Hi Physics Stack Exchange,
I’m a student with first year university chemistry knowledge and high school mathematics and physics knowledge (e.g. am familiar with complex numbers). I’m having some difficulty defining and understanding the the concept of “radial probability density” i.e. the square of the wavefunction. Its definition seems less intuitive to me than the definition for radial probability distribution. So, what I am going to do here is:
- Tell you what I know about I know about these quantities,
- Highlight the areas which I have difficulty understanding,
- Propose a definition for Psi^2 that makes sense to me, and
- Ask you to either validate or correct my thinking as necessary.
Just a note before we start, the scope of my course only introduces the general form of the wave equation but does not require us to calculate solutions. I’ve done a bit of background reading about the wave equation and wave function while researching this question but am not equipped with a detailed mathematical understanding of either, so if I make any oversimplifications or mistakes please correct me 😊
I am given the definition of radial probability density as “the probability of finding an electron in a sphere of distance r from the nucleus”. The plot of Psi^2 for the hydrogen 1s orbital indicates probability density increases toward the nucleus and decreases away from the nucleus.
However, this appears to conflict with what I know about radial probability distribution. The definition I have for this is “the probability of finding an electron on the surface of a spherical shell at distance r from the nucleus”. I am familiar that this is expressed as
i.e. probability density * an infinitesimally small difference in volumes between two spheres, which equates to 4pir^2.
The plot of this quantity can then be drawn by function multiplication of Psi^2 and 4pir^2.
Herein lies my problem: the plot of radial probability distribution has a maximum (which I understand to be at r = the Bohr radius). However, radial probability density is greater closer to the nucleus, which seems to suggest you are more likely to find an electron at the nucleus! I think the source of my confusion may be due to the wording of the Psi^2 definition.
My proposed solution to this issue is a more detailed explanation of Psi^2 I obtained by working backwards from the definition of probability distribution and using language that makes more sense to me:
“Radial probability density as a function of r is equal to the sum of all radial probability distribution values for the infinitesimally number of surfaces between the nucleus and the surfaces at r, divided by the volume of the sphere with radius r.” i.e. it is the probability of finding an electron expressed as a density of the volume being examined.
(As radial probability distribution is a PDS i.e. has integral value 1, this would suggest that you could find Psi^2 by integrating and then dividing the result by the volume of the sphere with radius r.)
Please clarify whether my understanding of this concept is correct or not, and if not, provide a definition of Psi^2 that helps explain the perceived contradiction highlighted earlier. Thank you so much for taking the time to look over this question.