Black Body Radiation and Einstein's Photon Theory

Black Body Radiation and Einstein's Photon Theory

Review of Black Body Radiation

• The average energy of a mode $f$ specified by $(n<em>x, n</em>y, n_z)$ is not equal to $kT$ due to the quantization of energy levels.
• Planck's quantum hypothesis states that energy levels are quantized, leading to certain ranges of the electromagnetic spectrum being less present than expected.
• For very high frequencies, the probability is proportional to $e^{-\frac{hf}{kT}}$, where even with $n = 1$, the energy is much greater than $kT$.
• If hf >> kT, the $n = 1$ energy level has a low probability.
• If energy weren't quantized, a small amount of energy in the mode would result in a non-zero probability, leading to $kT$.

Quantization of Energy Levels

• Energy levels are discrete, such as $E = nhf$ where $n = 0, 1, 2, …$
• Energy levels between $n = 0$ and $n = 1$ are not allowed, which would have made $e^{-\frac{E}{kT}}$ not close to zero and the average energy equal to $kT$.
• This quantization of energy levels explains the shape of the black body radiation graph.

Quantization in Electron Resonators

• The quantization of energy levels also applies to the electrons in the walls of the cavity (resonators), as they are in equilibrium with the electromagnetic substance.
• If the energy in the electromagnetic substance is quantized, the energy of the resonators will also be quantized.
• Electrons accelerating and radiating energy lose energy and exist at different quantized energy levels.

Modeling Electron Resonators

• A simplified model treats electrons as one-dimensional harmonic oscillators.
• The energy level of the electrons is quantized, leading to specific amplitudes of oscillation.
• Maxwell's model suggests that each resonator oscillates at a certain frequency which it cannot deviate from.
• The angular frequency $\omega = 2 \pi f$ equals $\sqrt{\frac{k}{m}}$, where $k$ relates to the electromagnetic forces and $m$ is the mass of the electron.
• Electrons jump between energy levels, emitting electromagnetic energy in the mode of $hf$.

Dynamic Equilibrium and Quantization

• Interactions between electrons and internal radiations complicate the model.
• Even considering these complexities, the quantization of the electromagnetic substance implies that any process exchanging thermal energies must also be quantized.
• If a process radiated a fraction of $kT$ into the electromagnetic substance, quantization wouldn't be observed.

Calculation of Average Energy

• Since the average energy is not $kT$, it is calculated as the sum over the probability of the energy level times the energy level, divided by the sum of the probabilities:
  $\bar{E} = \frac{\sum P(E) \cdot E}{\sum P(E)}$
• The denominator normalizes the probabilities to ensure they add up to one.
• Using Boltzmann factors, which provide relative probabilities, normalization is necessary.

Normalizing Probabilities

• An analogy is presented using a bucket of coins (dimes and quarters) to illustrate the need to normalize probabilities to obtain true probabilities.
• True probabilities are calculated as:
  $P<em>{true} = \frac{P</em>{unscaled}}{\sum P_{unscaled}}$

Derivation of Average Energy Formula

• The numerator involves summing terms like $n \cdot hf \cdot e^{-\frac{nhf}{kT}}$, and the denominator involves summing $e^{-\frac{nhf}{kT}}$.
• Define $x = e^{-\frac{hf}{kT}}$, which simplifies the series.
• The average energy is then given by:
  $\bar{E} = hf \cdot \frac{x + 2x^2 + 3x^3 + …}{1 + x + x^2 + x^3 + …}$
• The infinite series can be evaluated mathematically.
• The denominator can be shown to equal $\frac{1}{1 - x}$.
• The final result is the average energy:
  $\bar{E} = \frac{hf}{e^{\frac{hf}{kT}} - 1}$

Planck's Radiation Law

• The result is Max Planck's radiation law, giving the average energy in the mode $f$ considering the quantization of energy levels.
• Other modes are determined by quantization numbers $(n<em>x, n</em>y, n_z)$, which determine the frequency.
• Low $n$ values correspond to microwave range, thousands to infrared, and tens of thousands to ultraviolet.

Frequency Dependence

• The analysis applies to each mode $f(n<em>x, n</em>y, n_z)$.
• Graphing the function of average energy vs. frequency shows it starting at $kT$ and decreasing.
• At small $f$, the average energy is approximately $kT$, as the energy spectrum is essentially continuous and small energy levels are accessible.
• This consistency holds for frequencies smaller than infrared, like microwaves and radio waves.

Calculating Energy Density in the Cavity

• The energy density in the cavity is now given by:
  $u(f) df = \frac{8 \pi}{c^3} f^2 \cdot \frac{hf}{e^{\frac{hf}{kT}} - 1} df$
• This energy is between $f$ and $f + df$.
• As $f$ approaches zero, the function goes to $kT$, which can be shown using a Taylor expansion for $e^{\frac{hf}{kT}}$.

