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Homework Statement
The wave function for the three dimensional oscillator can be written
##\Psi(\mathbf r) = Ce^{-\frac{1}{2}(r/r_0)^2}##
where ##C## and ##r_0## are constants and ##r## the distance from the origen.
Calculate
a) The most probably value for ##r##
b) The expected value of ##r##
c) The expected value of ##1/r##.
Homework Equations
Expected value of function with a normed wave function
##<f(r)> = \int dr \Psi(r)^*f(r) \Psi ##.
The Attempt at a Solution
a) It is my understanding that the most probably value is the maximum value of ##\Psi^* \Psi## or equivalently in this case the maximum of ##\Psi## which is at ##r=0##.
The answer to the exercise however disagrees and says it's at ##r_0##.
If the oscillator is centered at ##r_0## this makes physical sense with the most probably value of course being the centre but it doesn't seem to agree with maximizing the function.
b) Norming the wave function
##1 = C^2 \int_0^\infty e^{-r^2/r_0^2}dr = C^2 \int_0^\infty r_0 e^{-s^2}ds = C^2\frac{r_0 \sqrt{\pi}}{2}## and hence that ##C^2 = \frac{2}{r_0 \sqrt{\pi}}##.
The expected value of ##r## is then
##<r> = \frac{2}{r_0\sqrt{\pi}} \int_0^\infty re^{-r^2/r_0^2} dr = \frac{r_0}{\sqrt{\pi}}##.
The answer however says ##\frac{2r_0}{\sqrt{\pi}}##.
c) For this one, I get a divergent integral ##<1/r> = \frac{2}{r_0\sqrt{\pi}} \int_0^\infty \frac{e^{-r^2/r_0^2}}{r} dr##.