

I think you are confusing several different concepts. "The proton" is assumed fixed in simple non-relativistic QM models of the electron's motion. However this is something we imposed on the proton to make the equations solvable. A more realistic model would include the proton moving, perhaps the proton-electron system with "reduced mass" moving around the CoM. This is more noticeable in positronium (e- and e+) rather than Hydrogen (s- and p). If you wanted to study the QM behavior of a proton beam and a potential barrier you would most certainly have an uncertainty relation for the proton. More specifically the uncertainty relation does not hold for specific particles, it holds for kinematic variables associated with any particle.

The uncertainty principle says: if there is a given amount of uncertainty in position, there must be at least a corresponding amount of uncertainty in momentum.

Now consider a specific amount of uncertainty in momentum. Momentum equals mass times velocity; uncertainty in momentum equals mass times uncertainty in velocity.

The greater the mass, the less uncertainty in velocity is required, in order to obtain the required amount of momentum uncertainty. So the heavier a particle is, the more localized its wavefunction can be, without violating the uncertainty principle.

Equations here: https://chemistry.stackexchange.com/a/39567

If you consider the Hydrogen atom (1 proton, 1 electron), you can reformulate the 2 bodies problem in two equations, one for the motion of the center of mass of the atom, and one for an effective mass representing the relative motion of the electron and proton around each others. Wikipedia : two bodies system (classical)

This is still true in quantum mechanics. With this reformulation you can consider the quantum aspects of the proton, together with the electron for this system.