Editing Path integral formulation (section)

=== The Schrödinger equation ===

{{Main|Relation between Schrödinger's equation and the path integral formulation of quantum mechanics}}

The path integral reproduces the Schrödinger equation for the initial and final state even when a potential is present. This is easiest to see by taking a path-integral over infinitesimally separated times.
: <math>\psi(y;t+\varepsilon) = \int_{-\infty}^\infty \psi(x;t)\int_{x(t)=x}^{x(t+\varepsilon)=y} e^{i\int_t^{t+\varepsilon} \bigl(\frac{1}{2}\dot{x}^2 - V(x)\bigr)dt} Dx(t)\,dx\qquad (1)</math>

Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of {{mvar|ẋ}}, the path integral has most weight for {{mvar|y}} close to {{mvar|x}}. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. (This separation of the kinetic and potential energy terms in the exponent is essentially the [[Lie product formula|Trotter product formula]].) The exponential of the action is
: <math>e^{-i\varepsilon V(x)} e^{i\frac{\dot{x}^2}{2}\varepsilon}</math>

The first term rotates the phase of {{math|''ψ''(''x'')}} locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to {{mvar|i}} times a diffusion process. To lowest order in {{mvar|ε}} they are additive; in any case one has with (1):
: <math>\psi(y;t+\varepsilon) \approx \int \psi(x;t) e^{-i\varepsilon V(x)} e^\frac{i(x-y)^2 }{ 2\varepsilon} \,dx\,.</math>

As mentioned, the spread in {{mvar|ψ}} is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase that slowly varies from point to point from the potential:
: <math>\frac{\partial\psi}{\partial t} = i\cdot \left(\tfrac12\nabla^2 - V(x)\right)\psi\,</math>
and this is the Schrödinger equation. The normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.