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 xa at time ta to xb at time tb, the time sequence
can be divided up into n + 1 little segments tj − tj − 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
where is the Lagrangian of the 1d-system with position variable x(t) and velocity v = ẋ(t) considered (see below), and dxj 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 ta in the initial state xa and at tb in the final state xb.
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 xn with the integration measure dx1...dxnx̃j is an arbitrary value of the interval corresponding to j, e.g. its center, (xj + xj − 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.
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