Editing Path integral formulation (section)

=== Free particle ===

The path integral representation gives the quantum amplitude to go from point {{mvar|x}} to point {{mvar|y}} as an integral over all paths. For a free-particle action (for simplicity let {{math|''m'' {{=}} 1}}, {{math|''ħ'' {{=}} 1}})
: <math>S = \int \frac{\dot{x}^2}{2}\, \mathrm{d}t,</math>
the integral can be evaluated explicitly.

To do this, it is convenient to start without the factor {{mvar|i}} in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions. The amplitude (or Kernel) reads:
: <math>K(x - y; T) = \int_{x(0) = x}^{x(T) = y} \exp\left(-\int_0^T \frac{\dot{x}^2}{2} \,\mathrm{d}t\right) \,\mathcal{D}x.</math>

Splitting the integral into time slices:
: <math>K(x - y; T) = \int_{x(0) = x}^{x(T) = y} \prod_t \exp\left(-\tfrac12 \left(\frac{x(t + \varepsilon) - x(t)}{\varepsilon}\right)^2 \varepsilon \right) \,\mathcal{D}x,</math>
where the {{mathcal|D}} is interpreted as a finite collection of integrations at each integer multiple of {{mvar|ε}}. Each factor in the product is a Gaussian as a function of {{math|''x''(''t'' + ''ε'')}} centered at {{math|''x''(''t'')}} with variance {{mvar|ε}}. The multiple integrals are a repeated [[convolution]] of this Gaussian {{mvar|G<sub>ε</sub>}} with copies of itself at adjacent times:
: <math>K(x - y; T) = G_\varepsilon * G_\varepsilon * \cdots * G_\varepsilon,</math>
where the number of convolutions is {{math|{{sfrac|''T''|''ε''}}}}. The result is easy to evaluate by taking the Fourier transform of both sides, so that the convolutions become multiplications:
: <math>\tilde{K}(p; T) = \tilde{G}_\varepsilon(p)^{T/\varepsilon}.</math>

The Fourier transform of the Gaussian {{mvar|G}} is another Gaussian of reciprocal variance:
: <math>\tilde{G}_\varepsilon(p) = e^{-\frac{\varepsilon p^2}{2}},</math>
and the result is
: <math>\tilde{K}(p; T) = e^{-\frac{T p^2}{2}}.</math>

The Fourier transform gives {{mvar|K}}, and it is a Gaussian again with reciprocal variance:
: <math>K(x - y; T) \propto e^{ -\frac{(x - y)^2}{2T}}.</math>

The proportionality constant is not really determined by the time-slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time slices the time evolution is quantum-mechanically unitary, but a more illuminating way to fix the normalization is to consider the path integral as a description of a [[stochastic process]].

The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the [[central limit theorem]], which can be interpreted as the first historical evaluation of a statistical path integral.

The probability interpretation gives a natural normalization choice. The path integral should be defined so that
: <math>\int K(x - y; T) \,dy = 1.</math>

This condition normalizes the Gaussian and produces a kernel that obeys the diffusion equation:
: <math>\frac{d}{dt} K(x; T) = \frac{\nabla^2}{2} K.</math>

For oscillatory path integrals, ones with an {{mvar|i}} in the numerator, the time slicing produces convolved Gaussians, just as before. Now, however, the convolution product is marginally singular, since it requires careful limits to evaluate the oscillating integrals. To make the factors well defined, the easiest way is to add a small imaginary part to the time increment {{mvar|ε}}. This is closely related to [[Wick rotation]]. Then the same convolution argument as before gives the propagation kernel:
: <math>K(x - y; T) \propto e^\frac{i(x - y)^2}{2T},</math>
which, with the same normalization as before (not the sum-squares normalization – this function has a divergent norm), obeys a free Schrödinger equation:
: <math>\frac{d}{dt} K(x; T) = i \frac{\nabla^2}{2} K.</math>

This means that any superposition of {{mvar|K}}s will also obey the same equation, by linearity. Defining
: <math>\psi_t(y) = \int \psi_0(x) K(x - y; t) \,dx = \int \psi_0(x) \int_{x(0) = x}^{x(t) = y} e^{iS} \,\mathcal{D}x,</math>

then {{mvar|ψ<sub>t</sub>}} obeys the free Schrödinger equation just as {{mvar|K}} does:
: <math>i\frac{\partial}{\partial t} \psi_t = -\frac{\nabla^2}{2} \psi_t.</math>