Question

The radii of atomic nuclei are of the order of 5.0 $\times$ 10${}^{-15}$ m. (a) Estimate the minimum uncertainty in the momentum of an electron if it is confined within a nucleus. (b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the relativistic relationship between energy and momentum, Eq. (37.39), to obtain an estimate of the kinetic energy of an electron confined within a nucleus. (c) Compare the energy calculated in part (b) to the magnitude of the Coulomb potential energy of a proton and an electron separated by 5.0 $\times$ 10${}^{-15}$ m. On the basis of your result, could there be electrons within the nucleus? ($Note$: It is interesting to compare this result to that of Problem 39.72.)

Short answer

No, electrons cannot exist in the nucleus due to high kinetic energy exceeding potential energy.

Step-by-step solution

Estimate Minimum Uncertainty in Momentum

Use the Heisenberg Uncertainty Principle: $$\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$$where $\mathrm{\Delta }x$ is the uncertainty in position, approximately the radius of the nucleus, $5.0\times {10}^{-15}$ m, and $\hslash \approx 1.05\times {10}^{-34}$ J·s.Thus, the minimum uncertainty in momentum ($\mathrm{\Delta }p$) is: $$\mathrm{\Delta }p\ge \frac{\hslash }{2\mathrm{\Delta }x}=\frac{1.05\times {10}^{-34}\text{J}\cdot \text{s}}{2\times 5.0\times {10}^{-15}\text{m}}\approx 1.05\times {10}^{-20}\text{kg}\cdot \text{m/s}$$

Estimate the Magnitude of the Momentum

According to part (b) of the problem, take the uncertainty in momentum calculated as the approximate magnitude of the momentum, i.e.,$$p\approx 1.05\times {10}^{-20}\text{kg}\cdot \text{m/s}$$

Calculate Kinetic Energy Using Relativistic Relationship

Use the relativistic energy-momentum relation:$$E^2=(pc{)}^2+(m_0{c}^2{)}^2$$where $E$ is the total energy, $p$ is momentum, and $m_0$ for electron is $9.11\times {10}^{-31}$ kg. Assuming $m_0{c}^2$ is negligible, kinetic energy $K\approx E$ can be estimated as:$$K\approx pc=1.05\times {10}^{-20}\text{kg}\cdot \text{m/s}\times 3\times {10}^8\text{m/s}\approx 3.15\times {10}^{-12}\text{J}$$

Calculate Magnitude of Coulomb Potential Energy

The Coulomb potential energy between a proton and an electron separated by distance ($r=5.0\times {10}^{-15}$ m) is given by:$$U=\frac{k{e}^2}{r}$$where: - $k=8.99\times {10}^9\text{N}\cdot {\text{m}}^2/{\text{C}}^2$ - $e=1.60\times {10}^{-19}\text{C}$Hence,$$U=\frac{8.99\times {10}^9\text{N}\cdot {\text{m}}^2/{\text{C}}^2\times (1.60\times {10}^{-19}\text{C}{)}^2}{5.0\times {10}^{-15}\text{m}}\approx 4.6\times {10}^{-14}\text{J}$$

Compare Energies and Conclude

Comparing the kinetic energy ($3.15\times {10}^{-12}\text{J}$) to the Coulomb potential energy ($4.6\times {10}^{-14}\text{J}$): the kinetic energy is significantly higher.Since this high kinetic energy far exceeds the potential energy binding an electron to a proton, electrons within a nucleus cannot exist under these circumstances.

Key concepts

Atomic Nucleus

Atomic nuclei are incredibly tiny structures at the heart of every atom. Their size is often around 5.0 $\times$ 10${}^{-15}$ meters, which is approximately a million times smaller than the atom itself. Inside this compact space reside protons and neutrons, collectively known as nucleons. The smallest radius of nuclei makes them densely packed with mass and energy.

The Heisenberg Uncertainty Principle plays a key role when estimating the behavior of particles within the nucleus. According to this principle, it is impossible to know precisely both the position and momentum of a particle. The principle is commonly expressed by the relation $\mathrm{\Delta }x\mathrm{\Delta }p\ge \frac{\hslash }{2}$, where $\mathrm{\Delta }x$ is the uncertainty in position, and $\mathrm{\Delta }p$ is the uncertainty in momentum. For electrons confined within a nucleus, this principle helps calculate a minimum uncertainty in momentum due to limited space.

For example, using the nucleus's radius as the uncertainty in position, physicists can estimate the momentum's uncertainty. It's fascinating to consider how such quantum principles influence our understanding of these invisible yet fundamental components of matter.

Relativistic Energy-Momentum Relation

The relativistic energy-momentum relation is essential for understanding the energy of particles moving at high speeds. This equation links the total energy $E$, momentum $p$, and rest mass $m_0$ of a particle with the speed of light $c$: $$E^2=(pc{)}^2+(m_0{c}^2{)}^2$$This formula illuminates the fact that as particles approach the speed of light, their kinetic energy increases significantly, potentially surpassing their rest mass energy.

When studying an electron within an atomic nucleus, where the estimates for momentum are exceedingly high, using this relation provides insight into its kinetic energy. In many cases, the rest mass energy term $(m_0{c}^2{)}^2$ becomes negligible compared to $(pc{)}^2$. Thus, the kinetic energy $K$ can be approximated by $pc$, yielding a straightforward calculation for electrons confined in tiny spaces.

This high kinetic energy of electrons calculated through the relativistic energy-momentum equation often exceeds the binding energies found in nuclear environments, indicating that such electrons would not be stably confined within the nucleus. Understanding this concept bridges the gaps between special relativity and quantum mechanics.

Coulomb Potential Energy

Coulomb potential energy describes the energy of interaction between two charged particles. It is given by the formula:$$U=\frac{k{e}^2}{r}$$where:

• $k$ is Coulomb's constant, valued at 8.99 $\times$ 10${}^9$ N$\cdot$m${}^2$/C${}^2$
• $e$ stands for the elementary charge, approximately 1.60 $\times$ 10${}^{-19}$ C
• $r$ represents the separation distance between the charges

When considering a proton and an electron separated by a nuclear-sized distance, this equation helps estimate the potential energy binding these particles together.

In many cases, this potential energy is much smaller than the kinetic energies calculated for confined particles, such as electrons within an atomic nucleus. This disparity means that the binding forces due to electrostatic attraction are insufficient to keep particles like electrons stably bound at such close interactions in the nucleus.

By comparing calculated kinetic and potential energies, we gain insights into why electrons do not exist within the nucleus. The conditions required for such existence would demand unrealistically low kinetic energies, making stable electron presence impossible within such a dense space.