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=== Simple harmonic oscillator ===
{{see also|Propagator#Basic examples: propagator of free particle and harmonic oscillator| Mehler kernel}}▼
The Lagrangian for the simple harmonic oscillator is<ref>{{cite web |last1=Hilke |first1=M. |title=Path Integral |work=221A Lecture Notes |url=http://hitoshi.berkeley.edu/221A/pathintegral.pdf}}</ref>
: <math>\mathcal{L} = \tfrac12 m \dot{x}^2 - \tfrac12 m \omega^2 x^2.</math>
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Comparison to the above eigenstate expansion yields the standard energy spectrum for the simple harmonic oscillator,
: <math>E_n = \left( n + \tfrac12 \right) \hbar \omega~.</math>
▲{{see also|Propagator#Basic examples: propagator of free particle and harmonic oscillator| Mehler kernel}}
=== Coulomb potential ===
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