Lecture 31: Light Quanta

Introduction to Light Quanta

• Light has previously been established as having wave-like properties, demonstrated through phenomena such as interference, diffraction, and polarization.
• Polarization specifically indicates that light functions as an electromagnetic wave.
• Quantum mechanics challenges this wave-only understanding by suggesting light can also be treated as a particle in certain scenarios.
• Concepts covered in this discussion include quantum physics, wave-particle duality, matter waves, the double-slit experiment, and the uncertainty principle.

The Transition to Quantum Physics

• In the late 19th and early 20th centuries, new phenomena were discovered that classical physics could not explain.
• Classical Physics: Comprises physics based on Newton’s laws of motion, Maxwell’s equations, and thermodynamics. These laws are highly effective for objects larger than atomic-size.
• Quantum Physics: A new branch of physics developed to describe phenomena occurring on a very small scale, specifically involving light and subatomic particles like atoms, electrons, and protons.

The Quantization of Properties

• The fundamental concept of quantum physics is that properties appearing continuous on a macroscopic scale become quantized at the extreme small scale.
• Quantized: A quantity that only takes on certain well-defined, discrete values.
• Examples of Quantized Quantities:
  • The number of people in a room (e.g., $40$ or $28$, but never $4.5$).
  • The number of pennies in a jar.
  • The number of atoms in a person.
  • The electrical charge of an atom.
  • The energy levels of electrons within an atom.
• Quantum physics is universally applicable to all scales (small and large), but classical physics (Newtonian laws) is used for macroscopic calculations (velocity, acceleration, momentum) because it is mathematically simpler and sufficiently accurate for objects visible to the eye.
• Current theoretical models—quantum physics, Newtonian physics, and general relativity—are rigorous but subject to change as scientific understanding evolves.

The Photoelectric Effect

• In 1905, Albert Einstein published a paper on the photoelectric effect, which led to his Nobel Prize in 1921. This paper provided evidence that light acts as a particle.
• Definition: The photoelectric effect is the emission of electrons when electromagnetic radiation (light) strikes a metal.
• Experimental Demonstration (Zinc and Electroscope):
  • A zinc plate is negatively charged (loaded with electrons) using a charged piece of plastic. An electroscope confirms the presence of the charge.
  • UV Light: When ultraviolet light is shone on the zinc, electrons are ejected, and the electroscope quickly returns to a neutral state.
  • Classical Prediction: Classical physics predicted that the intensity (brightness) of light would determine electron ejection. Dim light would eventually eject electrons after a time delay as energy built up.
  • Experimental Test of Classical Model: Shining a bright white light (much brighter than the UV source) on the zinc results in no change. No matter the duration, no electrons are ejected.

The Quantum Model of Light: Photons

• The quantum model explains that for the photoelectric effect, frequency matters rather than intensity.
• Light consists of individual, discrete, quantized particles known as photons.
• The energy of each individual photon is calculated using the formula:   $E = h \times f$   where $h$ represents Planck’s constant and $f$ represents the frequency.
• Electron ejection only occurs if individual photons meet or exceed a specific threshold energy.
• White Light Observation: White light fails to eject electrons from zinc because it is composed of lower frequencies than UV light. Therefore, each individual white light photon lacks the energy required to eject an electron, regardless of how many photons (intensity) hit the metal.

The Photovoltaic Effect

• Closely related to the photoelectric effect is the photovoltaic effect.
• Definition: This occurs when a photon hits a material and excites an electron to a state where it can cause an electrical current to flow.
• Application: The photovoltaic effect is the functional basis for generating electric power in solar farms.

Wave-Particle Duality

• Definition: The concept that the nature of light cannot be simplified to just a wave or just a particle; both definitions are accurate in different circumstances.
• General Rule:
  • When light travels through space, it acts like a wave (evidenced by interference and polarization).
  • When light interacts with matter, it acts like a particle (evidenced by the photoelectric and photovoltaic effects).
• Metaphor: Asking if light is a particle or a wave is compared to asking if a "spork" is a fork or a spoon; it is both.
• Einstein’s Perspective: He noted that we have two contradictory pictures of reality that individually fail to explain light phenomena entirely, but together they provide a complete explanation.

Matter Waves and the de Broglie Equation

• Wave-particle duality applies to matter as well; objects traditionally considered particles (like electrons) exhibit wave-like properties on a small scale.
• On a macroscopic scale, these wave-like properties are negligible and can be ignored.
• Matter Waves: Proposed by Louis de Broglie.
• de Broglie Equation:   $\lambda = \frac{h}{p}$   where $\lambda$ is the wavelength, $h$ is Planck’s constant, and $p$ is the momentum ($p = m \times v$).

Calculating Macroscopic vs. Subatomic Wavelengths

• Case 1: Macroscopic Baseball:
  • Mass ($m$): $150\,g = 0.15\,kg$
  • Velocity ($v$): $45\,m/s$
  • Momentum ($p$): $0.15\,kg \times 45\,m/s = 6.75\,kg \cdot m/s$
  • Planck’s Constant ($h$): $6.6 \times 10^{-34}\,J\cdot s$
  • Wavelength ($\lambda$): $\frac{6.6 \times 10^{-34}\,J\cdot s}{6.75\,kg \cdot m/s} = 9.8 \times 10^{-35}\,m$
  • Comparison: A baseball's diameter is about $0.07\,m$. The wavelength is essentially zero by comparison, making diffraction and interference unnoticeable.
• Case 2: Subatomic Electron:
  • Mass ($m$): $9.1 \times 10^{-31}\,kg$
  • Velocity ($v$): $1\%$ of the speed of light ($3 \times 10^6\,m/s$)
  • Momentum ($p$): $2.7 \times 10^{-24}\,kg \cdot m/s$
  • Wavelength ($\lambda$): $\frac{h}{p} = 2.4 \times 10^{-10}\,m = 0.24\,nm$
  • Comparison: This wavelength is on the scale of x-rays. Since the radius of an atom is much smaller than this wavelength, the wave nature of the electron is highly significant on this scale.

The Double-Slit Experiment

• The double-slit experiment provides evidence for the wave-nature of particles through diffraction and interference patterns.
• Wave Phenomenon: Diffraction occurs when waves pass through small openings. Two openings create an interference pattern of bright and dark spots.
• Single Photon Experiment: Even when single photons are sent through the slits one at a time, an interference pattern is observed on the screen over time, which classical physics cannot explain.
• Electron Experiment: When electrons are sent through a double slit, they also generate a diffraction pattern instead of a simple image of the slit openings. This confirms electrons exhibit wave-like behavior that persists even when sent through the slits individually.

The Uncertainty Principle

• The uncertainty principle is an inherent property of wave-like systems, not a limitation of technology.
• It defines the limits of how precisely we can know related pairs of properties.
• Position and Momentum:   $\Delta p \times \Delta x \ge \frac{\hbar}{2}$
  • $\Delta p$: Uncertainty in momentum.
  • $\Delta x$: Uncertainty in position.
  • $\hbar$ (h-bar): $\frac{h}{2\pi}$.
  • Implications: If the position of an object is known with high accuracy, the momentum becomes highly uncertain, and vice versa. Uncertainty can never be zero for both.
• Energy and Time:   $\Delta E \times \Delta t \ge \frac{\hbar}{2}$
  • $\Delta E$: Uncertainty in energy.
  • $\Delta t$: Uncertainty in the time at which the object had that energy.
• This principle exists on the macro scale, but because $\frac{\hbar}{2}$ is extremely small, the effects are trivial and unnoticeable for macroscopic objects.