Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Condensed matter physics → Band structure and transport theory ↓


Block theorem and the Kronig–Penney model


The Bloch theorem and the Kronig-Penney model are fundamental concepts in the study of band structure and transport theory in condensed matter physics. They are mainly used to understand the behavior of electrons in crystalline solids. In this discussion, we explore these two concepts in depth and relate them to real-world materials and phenomena.

Block's theorem

The properties of solids, especially their electronic behavior, can often be traced back to their periodic lattice structures. Block's theorem provides a powerful tool for understanding these properties by describing the wave functions of electrons in a periodic potential. The theorem is named after the physicist Felix Block.

In crystalline solids, atoms are arranged in a periodic lattice. This periodic potential has a profound effect on the behavior of the electrons, as it results in the formation of energy bands. Bloch's theorem states that the wave functions of electrons in a periodic potential can be expressed as plane waves modified by the periodic function. Mathematically, it is expressed as:

ψ_k(r) = u_k(r) * exp(i * k ⋅ r)

Where:

  • ψ_k(r) is the wavefunction of an electron with wavevector k.
  • u_k(r) is a periodic function with the same periodicity as the lattice.
  • exp(i * k ⋅ r) is a plane wave with wavevector k.

The implications of Bloch's theorem are important. It tells us that in a periodic potential, electrons retain wave-like properties, but with modifications due to the periodicity of the lattice. This leads to the concept of energy bands and band gaps, where certain energy levels are allowed or forbidden for electrons.

Visual representation

Consider a simple one-dimensional periodic potential:

In this simplified representation, the blue circles represent atoms in a lattice, and the line represents a periodic potential. Bloch's theorem shows that the wave functions of electrons will have periodic variations as they traverse this lattice, which fundamentally affects their allowed energy states.

Kroenig–Penney model

The Kronig–Penney model is an idealized model used to explore the implications of Blalock's theorem. It represents a one-dimensional lattice as a series of potential wells, which simplifies the complex problem of solving the Schrödinger equation for electrons in a periodic potential.

We consider a potential that has a rectangular shape, and that varies between high and low values as follows:

V(x) = { V_0, for 0 < x < a (potential barrier), 0, for a < x < a + b (free region), and repeats periodically with period a + b. }

The Schrödinger equation for an electron in such a potential is:

-ħ²/2m * d²ψ/dx² + V(x)ψ = Eψ

Solving this equation using Block's theorem gives the condition determining the allowed energy:

cos(ka) = cos(αa)cos(βb) - (p² + q²)/2pq * sin(αa)sin(βb)

Where:

  • α = sqrt(2m(E - V_0)/ħ²)
  • β = sqrt(2mE/ħ²)
  • p = αb q = βa
  • a + b is the period of the potential.

From the above equation, we can obtain the energy bands and gaps allowed by the atomic lattice where no electron states can exist. This forms the basis for understanding the electronic properties of materials.

Visual representation of the Kronig–Penney potential

Below is a graphical representation of the one-dimensional periodic potential used in the Kronig-Penney model:

A B

The potential varies between high and low values, indicating a periodic potential experienced by the electron. The shaded regions correspond to atomic positions in the lattice, and the electron's wave function should satisfy the Schrödinger equation in these regions.

Applications and implications

The results obtained from Bloch's theorem and the Kronig–Penney model have a profound impact in the field of solid state physics. They underlie the fundamental understanding of semiconductors, insulators, and conductors. The presence of energy bands explains why some materials can conduct electricity while others cannot. For example, in semiconductors, the band gap is small enough to allow electron excitation across the gap under certain conditions, which is responsible for their controllable conductivity.

In conclusion, Bloch's theorem and the Kronig-Penney model play important roles in condensed matter physics. They simplify complex potential scenarios into manageable problems, provide insight into material properties and guide the design of electronic components. By understanding these models, scientists and engineers can manipulate materials at the atomic level to advance technology.


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Block theorem and the Kronig–Penney model

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