Chapter 38: Quantization – Study Notes

Quantization of Energy for Light and Matter

Introduction

This chapter introduces the concept of quantization, focusing on how energy is quantized for both light and matter. The quantization of energy is a foundational idea in modern physics, leading to the development of quantum mechanics.

Units of Energy in the Microscopic World

Electron Volt (eV)

• Definition: An electron volt is the amount of energy gained by an electron when it moves through a potential difference of one volt.

• Conversion: $1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}$

• Usage: Electron volts are commonly used to express energies at the atomic and subatomic scale.

The Photon Model of Light

Basic Postulates

1. Light consists of discrete, massless units called photons, which travel at the speed of light in a vacuum.

2. Each photon has energy: $E_{\text{photon}} = hf$ where $f$ is the frequency of light and $h = 6.626 \times 10^{-34}\ \text{J} \cdot \text{s}$ (Planck's constant).

3. The superposition of many photons exhibits the characteristics of a classical light wave.

• Planck's constant in eV·s: $h = 4.136 \times 10^{-15}\ \text{eV} \cdot \text{s}$

Photon Momentum

• The momentum of a photon is given by: $p = \frac{E}{c} = \frac{h}{\lambda}$ where $c$ is the speed of light and $\lambda$ is the wavelength.

Energy and Intensity of Light

• Energy of a photon: Higher frequency (shorter wavelength) photons have more energy.

• Example: The energy of a photon of red light is smaller than that of blue light.

• Total energy for N photons: $E_{\text{light}} = Nhf$

• Power of light (rate of energy delivery): $P = \frac{dE_{\text{light}}}{dt} = \frac{dN}{dt}hf = Rhf$ where $R = \frac{dN}{dt}$ is the photon arrival rate (photons per second).

• Reducing intensity: If the intensity of light is reduced without changing wavelength, there are fewer photons per second, but each photon has the same energy.

Worked Examples

Example 1: Laser Emission

• A 1.56 W laser emits monochromatic red light ($\lambda = 650$ nm). To find the number of photons emitted per second, use: $P = Rhf \implies R = \frac{P}{hf}$

Example 2: Photon Properties

• Given a photon of green light ($\lambda = 520$ nm):

  • Frequency: $f = \frac{c}{\lambda}$

  • Momentum: $p = \frac{h}{\lambda}$

  • Energy: $E = hf$ (expressed in J and eV)

Example 3: Laser Pulse

• Laser with $\lambda = 652$ nm, pulse duration $= 20.0$ ms, average power $= 0.600$ W:

  • Energy per pulse: $E = P_{\text{av}} t$

  • Energy per photon: $E = \frac{hc}{\lambda}$

  • Number of photons per pulse: $N = \frac{E_{\text{pulse}}}{E_{\text{photon}}}$

The Photoelectric Effect

Discovery and Description

• Discovered by Phillip Lenard (~1900): Electric current is produced when ultraviolet light shines on a metal cathode.

• This phenomenon is called the photoelectric effect.

Key Characteristics

• Photoelectrons are emitted only if the light frequency $f$ exceeds a threshold frequency $f_0$.

• The value of $f_0$ depends on the metal (work function $E_0$).

• If the voltage between cathode and anode is made negative, the current decreases and reaches zero at the stopping potential $V_{\text{stop}}$.

The Work Function

• Definition: The minimum energy needed to free an electron from a metal is called the work function $E_0$.

• Different metals have different work functions.

Element	$E_0$ (eV)
Potassium	2.30
Sodium	2.36
Aluminum	4.28
Tungsten	4.55
Copper	4.65
Iron	4.70
Gold	5.10

The Stopping Potential

• Maximum kinetic energy of emitted electrons: $K_{\text{max}} = E_{\text{elec}} - E_0$

• When the potential difference causes the fastest electrons to have $K = 0$: $V_{\text{stop}} = \frac{K_{\text{max}}}{e}$

QuickCheck Questions

• Source of current in photoelectric effect: The light (not the battery or cathode) causes the current.

• Effect of reducing battery voltage: The current decreases slightly.

Light Quanta in Biology

• Human eyes have photoreceptors (cones) sensitive to different wavelengths (e.g., 437 nm, 533 nm, 575 nm).

• Energy of one quantum of light at these wavelengths can be calculated using $E = \frac{hc}{\lambda}$.

• Visible light corresponds to photon energies of roughly 2–3 eV.

Summary

• Energy is quantized for both light (photons) and matter (as will be seen in later sections).

• The photoelectric effect demonstrates the particle nature of light and the necessity of quantized energy exchange.

• Key equations:

  • $E_{\text{photon}} = hf = \frac{hc}{\lambda}$

  • $p = \frac{h}{\lambda}$

  • $P = Rhf$

  • $K_{\text{max}} = E_{\text{elec}} - E_0$

  • $V_{\text{stop}} = \frac{K_{\text{max}}}{e}$

Additional info: These notes cover the foundational aspects of quantization, photon energy, and the photoelectric effect, which are essential for understanding quantum physics and modern atomic theory.