Wallis meets quantum mechanics

Calculus Level 4

\[\int _{ -\infty }^{ \infty }{ { e }^{ 2x(4-x) }dx={ e }^{ 8 }\prod _{ c=1 }^{ \infty }{ \frac { 2c }{ \sqrt { \alpha { c }^{ 2 }-\beta } } } }\]

Find the positive, integral values of \(\alpha\) and \(\beta\) which satisfy the given equation. Write your answer as \(\alpha +\beta\).


This problem is original, but slightly inspired by Digvijay Singh.

Picture credits: QHO-squeezed-vacuum1dB-animation-color by Geek3, Wikipedia

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