Question

Quantum theory says that electromagnetic waves actually consist of discrete packets-photons-each with energy \(E=\hbar \omega,\) where \(\hbar=1.054572 \cdot 10^{-34} \mathrm{~J} \mathrm{~s}\) is Planck's reduced constant and \(\omega\) is the angular frequency of the wave. a) Find the momentum of a photon. b) Find the magnitude of angular momentum of a photon. Photons are circularly polarized; that is, they are described by a superposition of two plane- polarized waves with equal field amplitudes, equal frequencies, and perpendicular polarizations, one-quarter of a cycle \(\left(90^{\circ}\right.\) or \(\left.\pi / 2 \mathrm{rad}\right)\) out of phase, so the electric and magnetic field vectors at any fixed point rotate in a circle with the angular frequency of the waves. It can be shown that a circularly polarized wave of energy \(U\) and angular frequency \(\omega\) has an angular momentum of magnitude \(L=U / \omega\). (The direction of the angular momentum is given by the thumb of the right hand, when the fingers are curled in the direction in which the field vectors circulate.) c) The ratio of the angular momentum of a particle to \(\hbar\) is its spin quantum number. Determine the spin quantum number of the photon.

Short answer

Question: Using the provided information about the photon's energy and the relationship between a circularly polarized wave's angular momentum and energy, find a) the momentum of a photon, b) the magnitude of angular momentum of a photon, and c) the spin quantum number of the photon. Answer: a) The momentum of a photon is \(p = \frac{\hbar\omega}{c}\). b) The magnitude of angular momentum of a photon is \(L = \hbar\). c) The spin quantum number of the photon is \(s=1\).

Step-by-step solution

Write down the energy formula

We are given that the energy of a photon is \(E=\hbar\omega\).

Calculate the momentum

From the energy formula, you can use the energy-momentum relationship for photons: \(E=pc\) where \(E\) is the energy, \(p\) is the momentum, and \(c\) is the speed of light. Now, we can express the momentum as: \(p = \frac{E}{c}\) We already have the formula for the energy, so we can replace \(E\) with \(\hbar\omega\): \(p = \frac{\hbar\omega}{c}\) #b) Finding the magnitude of angular momentum of a photon#

Write down the formula for angular momentum

According to the problem, the magnitude of the angular momentum of a circularly polarized wave is given by \(L = \frac{U}{\omega}\), where \(U\) is the energy of the system and \(\omega\) is the angular frequency.

Calculate the angular momentum

Since the energy of a photon is equal to \(U\), we can use the formula for energy, \(E=\hbar\omega\). Therefore, we can replace \(U\) with \(\hbar\omega\): \(L = \frac{\hbar\omega}{\omega}\) Thus, the magnitude of angular momentum of a photon is given by: \(L = \hbar\) #c) Finding the spin quantum number of a photon#

Write down the formula for spin quantum number

The spin quantum number is the ratio of the angular momentum of a particle to \(\hbar\): \(s = \frac{L}{\hbar}\)

Calculate the spin quantum number

From the previous part, we know the angular momentum of a photon is \(L = \hbar\). We can now substitute this into our equation for the spin quantum number: \(s = \frac{\hbar}{\hbar}\) Thus, the spin quantum number of the photon is: \(s=1\)

Key concepts

Quantum Theory

Quantum theory revolutionized our understanding of how energy and matter interact at the smallest scales. In this conceptual framework, wave-like behavior and particle-like properties coexist, which is exemplified by the very nature of light. Classical physics described light solely as a wave, but quantum theory introduces light as consisting of particles called photons. Each photon carries energy quantized according to Planck's constant and the frequency of the wave, expressed by the familiar equation, \(E = \hbar \omega\).

Understanding this duality is crucial in physics, as it leads to fascinating phenomena like quantum entanglement and the uncertainty principle. By approaching problems with this quantum mindset, we can better equip ourselves to tackle exercises such as calculating a photon's momentum and angular momentum.

Energy-momentum Relationship

In the realm of quantum physics, energy and momentum are tightly interwoven concepts, especially when it comes to particles such as photons. Photons are massless, and yet they carry momentum which can exert pressure, known as radiation pressure. The relationship between a photon's energy \(E\) and its momentum \(p\) is given by the relativistic equation, \(E = pc\), where \(c\) represents the speed of light.

This formula is groundbreaking because it ties together two fundamental properties without involving mass directly. The concept allows us to derive the momentum of a photon simply by rearranging the formula to \(p = \frac{E}{c}\). By substituting \(E\) with \(\hbar\omega\), we obtain the momentum as a function of Planck's reduced constant and the angular frequency of the wave, \(p = \frac{\hbar\omega}{c}\).

Spin Quantum Number

Diving deeper into the quantum world, we encounter intrinsic properties of particles that have no classical equivalent, one of which is spin. The spin quantum number \(s\) is an integral or half-integral value that reflects a particle's inherent angular momentum. This quantum number is crucial because it greatly determines the behavior of particles, including how they interact with external fields and with one another.

For photons, we find that the magnitude of their angular momentum \(L\) is simply Planck's constant \(\hbar\). To find a photon's spin quantum number, we simply divide its angular momentum by \(\hbar\), yielding a spin quantum number of 1. This implies photons follow what's called Bose-Einstein statistics, and are identified as bosons, particles that allow multiple identical particles to occupy the same quantum state simultaneously.

Circular Polarization

When we describe the polarization of light, we refer to the orientation of its electric field vector. Circularly polarized light has an electric field that rotates in a helical pattern as the wave propagates. This type of polarization can be thought of as a superposition of two linearly polarized waves that are 90 degrees out of phase.

This leads to a fascinating property where the rotating electromagnetic field carries angular momentum. For a circularly polarized wave, the magnitude of this angular momentum \(L\) is proportional to the energy \(U\) of the wave and inversely proportional to its angular frequency \(\omega\), as expressed by \(L = \frac{U}{\omega}\). The direction of the angular momentum can be determined by the 'right-hand rule', helping us visualize this rotational aspect of photon dynamics.