quantum mechanics - Is there actually a 0 probability of finding an electron in an orbital node?

I have recently read that an orbital node in an atom is a region where there is a 0 chance of finding an electron.

However, I have also read that there is an above 0 chance of finding an electron practically anywhere in space, and such is that orbitals merely represent areas where there is a 95% chance of finding an electron for example.

I would just like to know if there truly is a 0 probability that an electron will be within a region defined by the node.

Many thanks.

Answer

The probability of finding the electron in some volume $V$ is given by:

$$ P = \int_V \psi^*\psi\,dV \tag{1} $$

That is we construct the function called the probability density:

$$ F(\mathbf x, t) = \psi^*\psi $$

and integrate it over our volume $V$, where as the notation suggests the probability density is generally a function of position and sometimes also of time.

There are two ways the probability $P$ can turn out to be zero:

1. 
   

   $F(\mathbf x, t)$ is zero everywhere in the volume $V$ - note that we can't get positive-negative cancellation as $F$ is a square and is everywhere $\ge 0$.

   
   



2. 
   

   we take the volume $V$ to zero i.e. as for the probability of finding the particle at a point

   
   

Now back to your question.

The node is a point or a surface (depending on the type of node) so the volume of the region where $\psi = 0$ is zero. That means in our equation (1) we need to put $V=0$ and we get $P=0$ so the probability of finding the electron at the node is zero. But (and I suspect this is the point of your question) this is a trivial result because if $V=0$ we always end up with $P=0$ and there isn't any special physical significance to our result.

Suppose instead we take some small but non-zero volume $V$ centred around a node. Somewhere in our volume the probability density function will inevitably be non-zero because it's only zero at a point or nodal plane, and that means when we integrate we will always get a non-zero result. So the probability of finding the electron near a node is always greater than zero even if we take near to mean a tiny, tiny distance.

So the statement the probability of finding the electron at a node is zero is either vacuous or false depending on whether you interpret it to mean precisely at a node or approximately at a node.

But I suspect most physicists would regard this as a somewhat silly discussion because we would generally mean that the probability of finding the elecron at a node or nodal surface is nebligably small compared to the probability of finding it elsewhere in the atom.