Path Integral Formulation
Feynman's Principle
Quantum amplitude is sum over all possible paths:
$$ \langle x_f|e^{-iHt/\hbar}|x_i\rangle = \int \mathcal{D}[x(t)] e^{iS[x]/\hbar} $$
Where $S[x]$ is the classical action:
$$ S[x] = \int_0^t L(x, \dot{x}, t') dt' $$
Key Idea
- Classical mechanics: One path (least action)
- Quantum mechanics: All paths contribute
- Paths near classical path contribute most
Applications
- Quantum field theory
- Statistical mechanics
- Quantum computing
Further Reading
Related Snippets
- Quantum Computing Basics
Qubits, gates, and quantum algorithms - Quantum Mechanics Basics
Wave functions, operators, and measurements - Quantum Operators
Position, momentum, and Hamiltonian operators - Schrödinger Equation
Time-dependent and time-independent formulations