PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Quantum mechanics → Schrödinger Equation ↓


Path integrals in quantum mechanics


The concept of path integrals in quantum mechanics is an innovative approach that extends the traditional Schrödinger equation. Originally proposed by Richard Feynman, the path integral formulation offers a different perspective on quantum mechanics, providing insights and computational methods that can be beneficial in a variety of physical problems. This formulation is particularly powerful in systems where the potential changes with time or when dealing with particles in field theory.

Original idea

In classical mechanics, the path of a particle is well defined; it travels on a path that minimizes action, according to the principle of least action. However, quantum mechanics requires a very different approach because of the inherent uncertainty in the position and momentum of the particle.

Feynman's path integral formulation assumes that a particle does not travel along a single path, but rather along all possible paths simultaneously. Each path is assigned a probability amplitude, and the actual path taken by the particle is the sum of all these amplitudes. Mathematically, the probability amplitude for a particle to travel from point A to point B is given as:

⟨B|A⟩ = ∫ D[paths] e^(iS[path]/ħ)

Here, ⟨B|A⟩ denotes the amplitude of the journey from A to B, S[path] is the classical action associated with a particular path, and ħ is the reduced Planck constant. D[paths] indicates that we integrate over all possible paths.

Relation with the Schrödinger equation

The path integral formulation is deeply connected to the Schrödinger equation. While the Schrödinger equation provides a differential equation for the wave function, the path integral formulation provides an alternative, integral method to calculate the uniform evolution of quantum states.

In the Schrödinger framework the time evolution of a quantum state from time t_0 to t is given by the unitary time evolution operator:

ψ(t) = U(t, t_0)ψ(t_0)

When considered in terms of path integrals, this evolution corresponds to the sum of the contributions from each possible path the particle could take during that time interval. Both methods ultimately provide the same probability distribution, although they do so through very different conceptual frameworks.

A visual example of path integrals

A B

In the illustration above, we see multiple paths from point A to point B. In the path integral formulation, the amplitude for a particle to travel from A to B is the sum of the contributions from all possible paths, highlighted here by different colors.

Mathematical background

A key element in the path integral is the classical action, S, which is defined as:

S = ∫[t0, t] L(x(t), x'(t), t) dt

where L is the Lagrange function given as:

L = T - V

T is the kinetic energy and V the potential energy. The path integral approach reformulates quantum mechanics by considering contributions from different paths weighted by a factor e^(iS/ħ). This approach connects deeply with classical physics through the classical limit, ħ → 0, when the classical path dominates.

Applications of path integrals

The path integral formulation reveals its power in several advanced applications:

1. Quantum field theory

Path integrals play a key role in the development of quantum field theories, particularly gauge theories and the Standard Model of particle physics. They facilitate calculations involving interactions in fields, allowing scientists to evaluate the processes that generate particles from fields and a variety of other phenomena.

2. Statistical mechanics

The connection between path integrals and statistical mechanics emerges naturally through the Euclidean path integrals used in statistical field theory. Here, path integrals help evaluate partition functions by providing a method to connect all possible microscopic states of a system.

3. Quantum computing

Insights gained from path integrals contribute to quantum computing through the understanding of quantum superpositions and state transitions. As quantum computing develops, path integrals may provide new ways to process quantum algorithms.

Another example with action

A B

This illustration again highlights the paths from point A to point B, with one distinguishable classical path (dashed line). The action associated with each path affects its contribution to the final amplitude, specifically in the form of the factor e^(iS/ħ) which physically represents interference effects.

Closing thoughts

The path integral formulation of quantum mechanics provides a profound but elegant method for understanding quantum phenomena beyond the capabilities of the traditional Schrödinger equation. By accepting uncertainty and integrating over all possible paths, this formulation ties closely with the probabilistic nature of quantum mechanics. Its myriad applications in quantum fields, statistical mechanics, and emerging technologies make it an invaluable tool in modern theoretical physics.


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Path integrals in quantum mechanics

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