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 → Quantum mechanics → Advanced wave mechanics ↓


Path integral formulation


The path integral formulation is a cornerstone concept in advanced wave mechanics and quantum mechanics, providing a new perspective on the behavior of particles at the quantum level. Originally proposed by physicist Richard Feynman, the path integral formulation reformulates quantum mechanics in such a way that it beautifully connects concepts from action and classical mechanics to quantum behavior. At its core, the path integral formulation allows us to calculate the probability of a particle traveling from one point to another by considering not just a single "classical" path, but the entire spectrum of possible paths.

Basic concepts

To begin to understand the path integral formulation, it is helpful to revisit the principle of least action in classical mechanics. In classical mechanics, the path taken by a particle between two points is the one for which the action is minimal. The action, usually denoted as S, is an integral of the Lagrangian L of the system over time.

S = ∫ L dt

Here, the Lagrangian L is defined as the difference between kinetic and potential energy. Given the initial and final states, classical mechanics will choose the path that minimizes this action S

Quantum mechanics perspective

However, quantum mechanics allows particles to behave in ways not allowed by classical dynamics. This is where Feynman's brilliant insight comes into play. Instead of a single trajectory, Feynman suggested that we consider all the possible paths the particle could take when transitioning from point A to point B. Each of these paths is given a complex number, known as a "phase", which is derived from the action of the path.

Path integral expression

The probability (or amplitude) of a particle traveling from point A to point B is calculated by summing these complex numbers. This integral over all possible paths is known as the path integral. Mathematically, we express it as:

Z = ∫ e^(iS/ħ) D[x(t)]

Here, ħ is the reduced Planck constant, and D[x(t)] symbolizes integration over all possible paths x(t). Note the presence of the complex exponent, e^(iS/ħ), which gives each path a phase factor determined by the action S

Analytical example

Consider a free particle moving from position x_i at time t_i to x_f t_f time t_i in one dimension. Classically, it would move in a straight line. However, in quantum mechanics, every path connecting these two points contributes to the probability amplitude. This means that not only straight line paths but also curved, zigzag and looped paths are conceivable.

Calculating the exact path integral is complicated, but by breaking it down into simpler examples or approximations (using techniques such as interference theory or lattice discretization), one can derive meaningful physical predictions.

Visualization along paths

To visualize this, consider a particle moving through a landscape marked by a series of peaks and valleys:

Path A - Zigzag Path B - Straight

In this scheme, the path integral approach considers not only the blue straight line path B, but also the zigzag red path A and other paths like it. Each path contributes an amount determined by the action computed on that path.

Interference pattern

An important insight from the path integral formulation is how it leads naturally to the concept of interference and the quantum principle of superposition. Because each path contributes a complex amplitude, when we sum over all possible paths, we add up all of these contributions. Due to the oscillatory nature of the exponential factor e^(iS/ħ), paths can interfere constructively or destructively.

constructive interference

In constructive interference, two paths that differ negligibly in action will have their amplitudes increase, reinforcing each other. This explains why some paths are more probable than others - it's similar to how waves can interfere.

Importance and applications

The path integral formulation provides deep insights and important computational approaches in quantum field theories and statistical mechanics. It is naturally suited to gauge theories and has influenced fields far beyond quantum mechanics, even reaching fission theory.

In quantum field theory

In the field of quantum field theory, the path integral approach provides a versatile framework necessary for the construction of modern particle physics. It facilitates the definition of transition amplitudes between states and the calculation of Feynman diagrams, which underpin theoretical calculations that match experimental discoveries at high-energy physics facilities such as CERN.

Statistical mechanics

The path integral formulation provides a close analogy to statistical mechanics, where path integrals can be used to calculate partition functions into statistical groups by treating time as an imaginary quantity, thereby allowing a better understanding of systems at finite temperature.

Conclusion

The path integral formulation reimagines how we understand particle trajectories within quantum mechanics, allowing for a much richer tapestry of possibilities than classical mechanics. Through its application, we gain important insights into the quantum realm, particularly through the principle of superposition and interference. Furthermore, it opens up new avenues in understanding the fundamental processes that govern the universe, providing a toolset for discovering new physics beyond the standard paradigm.


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Path integral formulation

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