Question

Quantum theory says that electromagnetic waves actually consist of discrete packets-photons-each with energy \(E=\hbar \omega,\) where \(\hbar=1.054573 \cdot 10^{-34} \mathrm{~J} \mathrm{~s}\) is Planck's reduced constant and \(\omega\) is the angular frequency of the wave. a) Find the momentum of a photon. b) Find the 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 \(\pi / 2\) rad \()\) 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

Answer: The momentum of the photon is \(p = \frac{\hbar \omega}{c}\), the angular momentum is \(L = \hbar\), and the spin quantum number is \(s = 1\).

Step-by-step solution

(Step 1: Find the momentum of a photon)

To find the momentum of a photon, first, we need to recall the relation between the energy and momentum: \(E = pc\) where \(p\) is the momentum and \(c\) is the speed of light. Given the energy of a photon: \(E = \hbar \omega\) Now, we can replace \(E\) with the momentum equation to obtain the momentum: \(\hbar \omega = pc\) \(p = \frac{\hbar \omega}{c}\)

(Step 2: Find the angular momentum of a photon)

As given, a circularly polarized wave of energy \(U\) and angular frequency \(\omega\) has an angular momentum of magnitude \(L=\frac{U}{\omega}\). In the case of a photon, the energy is given by \(E = \hbar \omega\): \(L = \frac{E}{\omega} = \frac{\hbar \omega}{\omega} = \hbar\) So, the angular momentum of a photon is equal to Planck's reduced constant \(\hbar\).

(Step 3: Determine the spin quantum number)

The spin quantum number can be determined by finding the ratio of the angular momentum of a particle to \(\hbar\): \(s = \frac{L}{\hbar}\) We've found the angular momentum of a photon to be equal to \(\hbar\). Now, we can substitute this value into the formula: \(s = \frac{\hbar}{\hbar} = 1\) Hence, the spin quantum number of the photon is 1.

Key concepts

Quantum Theory and Photons

Quantum theory revolutionized our understanding of matter and energy by proposing that energy is not continuous, but quantized. This means that energy can only exist in specific, discrete amounts, similar to how coins make change rather than a smooth flow of cash. Central to quantum theory is the concept of the photon: the fundamental particle of light.

A photon carries energy which is dependent on its frequency, described by the equation \(E = \text{h} u\) or \(E = \text{h} \bar\omega\), where \(\text{h}\) is the Planck constant, \(u\) is the frequency, and \(\bar\omega\) is the angular frequency. Understanding photons is crucial because they interact with matter and can exhibit both particle-like and wave-like properties. Their study not only forms the basis of quantum mechanics but also has practical implications in technologies such as lasers and solar cells.

Understanding Angular Momentum in Quantum Systems

Angular momentum is a fundamental concept in physics, reflecting the amount of rotation an object has. In quantum mechanics, it acquires a more nuanced meaning.

For particles like photons, their wave-like nature means they can have angular momentum even without mass. This idea expands into fascinating territory when considering circularly polarized light, where the electric and magnetic fields rotate, creating angular momentum. The magnitude of angular momentum for these waves or particles is given by \(L = \frac{U}{\omega}\), where \(U\) is the energy and \(\omega\) is the angular frequency.

Moreover, in quantum mechanics, angular momentum is quantized, meaning it can only take on certain discrete values. The quantization of angular momentum has profound implications for the structure of atoms and the behavior of particles on a microscopic scale, influencing the entire framework of quantum physics.

Spin Quantum Number and Its Role

The spin quantum number is an intrinsic form of angular momentum carried by particles. Unlike the angular momentum which might arise from a particle's movement through space, spin is an inherent property like charge or mass.

For a photon, the spin quantum number is a way of expressing its angular momentum in units of \(\hbar\), which is the reduced Planck constant. When we consider the angular momentum of a photon (\(L = \hbar\)) and compare it with \(\hbar\), we find that the spin quantum number (\(s\)) is 1. This indicates that the photon carries one unit of angular momentum in terms of \(\hbar\).

Spin is not just a theoretical concept; it determines the magnetic and optical properties of materials, influences the outcome of quantum experiments, and explains why certain particles behave as they do. For photons, their spin quantum number is essential in understanding the polarization of light and has implications in quantum information technologies like quantum computing and communications.