Quantum Physics Notes

To Wave or Not to Wave

• Physicists have long debated whether light is composed of particles or waves.
• Light exhibits both wave-like and particle-like behaviors.
• By 1900, experimental evidence suggested light was comprised of electromagnetic waves, but this theory had shortcomings.

The Ultraviolet Catastrophe

• The light-as-wave theory faced a problem called the ultraviolet catastrophe.
• Objects emit electromagnetic (EM) radiation due to vibrating electrons.
• The wave theory predicted that as the frequency of emitted EM waves increases (and wavelength decreases), energy should increase steadily.
• This prediction holds true for visible light frequencies but fails in the ultraviolet range.
• The "ultraviolet catastrophe" refers to the divergence of this prediction from reality at high frequencies.
• If the prediction were true, hot objects like stars would instantly emit all their energy.

The Photoelectric Effect

• Another issue with the light-as-wave theory is the photoelectric effect.
• Photosensitive metallic materials emit electrons (photoelectrons) when interacting with EM radiation.
• Ejection of electrons requires overcoming a work function, giving the electron a certain energy (K).
• Wave properties suggest that higher intensity should lead to more energy for electron ejection.
• However, experiments show that below a certain threshold frequency, no electrons are ejected, regardless of EM radiation intensity.

Planck's Quantum Theory

• In 1900, Max Planck proposed a new model for electromagnetic radiation that could explain these problems.
• Planck suggested that EM radiation is not emitted continuously but in discrete packets of energy $E_n$, called quanta.
• This theory implied that light was not a wave, which was initially confusing.

Einstein and the Photon

• In 1905, Albert Einstein proposed that the quantum energy packet is a particle-like photon.
• Photons are the smallest unit of electromagnetic radiation.
• EM radiation consists of photons with specific frequencies and quantized amounts of energy.
• A photon's energy is the product of its frequency and Planck's constant: $E = h \cdot v$ Where:
  • $E$ = energy of a photon
  • $v$ = frequency
  • $h$ = Planck's constant ($6.63 \times 10^{-34} Js$)

Wave-Particle Duality

• The discovery of photons revolutionized physics, leading to the understanding that light exhibits both particle and wave properties (wave-particle duality).
• Whether light behaves as a particle or a wave depends on the situation.
• All particles can also be described as waves.
• These small units are often referred to as quanta.

Photon Energy Example

• Calculating the energy of a photon with a wavelength of 10 nm:

  1. Find the frequency $v$:
     $c = \lambda \cdot f$
     $3 \times 10^8 m/s = (10 \times 10^{-9} m) \cdot f$
     $f = 3 \times 10^{16} Hz$

  2. Calculate the energy $E$:
     $E = h \cdot v$
     $E = (6.63 \times 10^{-34} Js) \cdot (3 \times 10^{16} Hz) = 1.989 \times 10^{-17} J$

  3. Convert to electron volts (eV):
     $E = (1.989 \times 10^{-17} J) / (1.6 \times 10^{-19} J/eV) = 124 eV$

Wave-like Behavior of Quanta

• Quanta exhibit wave-like behavior in certain situations, especially related to momentum.
• A quantum’s momentum can be calculated through its wavelength.
• Momentum (p) is equal to Planck's constant (h) divided by the wavelength (λ):
  $p = \frac{h}{\lambda}$

Momentum of a Photon

• Photons, despite being massless, possess momentum.
• Collisions between photons and electrons have been observed, and momentum is conserved during these collisions.
• The momentum of a massless quantum (photon) is equal to its energy (E) divided by the speed of light (c):
  $p = \frac{E}{c}$
• Since $E = h \cdot v$:,
  $p = \frac{h \cdot v}{c}$

Matter Waves

• Quanta with mass, like electrons, also exhibit wave-like behavior.
• Matter behaves like waves sometimes; these are called matter waves.
• This concept was proposed by Louis de Broglie and later supported by experiments.