Modern Physics — Quantum and Solid State

All bodies at temperature $T > 0$ emit thermal radiation (blackbody radiation). The radiated power and spectral distribution depend strongly on temperature.

Stefan-Boltzmann law: the total power emitted per unit surface area is proportional to $T^4$: $$P = \sigma T^4 A, \qquad \sigma = 5.67\times10^{-8},\mathrm{W/(m^2 K^4)}$$ Doubling the temperature increases the radiated power by a factor of 16.

Wien's law: the peak emission wavelength is inversely proportional to $T$: $$\lambda_{max}, T = 2.898\times10^{-3},\mathrm{m\cdot K}$$ Hotter bodies appear "bluer" (peak at shorter $\lambda$); cooler bodies appear "redder".

Planck's hypothesis (1900): to explain the blackbody spectrum, Planck proposed that the energy of atomic oscillators is quantized, i.e., it can only take discrete values that are integer multiples of a fundamental quantum: $$E = h\nu, \qquad h = 6.626\times10^{-34},\mathrm{J\cdot s}$$ This radical hypothesis marked the birth of quantum physics, introducing the constant $h$ now known as Planck's constant.

Real-world applications: infrared thermal cameras, incandescent light bulbs, stellar temperature estimation, microwave ovens, cosmology (cosmic microwave background at 2.7 K).

Photoelectric effect (Einstein, 1905): when light shines on a metal surface, electrons may be emitted if the light frequency exceeds a material-dependent threshold. Einstein interpreted the phenomenon by postulating that light consists of photons, each carrying energy $E = h\nu$. The photon's energy is entirely transferred to an electron, which must first overcome the work function $\phi$ (binding energy) before acquiring kinetic energy: $$KE_{max} = h\nu - \phi$$ The threshold frequency is $\nu_0 = \phi/h$: below it, no electrons are emitted regardless of light intensity.

Compton scattering (1923): a photon scatters inelastically off a free electron, transferring part of its energy. The scattered photon's wavelength is longer than the initial one: $$\Delta\lambda = \lambda_C (1-\cos\theta), \qquad \lambda_C = \frac{h}{m_e c} = 2.426\times10^{-12},\mathrm{m}$$ where $\lambda_C$ is the Compton wavelength of the electron and $\theta$ is the scattering angle.

Photon properties: energy $E = h\nu$, momentum $p = h/\lambda$. The photon behaves as a particle despite being an electromagnetic wave — the first evidence of wave-particle duality.

Real-world applications: solar cells (photovoltaic panels), photodiodes and photomultipliers, light sensors, photoelectron spectroscopy (XPS, UPS), medical imaging (Compton scattering tomography).

de Broglie hypothesis (1924): if light, which is a wave, can exhibit particle-like behavior (photons), then massive particles such as electrons should also exhibit wave-like behavior. Every particle with momentum $p$ has an associated de Broglie wavelength: $$\lambda = \frac{h}{p} = \frac{h}{mv}$$ For an electron accelerated to $100,\mathrm{eV}$, $\lambda \approx 0.12,\mathrm{nm}$, comparable to interatomic distances in crystals — indeed, electron diffraction by crystals was observed by Davisson and Germer (1927), confirming the hypothesis.

Heisenberg uncertainty principle (1927): there exist pairs of physical quantities (called complementary) that cannot be measured simultaneously with arbitrary precision. The most important ones: $$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}, \qquad \Delta E \cdot \Delta t \geq \frac{\hbar}{2}$$ where $\hbar = h/(2\pi) = 1.055\times10^{-34},\mathrm{J\cdot s}$ is the reduced Planck constant. The first relation says that if we localize a particle within $\Delta x$, its momentum is at least uncertain by $\hbar/(2\Delta x)$. The second implies that the energy of a state can fluctuate by $\Delta E$ for a time $\Delta t \sim \hbar/(2\Delta E)$.

Real-world applications: electron microscope (exploits short electron wavelength for high resolution), quantum tunneling (basis of STM and flash transistors), quantum computers.

The Bohr model (1913) combines quantized energy levels with classical orbital motion. The electron in a hydrogen atom can only occupy orbits with quantized angular momentum $L = n\hbar$, corresponding to discrete energies: $$E_n = -\frac{13.6,\mathrm{eV}}{n^2}, \qquad n = 1, 2, 3, \ldots$$ The $n=1$ state is the ground state (lowest energy); states with $n>1$ are excited states.

When an electron transitions from $n_i$ to $n_f$ with $n_i > n_f$, it emits a photon whose energy equals the difference: $$h\nu = E_{n_i} - E_{n_f}$$ The emitted wavelength follows the Rydberg formula: $$\frac{1}{\lambda} = R_H \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right), \qquad R_H = 1.097\times10^7,\mathrm{m^{-1}}$$ The main spectral series of hydrogen are: Lyman ($n_f=1$, ultraviolet), Balmer ($n_f=2$, visible), Paschen ($n_f=3$, infrared).

Real-world applications: atomic spectroscopy for element identification (spectral fingerprints), lasers (population inversion between energy levels), sodium and mercury vapor lamps, astronomy (chemical composition of stars).

When atoms come together to form a solid, their discrete energy levels broaden into bands due to interactions between neighboring atoms. The bands are separated by forbidden energy regions called the band gap $E_g$. The band structure determines the electrical properties of the material.

• Metals: the valence band is partially filled or overlaps with the conduction band. Electrons move freely, resulting in very high conductivity. Examples: Cu, Ag, Au.
• Insulators: $E_g > 5,\mathrm{eV}$. The valence band is full and the conduction band is empty; the required energy jump is too large for thermal excitation. Examples: diamond ($E_g \approx 5.5,\mathrm{eV}$), quartz.
• Semiconductors: $E_g$ is small enough ($\lesssim 2,\mathrm{eV}$) to allow a fraction of electrons to reach the conduction band via thermal excitation. Key examples: Si ($E_g=1.1,\mathrm{eV}$), Ge ($0.67,\mathrm{eV}$), GaAs ($1.42,\mathrm{eV}$).

Doping: controlled addition of impurities drastically changes conductivity. n-type (donors, e.g., P in Si) introduces extra electrons into the conduction band; p-type (acceptors, e.g., B in Si) creates "holes" (missing electrons) in the valence band.

The p-n junction is the fundamental building block of solid-state electronics: it allows current flow in only one direction (rectification). It is at the heart of diodes, transistors, LEDs, solar cells, and integrated circuits.

Real-world applications: microprocessors (millions of transistors on a chip), photovoltaic solar panels, LED lighting, image sensors (CCD, CMOS), flash memory, thermistors (temperature sensors).