Editing Propagator (section)

==Non-relativistic propagators==
In non-relativistic quantum mechanics, the propagator gives the probability amplitude for a [[Elementary particle|particle]] to travel from one spatial point (x') at one time (t') to another spatial point (x) at a later time (t).

The [[Green's function]] G for the [[Schrödinger equation]] is a function
<math display="block">G(x, t; x', t') = \frac{1}{i\hbar} \Theta(t - t') K(x, t; x', t')</math>
satisfying
<math display="block">\left( i\hbar \frac{\partial}{\partial t} - H_x \right) G(x, t; x', t') = \delta(x - x') \delta(t - t'),</math>
where {{math|''H''}} denotes the [[Hamiltonian (quantum mechanics)|Hamiltonian]], {{math|''δ''(''x'')}} denotes the [[Dirac delta-function]] and {{math|Θ(''t'')}} is the [[Heaviside step function]]. The [[Integral transform|kernel]] of the above Schrödinger differential operator in the big parentheses is denoted by {{math|''K''(''x'', ''t'' ;''x′'', ''t′'')}} and called the '''propagator'''.<ref group=nb> While the term propagator sometimes refers to {{mvar|G}} as well, this article will use the term to refer to {{mvar|K}}.</ref>

This propagator may also be written as the transition amplitude
<math display="block">K(x, t; x', t') = \big\langle x \big| U(t, t') \big| x' \big\rangle,</math>
where {{math|''U''(''t'', ''t′'')}} is the [[unitary operator|unitary]] time-evolution operator for the system taking states at time {{mvar|t′}} to states at time {{mvar|t}}.{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|pp=314,337}} Note the initial condition enforced by
<math display="block">\lim_{t \to t'} K(x, t; x', t') = \delta(x - x').</math>
The propagator may also be found by using a [[Path_integral_formulation#Path_integral_in_quantum_mechanics|path integral]]:
: <math>K(x, t; x', t') = \int \exp \left[\frac{i}{\hbar} \int_{t'}^{t} L(\dot{q}, q, t) \, dt\right] D[q(t)],</math>
where {{mvar|L}} denotes the [[Lagrangian mechanics|Lagrangian]] and the boundary conditions are given by {{math|''q''(''t'') {{=}} ''x'', ''q''(''t′'') {{=}} ''x′''}}. The paths that are summed over move only forwards in time and are integrated with the differential <math>D[q(t)]</math> following the path in time.{{sfn|Cohen-Tannoudji|Diu|Laloë|2019|p=2273}}

The propagator lets one find the wave function of a system, given an initial wave function and a time interval. The new wave function is given by
: <math>\psi(x, t) = \int_{-\infty}^\infty \psi(x', t') K(x, t; x', t') \, dx'.</math>
If {{math|''K''(''x'', ''t''; ''x''&prime;, ''t''&prime;)}} only depends on the difference {{math|''x'' − ''x′''}}, this is a [[convolution]] of the initial wave function and the propagator.

===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>
|border colour = #0073CF
|bgcolor = #F9FFF7}}

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>
|border colour = #0073CF
|bgcolor = #F9FFF7}}
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>