# Tag Info

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 Schrodinger), and Variational Mechanics (Due to Dirac). DO NOT USE THIS TAG for line/contour integrals.

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 Schrodinger), 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.

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

Where $x$ is actually the position of the particle in spacetime.

The intuition behind this is summarised by the following diagram:

The wavefunction, finally is given by:

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

The intuiton behind this is summarised by the following diagram:

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 it's future position. However, the next moment, you know absolutely nothing about it's 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) Schrodinger's Equation.

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