Why is the probability of finding an electron in an orbital not 100 %?

Here is what I know and please correct me if I am wrong:  

An orbit can have 2 electrons only but electrons do not exist in defined orbits. An orbital is like a hollow sphere where the 2 electrons are most likely to be found. In other words the 2 electrons of an orbit are not confined to that pathway and can be found anywhere in the orbital.


Now my question is:  

Why isnt the probability of finding the two electrons of an orbital $100~\%$? Most websites say the probability is $95~\%$, where else could they be and why?




There is a nonzero probability of finding an electron anywhere except for at the nodes, where the probability is 0 by definition. An orbital might be better thought of as an infinitely large cloud that is more dense in some areas than others. Integrated over all space, there is also by definition a 100% chance of finding an electron. This should make sense intuitively (if if doesn't exist somewhere then where is it and what does the integral even mean?).


Orbitals really aren't so well-defined as the nice little shapes we represent them with (think cloud), but we can make them well-defined by arbitrarily deciding how much space we wish to enclose. It is often decided for the practicality of the representation to show a space such that the probability of finding an electron in it is 95%, though mathematically it's true that there is a chance of finding it outside the space.


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The concept of an *orbital* directly derives from the concept of a free rotator in simple quantum mechanics. Before students learn about a free rotator, they typically learn about the harmonic (potentially including the aharmonic) oscillator and — first of all — the particle in a box. These are the three probably most fundamental quantum mechanical concepts which can be used as a first basis to understand everything else.


The particle in a box is a somewhat idealised case because the box is defined to have walls of infinite potential energy. Only in that case is the particle actually *confined* to the box. All other fundamental concepts in principle have a wave function contribution from $-\infty$ to $+\infty$ in all relevant coordinate axes. Which means that there is a nonzero probability of finding the particle *anywhere* in the coordinate system.


For the free rotator — which in principle gives us hydrogen orbitals — this means we need to define an arbitrary cutoff point if we want to display anything at all. Because 5 is a nice number, most of the time people choose a cutoff value of $95~\%$ and draw a line around the entire area (or volume) inside of which the electron has a $95~\%$ chance of appearing. Depending on your application, you might want to choose $90~\%$ (for less crowding) or $99~\%$ (to show more interactions) instead.


$100~\%$ cannot be used since that would put *everything* into an orbital and therefore the concept of an orbital would lose its usefulness. In general, an electron belonging to an atom somewhere in New Zealand has a nonzero probability of being found right here where I am sitting in Germany — and likewise a nonzero probability of being found at the other end of the Andromeda galaxy. (Note that while they are nonzero, they average out to zero at quite a large number of significant digits.)


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