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A well known result for the $l=0$ hydrogenic functions is that $$\psi_{nlm_l}=R_{nl}(r)Y_{lm_l}$$

$$|\psi_{n00}|^2=\frac{Z^3}{\pi a_0^3n^3}$$ where $R_{nl}$ and $Y_{lm_l}$ are the radial function and spherical harmonics. The above expression is a direct consequence of $R_{n0}(0) \neq 0$, which my lecturer used to conclude that there is a non-zero probability density of finding the electron at the origin. Another argument would be that the effective potential for the $l=0 \to -\infty$ as $r \to 0$. This also leads to $s$-core penetration in Helium.

However, what confuses me is that if the probability density is defined as $$\rho(r, \theta, \phi)drd\theta d\phi=|R_{nl}(r)|^2|Y_{lm_l}(\theta,\phi)|^2 r^2 sin(\theta)drd\theta d\phi.$$ Shouldn't the presence of $r^2$ in the volume element lead to zero whenever $r=0$? (i.e. the probability density of finding an electron near the origin is always zero regardless of whether $l=0$ or not)

Qmechanic
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Chern-Simons
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    The probability at a single point doesn't make any sense. If you want to talk about probability you have to integrate the density over some volume – FrodCube May 06 '22 at 16:56
  • @FrodCube yeah thanks just realised it was badly phrased, edited. – Chern-Simons May 06 '22 at 17:00
  • Related: Hydrogen radial wave function infinity at $r=0$ – Qmechanic May 06 '22 at 17:07
  • Why should the probability density at r=0 always vanish? – my2cts May 06 '22 at 17:22
  • @my2cts because $|R_{nl}(r)|^2r^2$ and $r^2=0$? – Chern-Simons May 06 '22 at 17:35
  • @Chern-Simons The probability of finding anything in a point is zero, unless the probability density diverges there. – my2cts May 06 '22 at 17:53
  • @my2cts yeah so my question is more about the probability density itself instead of the probability. Like, the s-core penetration of an excited electron in helium occurs for $l=0$ only and my lecturer has said that this is because $|R_{n0}(0)| \neq 0$ but I thought that the probability density is defined as $|R_{nl}(r)|^2r^2$ not $|R_{nl}(r)|^2$? – Chern-Simons May 06 '22 at 18:00
  • Aha that explains it. The $r^2$ factor is part of the volume element, not of the volume probability density. – my2cts May 06 '22 at 18:02
  • @my2cts I see, so the probability density is always just the square magnitude of the wavefunction excluding any volume element. Thanks! – Chern-Simons May 06 '22 at 18:08
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    @Chern-Simons That is correct. – my2cts May 06 '22 at 18:21
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    there is a difference between probability, and probability density in definition , see https://aapt.scitation.org/doi/10.1119/1.5092484 – anna v May 06 '22 at 18:34
  • Your expression (,,) in your past formula is pathologically and unconventionally defined. Normally, it is $|\psi(r,\theta,\phi)|^2$ but clearly not here! It is abusive of spherical coordinate notation. – Cosmas Zachos May 06 '22 at 19:12
  • @Cosmas Zachos so do I write $\int_V |\psi(r, \theta, \phi)|^2r^2sin(\theta)drd\theta d\phi$ or $\int_V |\psi(r, \theta, \phi)|^2 drd\theta d\phi$? – Chern-Simons May 06 '22 at 19:21
  • *Obviously* the first, since it involves the volume element in spherical coordinates! The second expression is crank gobbledygook, and should have never been written down! – Cosmas Zachos May 06 '22 at 19:23
  • @Cosmas Zachos I see. I think the major source of confusion is that there was a question on a past college exam which asked me to sketch the probability density of the 1s hydrogenic function in helium and went on to ask for the distance $r_{max}$ at which the probability density is a maximum. So I had to use $\partial |R_{10}(r)|r^2 /\partial r=0$ to get $r_{max}=a_0/2$ (correct answer), instead of $\partial |R_{10}(r)|^2/\partial r$ which is $r_{max}=0$ (wrong)... – Chern-Simons May 06 '22 at 19:29
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    The exam should have been careful in specifying probability of a sperical shell, I suppose; this is normally emphasized in first courses.... – Cosmas Zachos May 06 '22 at 19:35
  • @Cosmas Zachos This is indeed a first course and my lecturer has never mentioned it. That is really helpful, thanks! – Chern-Simons May 06 '22 at 20:26

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