Editing Quantum harmonic oscillator (section)

===Highly excited states===
{{multiple image
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| image1 = Excited_state_for_quantum_harmonic_oscillator.svg
| image2 = QHOn30pdf.svg
| footer = Wavefunction (top) and probability density (bottom) for the {{math|1=''n'' = 30}} excited state of the quantum harmonic oscillator. Vertical dashed lines indicate the classical turning points, while the dotted line represents the classical probability density.
}}
When {{mvar|n}} is large, the eigenstates are localized into the classical allowed region, that is, the region in which a classical particle with energy {{math|''E''<sub>''n''</sub>}} can move. The eigenstates are peaked near the turning points: the points at the ends of the classically allowed region where the classical particle changes direction. This phenomenon can be verified through [[Hermite_polynomials#Asymptotic_expansion|asymptotics of the Hermite polynomials]],  and also through the [[WKB approximation]]. 

The frequency of oscillation at {{mvar|x}} is proportional to the momentum {{math|''p''(''x'')}} of a classical particle of energy {{math|''E''<sub>''n''</sub>}}  and position {{mvar|x}}.  Furthermore, the square of the amplitude (determining the probability density) is ''inversely'' proportional to  {{math|''p''(''x'')}},  reflecting the length of time the classical particle spends near {{mvar|x}}. The system behavior in a small neighborhood of the turning point does not have a simple classical explanation, but can be modeled using an [[Airy function]]. Using properties of the Airy function,  one may estimate the probability of finding the particle outside the classically allowed region,  to be approximately 
<math display="block">\frac{2}{n^{1/3}3^{2/3}\Gamma^2(\tfrac{1}{3})}=\frac{1}{n^{1/3}\cdot  7.46408092658...}</math>
This is also given, asymptotically, by the integral 
<math display="block">\frac{1}{2\pi}\int_{0}^{\infty}e^{(2n+1)\left (x-\tfrac{1}{2}\sinh(2x)  \right )}dx ~.</math>