Problem 9 - An Introduction to Modern Astrophysics

Question

Use the laws of conservation of relativistic energy and momentum to prove that an isolated electron cannot absorb a photon.

Step-by-Step Solution

Answer

An isolated electron cannot absorb a photon because it violates momentum conservation.

Step 1: Understand the Scenario

We need to prove that an isolated electron cannot absorb a photon using the principles of conservation of relativistic energy and momentum. Initially, we consider an electron with rest mass energy and no kinetic energy.

Step 2: Initial Conditions

Consider an electron at rest with energy given by its rest mass energy: \[ E_i = mc^2 \]and momentum:\[ p_i = 0 \].A photon has energy:\[ E_{photon} = hf \]and momentum:\[ p_{photon} = \frac{hf}{c} \].Where \(m\) is the rest mass of the electron, \(c\) is the speed of light, \(h\) is Planck's constant, and \(f\) is the frequency of the photon.

Step 3: Final Conditions After Absorption

After absorbing the photon, the electron would have new energies and momenta:\[ E_f = mc^2 + hf \]\[ p_f = \frac{hf}{c} \].The rest mass energy plus the photon energy becomes the new energy of the electron, while the momentum becomes that of the photon alone, assuming additional kinetic energy is zero.

Step 4: Apply Conservation of Energy

According to the conservation of energy, the initial energy of the system is equal to the final energy:\[ mc^2 + hf = E_f = mc^2 + hf \].So the energy is fine in this case.

Step 5: Apply Conservation of Momentum

According to the conservation of momentum, the initial momentum of the system is equal to the final momentum:\[ 0 + \frac{hf}{c} = p_f = \frac{hf}{c} \].Momentum conservation results in a contradiction because an increase in energy due to the absorbed photon requires additional momentum for the moving electron, which is not accounted for.

Step 6: Conclusion

Since energy can be conserved but momentum cannot be so without additional terms due to kinetic energy, an isolated electron cannot absorb a photon. This imbalance indicates that without an external force or additional particles to conserve both, the process is not possible.

Key Concepts

Relativistic Energy

Relativistic energy is an extension of Einstein's energy-mass equivalence principle, allowing energy calculations for particles close to the speed of light. This principle tells us that the total energy of a particle like an electron is not just its rest mass energy, but also includes any kinetic energy it may have.
The formula for relativistic energy is given by:

• The rest mass energy: \[ E = mc^2 \]This is the energy an object has due to its mass when it is at rest. Here, \(m\) is the rest mass of the electron, and \(c\) is the speed of light.
• The total relativistic energy: \[ E = \gamma mc^2 \]Where \(\gamma\) is the Lorentz factor, \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), that accounts for the relativistic effects at high speeds, where \(v\) is the velocity of the particle.

When a stationary particle, like an electron, absorbs a photon, it gains energy from the photon, which should increase its total energy considering its rest mass and any newly acquired kinetic energy.

Momentum Conservation

Momentum conservation is one of the key concepts in physics, stating that the total momentum of a closed system remains constant over time unless acted upon by external forces. In the context of our discussion, this principle is crucial for understanding why an isolated electron cannot absorb a photon.
Initially, an isolated electron at rest has no momentum, meaning its momentum \( p_i = 0 \). However, a photon carries both energy and momentum, expressed as \( p_{photon} = \frac{hf}{c} \), where \( h \) is Planck's constant, \( f \) is the frequency, and \( c \) is the speed of light.
When the electron supposedly absorbs the photon, its momentum should also account for this increase:

• Initial momentum: \( 0 + \frac{hf}{c} = \frac{hf}{c} \)
• Final momentum (theoretically, if absorption were to occur): \( \frac{hf}{c} \)

However, conservation laws indicate an isolated electron cannot merely absorb a photon's momentum without a corresponding increase in velocity or involvement of another entity to aid in the process, thus creating an apparent contradiction.

Photon Absorption

Photon absorption involves transferring energy and momentum from a photon to another particle, such as an electron. The process typically requires that energy and momentum both balance before and after the interaction. When objects are not isolated, and in more complex systems, this balance can occur naturally.
A photon, being a light particle, carries specific energy \( E_{photon} = hf \) and momentum \( p_{photon} = \frac{hf}{c} \). Ordinarily, in an isolated event where a stationary electron absorbs a photon, we expect the electron’s energy to increase by the photon’s energy and similarly for momentum.
Despite this, in isolation, a stationary electron lacks the necessary conditions to meet both energy and momentum conservation after absorption because:

• No additional force or particle is available to balance the system.
• The absorption would require the electron to gain enough velocity to reflect the acquired photon's momentum, which isn't spontaneously feasible without external influence or particles.

Thus, photon absorption, in isolation, contradicts the established conservation laws.

Rest Mass Energy

Rest mass energy is a fundamental concept outlined by Einstein's famous equation \( E = mc^2 \), representing the intrinsic energy possessed by a particle purely due to its mass, independent of its motion or external energy inputs. For an electron at rest, this is its only form of energy and is why it initially starts with no momentum.
If an electron were to absorb a photon, its energy post-absorption would theoretically include:

• Its original rest mass energy plus the photon’s energy: \( E_f = mc^2 + hf \)
• The momentum would then, ideally, become the photon's: \( p_f = \frac{hf}{c} \)

However, rest mass energy doesn't account for any additional energy requirements brought about through motion or force interaction. The static state of a lone electron doesn’t allow for attaining the necessary kinetic energy to match the absorbed photon's momentum, therefore making the isolated absorption scenario impractical under these laws.