Questions tagged [path-integral]

Path integral formulation (Due to Feynman) is a major formulation of Quantum Mechanics along with Matrix mechanics (Due to Heisenberg and Pauli), Wave Mechanics (Due to Schrödinger), and Variational Mechanics (Due to Dirac).

DO NOT USE THIS TAG for line/contour integrals.

In the Path Integral formulation, a functional, called the phase, is associated with each path:

$$\phi = A e^\frac{iS}{\hbar}.$$

The Kernel, or the Matrix Element, is the path integral of this phase.

$$K(x ) =\int\phi\mbox{ } \mathcal{D}x.$$

The wavefunction, finally, is given by:

$$\Psi(x)=\int_{-\infty}^\infty \left(K(x,x_0)\Psi(x_0) \right) \mbox{d}x_0.$$

It is often surprising to many that the absolute value of the phase squared, $|\phi|^2$, is constant for all paths at $A^2$. However, this actually makes sense, as the position of the particle is initially completely well-defined, so Heisenberg's Uncertainty Principle tells us that we would have no idea about the momentum and thus no idea about its future position. However, the next moment, you know absolutely nothing about its momentum, and so on. This process coarse-grains a particular path, the classical path, which means it is much more probable than the other paths.

The mathematical description of this can be obtained by standard procedures (c.f. Feynman, Hibbs, Styer "Quantum Mechanics and Path Integrals", pg 77 - 79), and the final result is the (Time-Independent) Schrödinger's Equation.

$$\left(-\frac{\hbar^2}{2m}\nabla^2-i\hbar\frac{\partial }{\partial t }+U\right) \Psi = 0.$$