Editing Path integral formulation (section)

=== Time-slicing derivation ===
{{main|Relation between Schrödinger%27s equation and the path integral formulation of quantum mechanics}}
One common approach to deriving the path integral formula is to divide the time interval into small pieces. Once this is done, the [[Lie product formula|Trotter product formula]] tells us that the noncommutativity of the kinetic and potential energy operators can be ignored.

For a particle in a smooth potential, the path integral is approximated by [[zigzag]] paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position {{mvar|x<sub>a</sub>}} at time {{mvar|t<sub>a</sub>}} to {{mvar|x<sub>b</sub>}} at time {{mvar|t<sub>b</sub>}}, the time sequence
: <math>t_a = t_0 < t_1 < \cdots < t_{n-1} < t_n < t_{n+1} = t_b</math>
can be divided up into {{math|''n'' + 1}} smaller segments {{math|''t<sub>j</sub>'' − ''t''<sub>''j'' − 1</sub>}}, where {{math|''j'' {{=}} 1, ..., ''n'' + 1}}, of fixed duration
: <math>\varepsilon = \Delta t = \frac{t_b - t_a}{n + 1}.</math>

This process is called ''time-slicing''.

An approximation for the path integral can be computed as proportional to
: <math>\int\limits_{-\infty}^{+\infty} \cdots \int\limits_{-\infty}^{+\infty}
 \exp \left(\frac{i}{\hbar}\int_{t_a}^{t_b} L\big(x(t), v(t)\big) \,dt\right) \,dx_0 \, \cdots \, dx_n, </math>
where {{math|''L''(''x'', ''v'')}} is the Lagrangian of the one-dimensional system with position variable {{math|''x''(''t'')}} and velocity {{math|''v'' {{=}} ''ẋ''(''t'')}} considered (see below), and {{mvar|dx<sub>j</sub>}} corresponds to the position at the {{mvar|j}}th time step, if the time integral is approximated by a sum of {{mvar|n}} terms.<ref group=nb>For a simplified, step-by-step derivation of the above relation, see [http://www.quantumfieldtheory.info/website_Chap18.pdf Path Integrals in Quantum Theories: A Pedagogic 1st Step].</ref>

In the limit {{mvar|''n'' → ∞}}, this becomes a [[functional integral]], which, apart from a nonessential factor, is directly the product of the probability amplitudes {{math|{{bra-ket|''x<sub>b</sub>'', ''t<sub>b</sub>''|''x<sub>a</sub>'', ''t<sub>a</sub>''}}}} (more precisely, since one must work with a continuous spectrum, the respective densities) to find the quantum mechanical particle at {{mvar|t<sub>a</sub>}} in the initial state {{mvar|x<sub>a</sub>}} and at {{mvar|t<sub>b</sub>}} in the final state {{mvar|x<sub>b</sub>}}.

Actually {{mvar|L}} is the classical [[Lagrangian mechanics|Lagrangian]] of the one-dimensional system considered,
: <math> L(x, \dot x) = T-V=\frac{1}{2}m|\dot{x}|^2-V(x)</math>
and the abovementioned "zigzagging" corresponds to the appearance of the terms
: <math>\exp\left(\frac{i}{\hbar}\varepsilon \sum_{j=1}^{n+1} L \left(\tilde x_j, \frac{x_j - x_{j-1}}{\varepsilon}, j \right)\right)</math>
in the [[Riemann sum]] approximating the time integral, which are finally integrated over {{math|''x''<sub>1</sub>}} to {{mvar|x<sub>n</sub>}} with the integration measure {{math|''dx''<sub>1</sub>...''dx<sub>n</sub>''}}, {{mvar|x̃<sub>j</sub>}} is an arbitrary value of the interval corresponding to {{mvar|j}}, e.g. its center, {{math|{{sfrac|''x<sub>j</sub>'' + ''x''<sub>''j''−1</sub>|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.