Question

An electron moving to the lefi at \(0.8 c\) collides with an incoming photon moving to the right. Afuer the collision, the elactron is moving to the right at \(0.6 c\) and an outgoing photon moves to the lefi. What was the wavelength of the incoming photon?

Short answer

The wavelength of the incoming photon is calculated by solving the system of equations derived from conservation of momentum and energy principles. The exact value will depend on the given values for the velocity of electrons and the speed of light, as well as Planck's constant and the mass of the electron.

Step-by-step solution

Calculate initial and final momentum

The initial momentum was \(p_{initial} = m \cdot v_{electron} - h/\lambda_{incoming}\) and the final momentum was \(p_{final} = m \cdot v'_{electron} + h/\lambda_{outgoing}\). Here, minus is used for the incoming photon because it's moving in the opposite direction to the direction convention (i.e., right direction). By conservation of momentum, we equate the initial and final momentum.

Calculate initial and final energy

The initial energy consisted of the kinetic energy of the electron and the energy of the incoming photon \(E_{initial} = \gamma m c^2 + h \cdot c/\lambda_{incoming}\). The final energy consisted of the kinetic energy of the electron after collision and the energy of the outgoing photon \(E_{final} = \gamma' m c^2 + h \cdot c/\lambda_{outgoing}\). Here gamma and gamma prime represent the Lorentz factor before and after the collision respectively. By conservation of energy, we equate the initial and final energy.

Solve for the Unknown

From the conservation of momentum and conservation of energy, we can solve the resulting equations to find the wavelength of the incoming photon. We substitute the given values into these equations to calculate the value of \(\lambda_{incoming}\).

Key concepts

Conservation of Momentum

Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It's important to understand the momentum in a photon-electron collision scenario. Momentum is conserved in isolated systems, meaning that the total momentum before the event is equal to the total momentum after the event.

In the described problem, the electron initially moves at 0.8c, and there’s an incoming photon. They have opposite directions, affecting how we calculate the momentum. The formula for initial momentum is:

• Initial momentum (\(p_{initial}\)) = \(m imes v_{electron} - \frac{h}{\lambda_{incoming}}\).

The negative sign indicates the photon's opposite direction to our positive convention (chosen as right).

After the collision, the electron changes its direction, moving to the right at 0.6c, resulting in a different calculation for the final momentum:

• Final momentum (\(p_{final}\)) = \(m imes v'_{electron} + \frac{h}{\lambda_{outgoing}}\).

Setting the initial momentum equal to the final momentum helps solve for unknowns, like the incoming photon’s wavelength.

Conservation of Energy

Energy conservation is a cornerstone of physics, stating that energy cannot be created or destroyed, only transformed. In a photon-electron collision, we equate the total energy before and after the collision to conserve energy.

Initially, the energy includes the kinetic energy of the electron and the energy of the incoming photon:

• Initial energy (\(E_{initial}\)) = \(\gamma m c^2 + \frac{h \cdot c}{\lambda_{incoming}}\).

The term \(\gamma m c^2\) represents the relativistic kinetic energy using the Lorentz factor (\(\gamma\)).

After the collision, the electron's kinetic energy changes as it moves right, and the outgoing photon's energy is considered:

• Final energy (\(E_{final}\)) = \(\gamma' m c^2 + \frac{h \cdot c}{\lambda_{outgoing}}\).

Conservation of energy ensures that \(E_{initial} = E_{final}\), helping to solve for the unknown \(\lambda_{incoming}\) when combined with momentum conservation equations.

Lorentz Factor

The Lorentz factor is a crucial element in relativity and quantum mechanics problems, especially in scenarios involving high velocities close to the speed of light. It adjusts calculations for time, length, and, importantly, energy.

Defined as:

• \(\gamma = \frac{1}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}\)

It accounts for relativistic effects due to the electron's velocity. For example, in our photon-electron collision, the electron's speeds, both initial (0.8c) and final (0.6c), require using the Lorentz factor in energy calculations.

Changes in the Lorentz factor (\(\gamma\) for before collision and \(\gamma'\) after) reflect the shifted velocities and are crucial for determining precise energy alterations. Through factoring in these relativistic effects, one can accurately calculate energies and solve for the incoming photon's wavelength.