Question

The electron is known to have a radius no larger than \(10^{-18} \mathrm{~m}\) If actually produced by circulating mass, its intrinsic angular momentum of roughly \(\hbar\) would imply very high speed. even if all that mass were as far from the axis as possible. (a) Using simply \(r p\) (from \(|r \times p|\) ) for the angular momentum of a mass at radius \(r\). obtain a rough value of \(p\) and show that it would imply highly relativistic speed. (b) At such speeds, \(E=y m c^{2}\) and \(p=\gamma\) mu combine to give \(E \cong p c\) (just as for the speedy photon). How does this energy compare with the known intenial energy of the electron?

Short answer

The exercise demonstrates the limits of classical mechanics in correctly describing the observed properties of subatomic particles like electrons, reinforcing the necessity of quantum mechanics and special relativity. We found an estimate for the momentum of the electron that would imply a speed faster than light, and a predicted energy level much different than the actual energy level of an electron, both of which contradict the known theoretical and experimental results.

Step-by-step solution

Determine the Momentum of the Electron

The angular momentum of a particle circulating at a radius r with momentum p is given by the classic formula |r × p|. Since we know the upper limit for r (\(10^{-18}\) m), for simplicity's sake, and we know that the electron's intrinsic angular momentum is approximately \(\hbar\), we can make a straightforward estimate of p by using the simplified version of the momentum formula, rp. Therefore, p = \(\hbar / r\).

Analyze the Implications of This Momentum

The momentum p, obtained from step 1, can be used to calculate the classical speed v of the electron by dividing p by the electron mass m. That is, v = p/m. This yields a speed that is higher than the speed of light, which is not possible according to the theory of relativity. This means that the speed of electrons must be highly relativistic, reinforcing the need for relativistic treatment.

Applying Relativistic Formulas for Energy and Momentum

At relativistic speeds, the relationship between energy E, mass m, and speed squared c^2 is given by E=γmc^2, and the relationship between momentum p and speed v is given by p=γm. We also know that for high speeds, E≈pc. Therefore, to find the energy E, we simply multiply the momentum p obtained in step 1 by the speed of light c.

Compare the Energies

Finally, Compare this value of the energy E calculated from the high speed, relativistic formula with the known energy of an electron (approximately \(0.511 \, MeV\)). You will find that these values differ.

Key concepts

Electron Radius

The electron radius is a theoretical concept since electrons are considered point particles in the Standard Model of particle physics. However, when physicists talk about the 'radius' of an electron, they are often referring to the region within which the electron's electric field is significant. Experiments have set an upper bound on this radius at about \(10^{-18} \, \text{meters}\).
This is minuscule compared to atomic scales and suggests the electron's intrinsic properties are not easily confined to a spatial region.
Understanding electron radius helps with visualizing phenomena like scattering, where the electron interacts with other particles. It's crucial to realize that these theoretical radii help us comprehend physical limits, much like the size constraints of other subatomic particles.

• In practical experiments, the "radius" provides no definitive boundary like a solid sphere but indicates interaction ranges.
• Electron radius plays a role in quantum mechanics, influencing interpretations of phenomena like Heisenberg's uncertainty principle.

Angular Momentum

Angular momentum in relativistic physics describes an object's rotation. For an electron, this is inherently linked to its intrinsic spin, measured as the reduced Planck's constant \(\hbar\). The electron's spin is a quantum property, akin to microscopic rotation, vital for spectroscopy and understanding atomic structure.
Using the product \(r \times p\) helps calculate the electron's angular momentum, assuming a particular radius.
The formula \(rp\) signifies the magnitude of angular momentum for a particle moving in a circular path.

• Considering maximal electron radius \(r\), allows calculating momentum \(p = \hbar / r\).
• It shows angular momentum quantization: a foundational quantum mechanics concept.

Understanding these relationships gives insight into orbital shapes and chemical bonding originating from electron configurations.

Relativistic Speed

Relativistic speed implies that an object is traveling at a significant fraction of the speed of light, necessitating modifications to classical formulas. For an electron, achieving such high speeds leads to notable relativistic effects where classical equations of motion become insufficient.
This is because, as velocity approaches the speed of light, relativistic effects such as time dilation and length contraction start to dominate the dynamics.

• In our exercise, computing \(v = p / m\) reveals velocities greater than light speed, a theoretical impossibility solved by relativistic physics.
• The necessity of relativistic treatment stems from \(v/c\) nearing one, demanding the Lorentz factor \(\gamma\).

Thus, any interpretation of high-velocity electrons must incorporate these effects to maintain physical accuracy and consistency.

Electron Energy

Electron energy in a relativistic context is calculated with the equation \(E = \gamma m c^{2}\). The energy \(E\) here varies directly with the Lorentz factor \(\gamma\), which adjusts for relativistic speeds.
When speeds get close to the speed of light, the energy becomes comparable to \(pc\), the momentum times the speed of light.

• In practical applications, measuring electron energies helps probe other quantum physics phenomena.
• Energy calculations compare theoretical results with known values (0.511 MeV for electrons), ensuring real-world accuracy.

Understanding electron energy is critical in fields like particle physics and accelerators, where electrons reach relativistic speeds routinely. Modifying how we calculate energy provides deeper insights into the universe's building blocks and forces at these energy scales.