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I was given a graph of square of the wave function of a hydrogen atom, against the distance of the electron from the nucleus (denoted by r).

What I know is that the square of the wave function gives the probability of finding an electron at a particular position, but could anyone explain to me that why does the area under this graph has to do with the probablity of locating an electron? Because I thought that on the graph where the highest value of square of wave function is, the corresponding value of r will be the answer. Isn't that so??

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    $\begingroup$ Think about this first: what is the probability of finding a point particle at one particular point in space? $\endgroup$
    – Wouter
    Nov 9, 2013 at 14:03

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It's not quite true that "the square of the wave function gives the probability of finding an electron at a particular position." The square of the wave function gives you the probability density of find the the electron as a particular position. IN order to get the probability of finding the electron in a given region of space, you need to integrate the square of the (normalized) wave function over the volume you're interested in.

Another way to think about why you need to integrate over some volume is that the wave function is giving you the probability density at a given point in space. That point is infinitesimally small; in mathematical terms, it has measure zero. Because it is infinitesimally small, there is an infinitesimally small probability of finding the electron there. You need to integrate over a finite volume in order to get a finite probability.

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    $\begingroup$ To be fair, one can stop one step before the actual integration, in the sense that f(x) dx is the probability of the particle being in the interval [x,x+dx]. But that leans more towards pure mathematics than physics. $\endgroup$
    – legrojan
    Nov 9, 2013 at 18:58
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The square of the wave function does not directly give you the probability to find a particle at position x. To get that, you first have to divide by the integral of this squared wave function. And as you probably know, the integral of a curve is actually its area.

There is a tutorial on this particular issue at University of St Andrews. You will notice that the probability density they use there contains a factor of $r^2$. This has to do with the use of spherical coordinates, much in the same way that you need to multiply by $r^2 \sin \theta$ when you perform a spherical integral in three dimensions. Refer to calculus books if you don't know what I am talking about here.

As a little exercise, think about why you should divide by the integral.

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