Spherical harmonics Calculator

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Overview

Spherical harmonics are simultaneous eigenfunctions of $L^2$ and Lz, so they carry the l and m quantum numbers.

Apply it well

When To Use

When to use: Defines the angular functions used for rigid rotors and atomic orbitals.

Why it matters: Spherical harmonics are simultaneous eigenfunctions of L^2 and Lz, so they carry the l and m quantum numbers.

Avoid these traps

Common Mistakes

• Forgetting that m must satisfy -l <= m <= l.
• Confusing spherical harmonics with radial wavefunctions.

One free problem

Practice Problem

For l = 2, what values of m are allowed?

Solve for: $Y_l^m(θ, φ)

Hint: Focus on what the formula is telling you physically.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Chemistry LibreTexts, Rotational Motions of Rigid Molecules; Chemistry LibreTexts, Selection Rule for the Rigid Rotator
2. Chemistry LibreTexts, Rotational Motions of Rigid Molecules
3. Chemistry LibreTexts, Selection Rule for the Rigid Rotator
4. Wikipedia: Spherical harmonics
5. NIST CODATA: Fundamental Physical Constants
6. Griffiths, David J. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press.
7. Wikipedia, "Spherical harmonics"