Detailed Study Notes on Quantum Theory and Particle Model of Waves

A Particle Model of Waves

Quantum Theory 1 - Introduction to Particle Model of Waves

The Compton Effect
  • The photoelectric effect demonstrates that light behaves like particles (photons).
  • Characteristics of photons:
      - Mass: Photons have no mass.
      - Energy: Given by the equation:
    E=hfE = hf
        where:
        - hh = Planck's constant, approximately 6.63imes10−34extJs6.63 imes 10^{-34} ext{ J s}
        - ff = frequency of the light.
      - Momentum: Calculated as:
    P=EcP = \frac{E}{c}
        or
    P=hfcP = h \frac{f}{c}
        where cc = speed of light.
      - Photons have associated properties:
        - Short wavelength → high energy.
        - Long wavelength → low energy.
Photon Energy Equations
  • The photon energy can also be expressed as:
    E=1240exteVnmextwavelengthinnmE = \frac{1240 ext{ eV nm}}{ ext{wavelength in nm}}

The Compton Experiment (1922)

  • Scientist: Arthur Holly Compton.
  • Experiment: Fired X-rays at a graphite target.
  • Results from the experiment:
      1. Some X-rays observed:
         - Same wavelength.
         - No energy loss measured.
      2. Other X-rays observed:
         - Longer wavelength.
         - Lower energy and momentum.
  • Observations:
      - If the wavelength increases, the energy EE decreases:
    E=hcextwavelengthE = \frac{h c}{ ext{wavelength}}
  • Conclusion: Photons behave like particles and appear to collide (similar to billiard balls).

The Compton Effect Explained

  • Definition: The Compton Effect describes the increase in the wavelength (and decrease in energy) of a photon when it collides with a stationary electron.
  • Key Features:
      - Photons lose energy and momentum during the collision, which indicates particle behavior.
      - Evidence that photons can collide and transfer energy to electrons.
Conservation Laws in Compton Effect
  • Conservation of Energy: Total energy before the collision = Total energy after the collision.
  • Conservation of Momentum: Total momentum before the collision = Total momentum after the collision.
Illustrative Diagram of Compton Effect
  • Before Collision:
      - Photon possesses initial energy and momentum.
      - Electron is initially at rest.
  • After Collision:
      - The photon loses energy and momentum.
      - The electron gains energy and momentum.
Practical Implications
  • Photons can transfer energy and momentum, adhering to the laws of physics similar to massive particles.

Distinishing Effects

  • Photoelectric Effect vs Compton Effect:
      - Scenario 1: An X-ray strikes a target and ejects an electron without additional radiation — indicates the photoelectric effect.
      - Scenario 2: An X-ray collides with an electron and scatters — indicative of the Compton effect.
Properties of Scattered Photons
  • When a photon collides with an electron during the Compton effect:
      - It loses energy, resulting in a speed decrease or possibly appears to vanish if its energy transfers entirely.
      - The scattered photon will exhibit shorter wavelengths and greater frequencies compared to the incident photon.

Matter Waves

  • De Broglie's Hypothesis: If light (wave) can act as particles, particles can also exhibit wave-like properties.
  • Every moving particle has an associated wave called the de Broglie wavelength:
    extdeBrogliewavelength: λ=hmvext{de Broglie wavelength: } \ \lambda = \frac{h}{mv}
      - Where:
        - exth=6.63imes10−34extJsext{h} = 6.63 imes 10^{-34} ext{ J s} (Planck's constant)
        - m=extmassoftheparticle(kg)m = ext{mass of the particle (kg)}
        - v=extspeedoftheparticle(m/s)v = ext{speed of the particle (m/s)}
  • Examples of Matter Waves:
      1. Electron: Small mass allows for measurable wavelengths and exhibits wave behavior like diffraction.
      2. Baseball: High mass results in extremely small wavelengths, with no observable wave behavior.

Calculations of De Broglie Wavelengths

  • Application of de Broglie's wavelength equation to various particles.
  • Example Calculation:
      An electron accelerated by a potential difference of 75V75 V:
      - Calculate: Speed, Momentum, de Broglie Wavelength.
  • For a 7 kg bowling ball travelling at 8.5 m/s, finding the de Broglie wavelength yields:
    λ=1.11×10−34m\lambda = 1.11 \times 10^{-34} m
  • Explanation of negligible wave behavior for massive objects.
Comparison of Rates of Particle Movements
  • Potential Difference Calculations for Specific Wavelengths:
      - Required to accelerate particles to achieve specific de Broglie wavelengths.
  • Example with Alpha Particle: An alpha particle moving at a specific speed can be calculated for its de Broglie wavelength:
    λ=8.37×10−10m\lambda = 8.37 \times 10^{-10} m

Summary of Key Relations and Principles

  • Confirmation that wave behavior diminishes with increased mass and velocities.
  • Uncertainty Principle emphasizes limits of knowledge regarding position and momentum.
  • Momentum relations provide insight into interactions between particles of varying mass.