Energy Flux Density

• Calculate how much energy exits the cavity through a small hole of area $a$ in time $\Delta t$.
• Only energy within a radius $c \Delta t$ can escape in that time.
• The spectral energy flux density is the energy per unit area per unit time.

Deriving Spectral Energy Flux Density

• The spectral energy flux density equals $\frac{1}{4}c \cdot u$.
• The derivation involves a triple integral in spherical coordinates:
  $\int<em>{0}^{c \Delta t} \int</em>{0}^{\frac{\pi}{2}} \int_{0}^{2 \pi} r^2 \sin(\theta) d\phi d\theta dr$
• The integrand includes a cosine theta factor due to the angle-dependent effective area of the hole.

Full Black Body Spectrum

• The spectral energy flux density of a black body is given by:
  $F(f) df = \frac{2 \pi h}{c^2} \frac{f^3}{e^{\frac{hf}{kT}} - 1} df$
• The units are joules per meter squared per second per frequency.

Significance of Black Body Problem

• The birth of quantum ideas arises from understanding thermal physics.
• Thermal physics, with its disorganized energy, contrasts with the neatly organized discrete levels of quantum theory.

Einstein's Photon Theory

• Einstein proposed that the light itself is quantized, not just the energy levels.
• He suggested that electromagnetic substance is quantized into photons.
• In the $n = 4$ state, there are four separate photons, each with energy $hf$.

Particle-like Nature of Photons

• Einstein viewed photons as having a particle-like nature with a location.
• Photons are localized in space, unlike extended waves.

Questions Raised by Photon Theory

• Several questions arise:
  • What size do photons have?
  • How do photons satisfy boundary conditions for standing waves?
  • How does a photon know its frequency and wavelength?

Challenges with Photon Theory

• The equation $E = hf = \frac{hc}{\lambda}$ combines particle and wave ideas, leading to conceptual problems.
• It is difficult to visualize how a particle-like creature can have a wavelength.

Wave Packets

• Quantum mechanics combines these ideas by representing photons as localized wave packets.
• This approach will be explored further in later chapters.

Experimental Verification: Photoelectric Effect

• Einstein proposed the photoelectric effect to verify the photon theory.
• Shining electromagnetic radiation (light) onto a metal surface leads to the ejection of electrons.
• The metal has a work function $\phi$, the energy needed to liberate an electron.
• Units of work function are joules or electron volts.

Shortcomings of Wave Theory

• Wave theory predicts that increasing the intensity of incoming light should lead to ejected electrons.
• However, even with high-intensity green light, no electrons are ejected.

Success of Photon Theory

• Photon theory states that each photon must have enough energy to eject an electron; otherwise, increasing the intensity is irrelevant.
• The solution is to increase the frequency, e.g., switch from green to ultraviolet light.
• A few ultraviolet photons can eject electrons, while a million green photons cannot.

Energy Balance in Photoelectric Effect

• The kinetic energy of the ejected electron equals the photon's energy minus the work function:
  $KE = hf - \phi$
• This shows conservation of energy in the photon-electron-atom system.
• Graphing kinetic energy as a function of $f$ shows a linear relationship above a certain frequency.

Cutoff Frequency

• The cutoff frequency is when $hf = \phi$, where the kinetic energy is zero.
• For most metals, the cutoff frequency is in the ultraviolet range.
• The photoelectric effect experiment is best conducted in a vacuum to avoid electron interactions with matter.
• Einstein received the Nobel Prize for explaining the photoelectric effect.

Wave-Particle Duality

• Unanswered questions remain about how electromagnetic substance exhibits both wave and photon properties.
• The two-slit experiment suggests wave-like behavior, while photoelectric effect indicates particle-like behavior.
• Experiments can best be understood by choosing either a wave or a particle view.

Generation of X-Rays: Bremsstrahlung

• Accelerating electrons through a voltage and crashing them into a metal leads to the emission of radiation at all frequencies.
• This is known as Bremsstrahlung (braking) radiation.
• X-rays are only generated if the electron has enough energy to equal the energy of an X-ray photon.

Maximum Frequency of Emitted Radiation

• The maximum frequency of emitted light occurs when nearly 100% of the electron's kinetic energy is converted into a single photon.
• The maximum frequency is given by:
  $KE = (\gamma - 1) m_0 c^2 = hf$
• This provides further confirmation of the photon theory.

Experiments Supporting Photon Theory

• The photoelectric effect and the generation of X-rays support the photon theory.
• Other experiments, like thin-film interference and Young's two-slit experiment, require the wave theory.
• Wave-particle duality is understood by choosing either a wave view or a particle view to explain different experiments.