Editing Propagator (section)

===Examples===
{{see also|Path integral formulation#Simple harmonic oscillator| Heat equation#Fundamental solutions}}
For a time-translationally invariant system, the propagator only depends on the time difference {{math|''t'' − ''t''′}}, so it may be rewritten as
<math display="block">K(x, t; x', t') = K(x, x'; t - t').</math>

The [[Wave packet#Free propagator|propagator of a one-dimensional free particle]], obtainable from, e.g., the [[Path integral formulation#Free particle|path integral]], is then
{{Equation box 1
|indent = :
|equation = <math>K(x, x'; t) = \frac{1}{2\pi} \int_{-\infty}^{+\infty} dk\, e^{ik(x-x')} e^{-\frac{i\hbar k^2 t}{2m}} = \left(\frac{m}{2\pi i\hbar t}\right)^{\frac{1}{2}} e^{-\frac{m(x-x')^2}{2i\hbar t}}.</math>
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Similarly, the propagator of a one-dimensional [[Quantum harmonic oscillator#Natural length and energy scales|quantum harmonic oscillator]] is the [[Mehler kernel]],<ref>E. U. Condon, [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076889/pdf/pnas01779-0028.pdf "Immersion of the Fourier transform in a continuous group of functional transformations"], ''Proc. Natl. Acad. Sci. USA'' '''23''', (1937) 158–164.</ref><ref>[[Wolfgang Pauli]], ''Wave Mechanics: Volume 5 of Pauli Lectures on Physics'' (Dover Books on Physics, 2000) {{ISBN|0486414620}}. Section 44.</ref> 
{{Equation box 1
|indent = :
|equation = <math>K(x, x'; t) = \left(\frac{m\omega}{2\pi i\hbar \sin \omega t}\right)^{\frac{1}{2}} \exp\left(-\frac{m\omega\big((x^2 + x'^2) \cos\omega t - 2xx'\big)}{2i\hbar \sin\omega t}\right).</math>
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The latter may be obtained from the previous free-particle result upon making use of van Kortryk's SU(1,1) Lie-group identity,<ref>Kolsrud, M. (1956). Exact quantum dynamical solutions for oscillator-like systems, ''Physical Review'' '''104'''(4), 1186.</ref>
<math display="block">\begin{align}
&\exp \left( -\frac{it}{\hbar} \left( \frac{1}{2m} \mathsf{p}^2 + \frac{1}{2} m\omega^2 \mathsf{x}^2 \right) \right) \\
&= \exp \left( -\frac{im\omega}{2\hbar} \mathsf{x}^2\tan\frac{\omega t}{2} \right) \exp \left( -\frac{i}{2m\omega \hbar}\mathsf{p}^2 \sin(\omega t) \right) \exp \left( -\frac{im\omega }{2\hbar} \mathsf{x}^2 \tan\frac{\omega t}{2} \right),
\end{align}</math>
valid for operators <math>\mathsf{x}</math> and <math>\mathsf{p}</math> satisfying the [[Canonical_commutation_relation|Heisenberg relation]] <math>[\mathsf{x},\mathsf{p}] = i\hbar</math>.

For the {{mvar|N}}-dimensional case, the propagator can be simply obtained by the product
<math display="block">K(\vec{x}, \vec{x}'; t) = \prod_{q=1}^N K(x_q, x_q'; t).</math>