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<math display="block">\int_{x_2}^{x_1} \sqrt{2m \left( E-V(x)\right)}\,dx = (n_{\text{odd}}+1/2)\pi \hbar</math>
Letting <math display="inline">n_{\text{odd}}=2n-1 </math> where <math display="inline">n = 1,2,3,\cdots </math>, solving for <math display="inline">E </math> with given <math>V(x) = mg|x|</math>, we get the quantum mechanical energy of a bouncing ball:<ref>{{Cite book |last=Sakurai |first=Jun John |title=Modern quantum mechanics |last2=Napolitano |first2=Jim |date=2021 |publisher=Cambridge University Press |isbn=978-1-108-47322-4 |edition=3rd |___location=Cambridge}}</ref>
<math display="block">E = {\left(3\left(n-\frac 1 4\right)\pi\right)^{\frac 2 3} \over 2}(mg^2\hbar^2)^{\frac 1 3}. </math>
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