Question

In an inertial frame \(\mathrm{S}\), two photons of frequencies \(v_{1}\) and \(v_{2}\) travel in the positive and negative \(x\)-directions respectively. Find the velocity of the CM frame of these photons. [Answer: \(v / c\) \(\left.=\left(v_{1}-v_{2}\right) /\left(v_{1}+v_{2}\right) .\right]\)

Short answer

Answer: The expression for the velocity of the center of mass frame (relative to the speed of light, c) of the two photons is given by: \(\frac{v}{c} = \frac{\nu_1 - \nu_2}{\nu_1 + \nu_2}\).

Step-by-step solution

Identify the velocities of the photons

Since photons always travel at the speed of light c, their velocities are given by: Photon 1: \(v_{1} = c\) Photon 2: \(v_{2} = -c\)

Find the momentum of each photon

The momenta of the photons can be expressed using their frequencies (\(\nu_1\) and \(\nu_2\)) and the Planck's constant \(h\): Photon 1: \(p_1 = \frac{h\nu_1}{c}\) Photon 2: \(p_2 = -\frac{h\nu_2}{c}\) (negative sign due to the opposite direction)

Compute the total momentum

The total momentum \(P\) is given by the sum of the momenta of the photons: \(P = p_1 + p_2 = \frac{h\nu_1}{c} - \frac{h\nu_2}{c} = \frac{h(\nu_1 - \nu_2)}{c}\)

Relate total momentum to the center of mass velocity

The principle of conservation of momentum states that the total momentum of the system can be related to the center of mass velocity, \(v_{CM}\), as follows: \(P = (\nu_1 + \nu_2)\frac{h}{c} v_{CM}\)

Solve for the center of mass velocity

Now, we can solve for the center of mass velocity by equating the expressions from Step 3 and Step 4: \(\frac{h(\nu_1 - \nu_2)}{c} = (\nu_1 + \nu_2)\frac{h}{c} v_{CM}\) Divide both sides by \(\frac{h}{c}\) to get: \(v_{CM} = \frac{\nu_1 - \nu_2}{\nu_1 + \nu_2}\) Finally, since the photons are traveling at the speed of light, the center of mass velocity relative to the speed of light can be expressed as: \(\frac{v}{c} = \frac{v_{CM}}{c} = \frac{\nu_1 - \nu_2}{\nu_1 + \nu_2}\) So, the velocity of the center of mass frame of the photons is given by \(\frac{v}{c} = \frac{\nu_1 - \nu_2}{\nu_1 + \nu_2}\).

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics formulated by Albert Einstein in 1905. At its core, it addresses the behavior of objects traveling at speeds approaching the speed of light in vacuum, denoted as 'c'. One crucial postulate of special relativity is that the speed of light is the same for all observers, regardless of their relative motion.

This has profound implications: as objects move faster, they experience time dilation (time slows down) and length contraction (lengths shorten), all based on the observer's frame of reference. In the context of our exercise, special relativity plays an essential role when we talk about photons moving at light speed and how their behavior is governed by relativistic principles.

Photon Momentum

Photons, which are quanta of light, carry momentum despite having no rest mass. The momentum of a photon is intrinsically linked to its frequency and is given by the formula \( p = \frac{hu}{c} \), where \( p \) is the momentum, \( u \) is the frequency, and \( h \) is Planck's constant.

Momentum in physics is generally a measure of the quantity of motion of a moving body and is vectorial in nature, meaning it has both magnitude and direction. This explains why the momentum of a photon traveling in the opposite direction has a negative sign, as seen in the exercise. The concept of photon momentum is vital to understanding the dynamics of light and other electromagnetic radiation in both classical and quantum mechanics.

Planck's Constant

Planck's constant, denoted by \( h \), is a fundamental physical constant that plays a pivotal role in quantum mechanics. Its value approximately equals \( 6.626 \times 10^{-34} \) joule seconds. This tiny number may seem insignificant; however, it is foundational for explaining the quantized nature of energy in the microscopic world.

It was Max Planck who first introduced this constant in 1900, which later led to the birth of quantum theory. The constant \( h \) links the energy of a photon (the quantum of electromagnetic radiation) to its frequency through the equation \( E = hu \), and as our exercise demonstrates, it likewise connects the photon's momentum to its frequency. Planck's constant bridges the gap between the discontinuous quantum world and the continuous classical physics.

Speed of Light

The speed of light in vacuum, often abbreviated as 'c', is a universal physical constant important in many areas of physics. Its value is roughly \( 299,792,458 \) meters per second (m/s). This speed is the ultimate speed limit for all matter and information in the universe, according to special relativity.

Understanding that light always travels at this constant speed allows us to dive deeper into phenomena like time dilation, length contraction, and the invariance of the speed of light across all frames of reference, which was pivotal for Einstein in developing the theory of special relativity. In the exercise at hand, the speed of 'c' helps us to calculate the momentum of photons, as well as to elucidate the relativistic behavior of light.

Conservation of Momentum

Conservation of momentum is a fundamental principle in physics stating that the total momentum of a closed system is constant if no external forces act on it. In essence, if you add up all the momenta (mass times velocity) of the objects within the system, this sum will not change unless external forces are applied.

When we analyze photon collisions or interactions like those in our exercise, conservation of momentum is key. It holds true even in a relativistic context, such as interactions involving photons moving at the speed of light. This principle allows us to deduce the velocity of the center of mass frame when we know the momentum of the individual photons involved, contributing to our understanding of more advanced concepts like particle collisions in high-energy physics.