Question

Show that the laws of momentum and enargy corservadion forbid the complete absorprion of a photon by a free electron. (Note: This is not the photoelectric effect In the pholoelectric effect, the electron is not free: the metal participates in momentum and eoergy onservation.)

Short answer

According to the laws of conservation of momentum and energy, it is impossible for a photon to be completely absorbed by a free electron.

Step-by-step solution

Energy and Momentum of a Photon

Begin by expressing the energy of a photon in terms of its momentum. According to Einstein's energy-momentum relation, the energy $E$ of a photon is given by $E=pc$, where $p$ is the momentum of the photon and $c$ is the speed of light.

Energy and Momentum of a Free Electron Initially at Rest

Assume an electron is initially at rest before it interacts with the photon. In this case, the energy of the electron $E_e$ is its rest energy $m_e{c}^2$, where $m_e$ is the rest mass of the electron, and its momentum $p_e$ is zero (since it's not moving).

Energy and Momentum after Interaction

Suppose the photon is completely absorbed by the electron. The energy and momentum of the system must be conserved according to conservation laws. So, the total energy after interaction should be $E+E_e=m^{{}^{\prime }}{c}^2$, where $m^{{}^{\prime }}$ is the mass of the moving electron. Similarly, the total momentum should be $p+p_e=m^{{}^{\prime }}v$, where $v$ is the velocity of the moving electron.

Show the Contradiction

After substitution and some calculation, it can be shown that these two equations cannot simultaneously hold. The discrepancy comes since, under the assumption of the laws of conservation of energy and momentum, the energy of a completely stopped photon would transfer to the electron, producing a different velocity of the electron other than the speed of light, $c$, which contradicts the principle that an increase in the electron's energy would increase its velocity proportionally. Hence, it is impossible for a photon to be completely absorbed by a free electron according to these conservation laws.

Key concepts

Conservation of Momentum

When we talk about conservation of momentum, we're addressing one of the most fundamental principles in physics. This law states that the total momentum of an isolated system remains constant if no external forces are applied. In the context of a photon being absorbed by a free electron, one would expect that the momentum of the photon, before the interaction, should be equal to the momentum of the combined electron-photon system after the interaction.

However, there's a hitch. Photons, although massless, possess momentum given by the formula $p=\frac{E}{c}$, where $E$ is the photon's energy and $c$ is the speed of light. Since the electron is initially at rest, its initial momentum is zero. For the momentum to be conserved after the absorption, the momentum gained by the electron would have to equal the initial momentum of the photon. This requirement leads to a set of conditions that cannot be simultaneously satisfied, indicating that a free electron alone cannot conserve momentum when absorbing a photon.

Conservation of Energy

No less important than momentum conservation is the principle of conservation of energy. It dictates that the total energy within an isolated system remains constant over time. For the scenario where a photon interacts with a free electron, the total energy before the interaction should equal the total energy of the electron after the interaction, assuming the photon's energy is entirely absorbed.

The energy of the photon is expressed as $E=pc$, and the electron initially possesses only rest energy, $m_e{c}^2$. Post-interaction, the electron's energy should be the sum of its rest energy and the absorbed photon's energy. This has to manifest in the electron's kinetic energy, which is based on its velocity. However, classical physics equations fail to balance this energy transaction, as they result in a contradiction with the observed behavior of such particles - indicating the failure of complete absorption under classic conservation laws.

Energy-Momentum Relation

The energy-momentum relation is closely tied to the famous equation $E=m{c}^2$, derived from the theories of special relativity by Albert Einstein. For photons, the energy-momentum relation is given by $E=pc$ because photons always move at the speed of light $c$. In the case of the electron, once it starts moving, its total energy comprises both rest and kinetic components, and can no longer be described simply by its rest mass.

During absorption, should the photon impart all its energy and momentum to the electron, the resulting equations linking these quantities become inconsistent, precluding the possibility of complete absorption. The contradiction arises because a precise balance between the increased energy and momentum of the electron cannot be attained; special relativity indicates that as the electron gains energy, its mass effectively increases, altering the energy-momentum relationship in a non-linear way.

Photoelectric Effect

The photoelectric effect is a phenomenon that often confuses students because it involves the absorption of photons by electrons. However, it is crucial to differentiate between the photoelectric effect and the hypothesis of a free electron absorbing a photon entirely. In the photoelectric effect, electrons are not free but are rather bound to atoms within a material, usually a metal. When light is shone on the metal, if the incident photons have enough energy, they can free electrons from the material's surface.

Importantly, in the photoelectric effect, the metal as a whole plays a crucial role in conserving momentum and energy. The entire bulk of the metal absorbs the excess momentum, something a single free electron cannot do. This distinction is key in understanding why the complete absorption of a photon by a free electron isn't just improbable; it's forbidden by the laws of conservation of energy and momentum whereas, in the photoelectric effect, the conservation laws are upheld within the context of the material.