Chapter 30: Quantum Physics – Study Notes

Quantum Physics

Introduction

Quantum physics explores the behavior of matter and energy at the smallest scales, where classical physics fails to explain certain phenomena. This chapter introduces the particle character of electromagnetic radiation, wave-particle duality, and foundational principles such as the photoelectric effect, Compton effect, de Broglie hypothesis, and Heisenberg uncertainty principle.

Particle Character of Electromagnetic Radiation

Blackbody Radiation and Planck's Law

Blackbody radiation refers to the emission of electromagnetic radiation by an idealized object that absorbs all incident light without reflection. The energy distribution of blackbody radiation is independent of material but depends on temperature.

• Quantization of Energy: Energy is emitted in discrete packets (quanta).

• Wien's Displacement Law: The peak frequency of radiation increases with temperature. $f_{\text{peak}} = \frac{1}{\lambda_{\text{peak}}} = (5.88 \times 10^{10} \text{~s}^{-1}\text{K}^{-1})T$

• Planck's Law: Electromagnetic radiation is quantized. $E = nhf$ where $n$ is an integer, $h$ is Planck's constant ($6.63 \times 10^{-34}$ Js), and $f$ is frequency.

Example: Wien's law is used in astronomy to determine the temperature of stars based on their color.

Energy Quantization

• Classical Physics Prediction: Rayleigh-Jeans law predicted infinite radiation at high frequencies, which disagreed with experiments.

• Planck's Solution: Energy is quantized, resolving the ultraviolet catastrophe.

Particles of Light: Photons

• Photon: A quantum of electromagnetic energy.

• Energy of a Photon: $E = hf$

• Momentum of a Photon: $p = \frac{h}{\lambda}$

• Rest Mass: $m = 0$ (photons are massless)

• Energy of EM Wave: $E = Nhf$ (for $N$ photons)

Example: UV light photons have more energy than infrared photons due to higher frequency.

The Photoelectric Effect

Explanation and Quantum Physics

The photoelectric effect is the ejection of electrons from a metal when light of sufficient frequency is incident on its surface.

• Cutoff Frequency: Electrons are ejected only if light frequency $f$ exceeds cutoff $f_0$.

• Work Function ($W_0$): Minimum energy required to release an electron.

• Kinetic Energy of Ejected Electrons: $KE = hf - W_0$

Applications: Photocells, solar energy panels.

ConcepTests – Photoelectric Effect

• Photon Energy: Blue light photons have more energy than red, yellow, or green due to higher frequency ($E = hf$).

• Work Function and Cutoff Frequency: Higher cutoff frequency means higher work function ($W_0 = hf_0$).

• Effect of Wavelength: If incident light wavelength exceeds cutoff, no electrons are emitted.

• Effect of Intensity: Increasing light intensity (at fixed wavelength) increases the number of emitted electrons, not their energy.

The Compton Effect

Photon Scattering by Electrons

The Compton effect describes the scattering of a photon by an electron, resulting in a change in the photon's wavelength.

• Compton Shift Formula: $\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos \theta)$ where $\lambda'$ is the outgoing photon wavelength, $\lambda$ is the incoming, $m_e$ is electron mass, $c$ is speed of light, $\theta$ is the scattering angle.

• Significance: Demonstrates particle-like properties of light and conservation of energy and momentum.

Wave-Particle Duality

de Broglie Hypothesis

Louis de Broglie proposed that particles can exhibit wave-like properties, with a wavelength given by:

• de Broglie Wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$

• Significance: Subatomic particles (e.g., electrons) have significant wavelengths; macroscopic objects have negligible wavelengths.

Example: An electron with 10 eV energy has $\lambda \approx 3.9 \times 10^{-10}$ m.

Electron Diffraction

Electron diffraction experiments (e.g., Davisson-Germer) show that electrons can produce interference patterns, confirming their wave-like nature.

• Condition for Constructive Interference: $2d \sin \theta = m \lambda$ where $d$ is crystal plane separation, $\theta$ is scattering angle, $m$ is integer, $\lambda$ is electron wavelength.

The Heisenberg Uncertainty Principle

Principle and Relationships

The Heisenberg Uncertainty Principle states that the precision of simultaneously measuring certain pairs of physical properties (such as position and momentum) is fundamentally limited.

• Momentum and Position: $\Delta x \Delta p_x \geq \frac{h}{2\pi}$

• Energy and Time: $\Delta E \Delta t \geq \frac{h}{2\pi}$

• Interpretation: Imprecision is due to wave-particle duality, not experimental error.

Uncertainty Principle Explained

• Diffraction of electrons through a slit demonstrates the uncertainty principle.

• The product of the uncertainty in position ($\Delta y$) and velocity ($\Delta v$) remains approximately constant for different slit widths.

• Opening Angle of Central Maximum: $a \sin \theta = \lambda = \frac{h}{p}$

Summary Table

Concept	Key Equation	Significance
Photon Energy	$E = hf$	Energy of a photon depends on frequency
Photon Momentum	$p = \frac{h}{\lambda}$	Photons have momentum despite zero rest mass
Photoelectric Effect	$KE = hf - W_0$	Explains electron ejection from metals
Compton Effect	$\Delta \lambda = \frac{h}{m_e c}(1 - \cos \theta)$	Photon wavelength shift after scattering
de Broglie Wavelength	$\lambda = \frac{h}{mv}$	Wave nature of particles
Uncertainty Principle	$\Delta x \Delta p_x \geq \frac{h}{2\pi}$	Limits precision of measurements
Electron Diffraction	$2d \sin \theta = m \lambda$	Condition for constructive interference

ConcepTest Review

• Photon Energy: Higher frequency (blue light) means higher energy.

• Photoelectric Effect: Higher cutoff frequency means higher work function; electrons are emitted only if photon energy exceeds work function.

• Wave-Particle Duality: For same momentum, electron and proton have same wavelength; for same speed, lighter particle (electron) has longer wavelength.

Applications

• Solar panels (photoelectric effect)

• Electron microscopes (wave-particle duality)

• Astronomy (blackbody radiation, Wien's law)

Additional info: These notes expand on the provided slides with definitions, equations, and examples for clarity and completeness.