Equations with meaning
Ψ₃₂m(r, θ, φ) = R₃₂(r)Y₂m(θ, φ)Every d orbital has angular quantum number ℓ = 2.
Probability density = |Ψ|²The square gives a nonnegative likelihood even where the wavefunction changes sign.
Radial probability = r²|R₃₂(r)|²Probability in a thin spherical distance interval.
Radial nodes = n − ℓ − 1 = 0Angular nodes = ℓ = 2Total nodes = n − 1 = 2E₃ = −13.6 eV / 3² = −1.51 eVIn isolated hydrogen, all five 3d states have the same energy.
The familiar dxy, dxz, dyz, and dx²−y² pictures are real combinations of mℓ states. The dz² orbital corresponds to mℓ = 0 for the selected z axis.
Model boundaries
This is a one-electron, nonrelativistic hydrogen 3d model. It is a probability model—not a planetary orbit, electron path, or exact description of a many-electron transition-metal atom.
The field-mapping chamber, detector ring, node guides, and scanner are illustrative teaching apparatus. Cloud points and measurement markers sample simplified analytic 3d probability densities.
Crystal-field splitting, spin-orbit coupling, shielding, electron-electron repulsion, hybridization, bonding, and time-dependent transitions are outside this simulation. One spatial d orbital may hold at most two opposite-spin electrons; the complete d subshell can hold ten.