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When studying wavefunctions in Chemistry, we modelled an oscillator in the form $e^{-ax^2}$.
When I squared this wave function (taking $a = 1$) and plotted the graph, I found that the line decayed faster than my original function.
This is shown below.
I was trying to work out mathematically why this was the case but I couldn't understand for as long as I tried.
Why is this the case?
$\endgroup$ 22 Answers
$\begingroup$Because, it is :
$$\left(e^{-x^2}\right)^2 = e^{-2x^2} < e^{-x^2}$$
The exponent of the squared expression is bigger (or equal for some distinct cases), thus with the minus sign, you have a decaying exponential function. Of course, the decay becomes bigger if the exponent grows bigger.
$\endgroup$ 0 $\begingroup$Simply for $\alpha\in (0,1)$:$$(1-\alpha)^2=1-(2\alpha-\alpha^2)<1-\alpha$$
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