Question

In the beginning of the twentieth century, some scientists thought that a nucleus may contain both electrons and protons. Use the Heisenberg uncertainty principle to show that an electron cannot be confined within a nucleus. Repeat the calculation for a proton. Comment on your results. Assume the radius of a nucleus to be $1.0\times {10}^{-15}\mathrm{m}$. The masses of an electron and a proton are $9.109\times {10}^{-31}\mathrm{kg}$ and $1.673\times {10}^{-27}\mathrm{kg},$ respectively. (Hint: Treat the radius of the nucleus as the uncertaintv in position.)

Short answer

Electrons cannot be confined in a nucleus due to high kinetic energy, while protons can be.

Step-by-step solution

Understand the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is expressed as $\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$, where $\mathrm{\Delta }x$ is the uncertainty in position and $\mathrm{\Delta }p$ is the uncertainty in momentum. $\hslash$ is the reduced Planck's constant, approximately $1.0546\times {10}^{-34}{{\mathrm{m}}^2\mathrm{kg}\mathrm{s}}^{-1}$. In this problem, the radius of the nucleus is treated as $\mathrm{\Delta }x$.

Calculate the Minimum Uncertainty in Momentum for an Electron

Using $\mathrm{\Delta }x=1.0\times {10}^{-15}\mathrm{m}$, calculate $\mathrm{\Delta }p$ for an electron:$$\mathrm{\Delta }p=\frac{\hslash }{2\mathrm{\Delta }x}=\frac{1.0546\times {10}^{-34}}{2\times 1.0\times {10}^{-15}}\approx 5.273\times {10}^{-20}\mathrm{kg}\mathrm{m}/\mathrm{s}$$.

Calculate the Minimum Kinetic Energy for an Electron

The kinetic energy $K$ can be calculated from momentum $p$ using $K=\frac{p^2}{2m}$. Substituting $\mathrm{\Delta }p$ and the electron mass $9.109\times {10}^{-31}\mathrm{kg}$:$$K=\frac{(5.273\times {10}^{-20}{)}^2}{2\times 9.109\times {10}^{-31}}\approx 1.527\times {10}^{-13}\mathrm{J}$$.

Interpret Electron Results

The calculated minimum kinetic energy for the electron is very high compared to typical energy scales binding electrons, suggesting that confining an electron within a nucleus is not feasible.

Calculate the Minimum Uncertainty in Momentum for a Proton

Repeat the computation of $\mathrm{\Delta }p$ with the same $\mathrm{\Delta }x$:$$\mathrm{\Delta }p=\frac{\hslash }{2\times 1.0\times {10}^{-15}}\approx 5.273\times {10}^{-20}\mathrm{kg}\mathrm{m}/\mathrm{s}$$.

Calculate the Minimum Kinetic Energy for a Proton

Using $K=\frac{p^2}{2m}$ for the proton with mass $1.673\times {10}^{-27}\mathrm{kg}$:$$K=\frac{(5.273\times {10}^{-20}{)}^2}{2\times 1.673\times {10}^{-27}}\approx 8.3\times {10}^{-14}\mathrm{J}$$.

Interpret Proton Results

The calculated minimum kinetic energy for the proton is reasonable for nuclear energy levels, so a proton can be confined within a nucleus. This is consistent with our understanding of the nucleus containing protons but not electrons.

Key concepts

Nuclear Physics

Nuclear physics is a fascinating branch of physics that explores the constituents and behavior of atomic nuclei. The nucleus, located at the center of an atom, is made up of protons and neutrons. Each of these particles plays a crucial role in the stability and properties of the atom. During the early twentieth century, scientists were curious about the internal structure of the nucleus, even considering that it might contain electrons. However, modern nuclear physics has unveiled that the nucleus is comprised of protons and neutrons, and interactions between these particles are responsible for nuclear forces, an essential concept in understanding nuclear reactions and energy generation. Nuclear physics not only aids in explaining the composition of atoms but also paves the way for nuclear energy and medical applications like radiation therapy.

Electron Confinement

The Heisenberg Uncertainty Principle gives us a fundamental limit on how precisely we can know the position and momentum of a particle simultaneously. When scientists considered the possibility of electrons within a nucleus, they turned to this principle to evaluate feasibility.For an electron, this principle suggests that as we attempt to confine it within a minuscule space like a nucleus, the uncertainty in its momentum becomes exceedingly large. This is expressed as the limitation: $$\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$$Here, $\mathrm{\Delta }x$ represents the uncertainty in position, approximated by the nuclear radius, $1.0\times {10}^{-15}\mathrm{m}$. The resulting large momentum uncertainty ($\mathrm{\Delta }p$) translates into a very high kinetic energy for the electron, calculated as approximately $1.527\times {10}^{-13}\mathrm{J}$. This energy is too high for typical nuclear binding energies and confirms that electrons cannot be feasibly confined within a nucleus.

Proton Confinement

Protons, unlike electrons, are indeed found within the nucleus, which poses an interesting question: how is it possible for protons but not electrons? Applying Heisenberg's principle using the same nuclear constraints provides insight. For protons, even though the uncertainty in momentum ($\mathrm{\Delta }p$) is the same as for electrons, their mass $(1.673\times {10}^{-27}\mathrm{kg})$ plays a significant role in calculating kinetic energy:$$K=\frac{(5.273\times {10}^{-20}{)}^2}{2\times 1.673\times {10}^{-27}}\approx 8.3\times {10}^{-14}\mathrm{J}$$This value is much more compatible with nuclear energy levels. Hence, protons can be confined within the nucleus due to their greater mass reducing the kinetic energy to a level that aligns with nuclear forces, allowing them to remain confined without such high energies as electrons would exhibit.

Kinetic Energy Calculation

Kinetic energy represents the energy of motion of particles, influenced by mass and velocity. In our context, using Heisenberg's principle, the kinetic energy is derived from the uncertainty in momentum, which correlates to the calculated high momenta outcomes.The relationship is defined by:$$K=\frac{p^2}{2m}$$Where $p$ is the uncertainty in momentum and $m$ is the mass of the particle. For physics students, understanding this calculation is crucial as it illustrates why certain particles can exist within the constraints of a nucleus while others cannot.By seeing how these mathematical principles compute kinetic energy, students can appreciate the balance between particle mass, momentum, and the feasibility of particle confinement within the nuclear domain. The distinctions in calculated energies demonstrate why electrons, lightweight with high kinetic energies when densely confined, act differently from protons under the same constraints.