Path Integral Formulation - Concrete Formulation

Time-slicing Definition

For a particle in a smooth potential, the path integral is approximated by zig-zag paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position x_a at time t_a to x_b at time t_b, the time sequence

can be divided up into n + 1 little segments t_j − t_{j − 1}, where j = 1,...,n + 1, of fixed duration

This process is called time-slicing.

An approximation for the path integral can be computed as proportional to

$\int\limits_{-\infty}^{+\infty}\,d x_1 \ldots \int\limits_{-\infty}^{+\infty}\,dx_n \ \exp \left(\frac{{\rm i}}{\hbar}\int\limits_{t_a}^{t_b} L(x(t),v(t), t)\,\mathrm{d}t\right)$

where is the Lagrangian of the 1d-system with position variable x(t) and velocity v = ẋ(t) considered (see below), and dx_j corresponds to the position at the j-th time step, if the time integral is approximated by a sum of n terms.

In the limit n → ∞, this becomes a functional integral, which - apart from a nonessential factor - is directly the product of the probability amplitudes - more precisely, since one must work with a continuous spectrum, the respective densities - to find the quantum mechanical particle at t_a in the initial state x_a and at t_b in the final state x_b.

Actually is the classical Lagrangian of the one-dimensional system considered, also

where is the Hamiltonian,

, and the above-mentioned "zigzagging" corresponds to the appearance of the terms:

In the Riemannian sum approximating the time integral, which are finally integrated over x₁ to x_n with the integration measure dx₁...dx_nx̃_j is an arbitrary value of the interval corresponding to j, e.g. its center, (x_j + x_{j − 1})/2.

Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.

Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the Coulomb potential e2/r at the origin. Only after replacing the time t by another path-dependent pseudo-time parameter

the singularity is removed and a time-sliced approximation exists, that is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by İsmail Hakkı Duru and Hagen Kleinert. The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically the Duru-Kleinert transformation.

Read more about this topic: Path Integral Formulation, Concrete Formulation

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Path Integral Formulation - Concrete Formulation - Time-slicing Definition

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