Hydrogen wavefunctions

electron cloud

Orbitals of multi-electron atoms are qualitatively similar to those of hydrogen

Now that we have the closed-form analytical expressions for the Hydrogen wavefunctions, we can turn to the problem of their visualisation.

They extend to much extent to all atoms, as we shall see in the coming lectures, where they are called "atomic orbitals". For hydrogen, the stationary orbitals are specified by three quantum numbers:

$$n,\quad l\quad\text{and}\quad m$$

The number $n$ or principal quantum number can take any positive integer value, $n=1, 2, \cdots$. It describes the energy and also the average distance of the electron from the nucleus. Orbitals with the same $n$ are, for this reason, collectively described as belonging to the same "shell". It already appears in the Bohr model.

The number $l$, or azimuthal quantum number, describes the rotation, or angular momentum, of the electron. It is an integer, possibly and always at least zero, otherwise strictly bounded by the principal quantum number: $0\le l\le n-1$. The orbitals with the same $n$ and $l$ are called "subshells". They are denoted $s$ (sharp) for $l=0$, $p$ (principal) for $l=1$, $d$ (diffuse) for $l=2$ and $f$ (fundamental) for $l=3$ due to historical reasons (?!) and proceed beyond that alphabetically at the exception of j which is skipped: $g, h, i, k, l, \cdots$ This gives rise to naming such as 1s ($n=1, l=0$, i.e., the ground state) or $2p$ ($n=2$, $l=1$).

The number $m$, or magnetic quantum number, can take negative values, and is bounded in absolute value by $l$: $-l\le m\le l$.

real orbitals

(linear combination)

(see for instance this Wikipedia page). A great piece of code is provided by Jacopo Bertolotti:

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\[Alpha]0 = 1;
\[Psi][n_, l_, m_, r_, \[Theta]_, \[Phi]_] := Sqrt[(2/(n \[Alpha]0))^3 (n - l - 1)!/(2 n ((n + l)!))] E^(-r/(n \[Alpha]0)) ((2 r)/(n \[Alpha]0))^l LaguerreL[n - l - 1, 2 l + 1, (2 r)/(n \[Alpha]0)] SphericalHarmonicY[l, m, \[Theta], \[Phi]];
p1 = Flatten@Table[
   f = TransformedField["Spherical" -> "Cartesian", \[Psi][n, l, m, r, \[Theta], \[Phi]], {r, \[Theta], \[Phi]} -> {x, y, z}];
   DensityPlot3D[Abs[f]^2
    , {x, -30, 30}, {y, -30, 30}, {z, -30, 30}, ColorFunction -> Hue, ColorFunctionScaling -> True, Boxed -> False, Axes -> False, PlotLabel -> Style[StringForm["Hydrogen atom orbitals\n |\[Psi]\!\(\*SuperscriptBox[\(|\), \\(2\)]\) : n=`` l=`` m=``", n, l, m], Medium, FontFamily -> "DejaVu Serif"], LabelStyle -> {Black, Bold}, RegionFunction -> Function[{x, y, z}, x < 0 || y > 0], PlotLegends -> Automatic], {n, 1, 4}, {l, 0, n - 1}, {m, -l, l}]

= Links =

 [http://falstad.com/qmatom/ Applet].

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