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Quantum Mechanical Graphs of Orbitals

Quantum Mechanical Graphs of Orbitals | ChemCa.in
Physical Chemistry / Structure of Atom

Graphing Quantum Orbitals

Interpreting Wave Function, Probability Density, and Radial Distribution graphs.

The quantum mechanical model describes electrons as 3D standing waves. By graphing mathematical solutions to Schrödinger's equation against the distance from the nucleus ($r$), we can visualize electron locations. Let's explore the three crucial types of graphs.

1 Wave Function ($\Psi$) vs. Distance ($r$)

The wave function, denoted by $\Psi$ (Psi), represents the amplitude of the electron wave at a given distance $r$ from the nucleus. Mathematically, $\Psi$ can be positive, negative, or zero.

  • The point where $\Psi = 0$ (and changes sign) is called a Radial Node.
  • The number of radial nodes is given by the formula: $n - l - 1$.

$\Psi$ vs. $r$ (s-orbitals)

Notice how the 2s orbital crosses the zero axis.

Wave Function (Psi) vs Radius Graph of Psi vs r. 1s is a curve decaying from a max value. 2s starts high, drops below zero (radial node), then approaches zero. r $\Psi$ 1s 2s Radial Node

2 Probability Density ($\Psi^2$) vs. Distance ($r$)

While $\Psi$ has no physical meaning, its square, $\Psi^2$, represents the Probability Density. It tells us the probability of finding an electron in a tiny, specific point volume around the nucleus.

  • Because it is squared, $\Psi^2$ is always positive.
  • For all $s$-orbitals, $\Psi^2$ is maximum at the nucleus ($r=0$).
  • For $p, d,$ and $f$-orbitals, $\Psi^2$ is exactly zero at the nucleus.

$\Psi^2$ vs. $r$ (s-orbitals)

Values are strictly positive. Nodes touch zero.

Probability Density (Psi squared) vs Radius Graph of Psi squared vs r. 1s is a steep positive decay. 2s drops sharply to zero, then forms a secondary positive peak. r $\Psi^2$ 1s 2s Node

3 Radial Distribution ($4\pi r^2 \Psi^2$)

We usually care about finding the electron in a spherical shell at a distance $r$, rather than a single point. This is the Radial Probability Distribution Function.

$$\text{Prob.} = 4\pi r^2 \Psi^2 dr$$
  • Because the volume of a shell at $r=0$ is zero, the radial probability is always zero at the nucleus for all orbitals.
  • The peak of this graph indicates the distance where the electron is most likely to be found ($r_{max}$).
  • The total number of peaks on this graph is equal to: $n - l$.

Radial Probability vs. Radius ($r$)

Observe the number of peaks ($n-l$) for 1s, 2s, and 3s.

Radial Probability Distribution vs Radius Graph of 4 pi r squared Psi squared. The 1s curve starts at 0, has 1 peak. 2s starts at 0, has 2 peaks. 3s starts at 0, has 3 peaks. r $4\pi r^2 \Psi^2$ 1s 2s 3s

Quick Formula Summary

Orbital Radial Nodes
($n-l-1$)		 Angular Nodes
($l$)		 Total Nodes
($n-1$)		 Number of Peaks
in Graph ($n-l$)
1s 0 0 0 1
2s 1 0 1 2
2p 0 1 1 1
3s 2 0 2 3
3d 0 2 2 1

Knowledge Check

10 Practice MCQs on Quantum Orbital Graphs

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