Spherical harmonics

Core idea

Overview

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

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.

Walkthrough

Derivation

Derivation of Spherical harmonics

Defines the angular functions used for rigid rotors and atomic orbitals.

The symbols use the standard quantum-chemistry convention for this topic.
The expression is used within the model named in the entry.

1

Start from the model

Interpret the displayed relation as a rule, definition, or operator statement.

Y_{l}^{m} (θ, ϕ) = (- 1)^{m} \frac{2 l + 1}{4 π} \frac{( l - m )!}{( l + m )!} P_{l}^{m} (cos θ) e^{im ϕ}

2

Identify the physical pieces

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

Y_{l}^{m} (θ, ϕ) = (- 1)^{m} \frac{2 l + 1}{4 π} \frac{( l - m )!}{( l + m )!} P_{l}^{m} (cos θ) e^{im ϕ}

3

Use the result carefully

Apply the expression only when the assumptions of the model are satisfied.

Y_{l}^{m} (θ, ϕ) = (- 1)^{m} \frac{2 l + 1}{4 π} \frac{( l - m )!}{( l + m )!} P_{l}^{m} (cos θ) e^{im ϕ}

Result

Y_{l}^{m} (θ, ϕ) = (- 1)^{m} \frac{2 l + 1}{4 π} \frac{( l - m )!}{( l + m )!} P_{l}^{m} (cos θ) e^{im ϕ}

Source: Chemistry LibreTexts, Rotational Motions of Rigid Molecules; Chemistry LibreTexts, Selection Rule for the Rigid Rotator

Free formulas

Rearrangements

Solve for $_{s} t a t u s$

b l oc k e d_{n} o n_{i} n v er t ib l e

Solve for reason

T h ee q u a t i o ni s a d e f ini t i o n o f a co m pl e x m u l t i - v a r iab l e f u n c t i o nin v o l v in g t r an sce n d e n t a l f u n c t i o n s (e x p o n e n t ia l), r ec u r s i v e p o l y n o mia l s (L e g e n d r e), an df a c t or ia l a r g u m e n t s in s i d e a s q u a r er oo t . I t c ann o t b er e a r r an g e d t o i so l a t e t h e q u an t u mn u mb er s^{'} l^{'} o r^{'} m^{'}, n or t h e an g l e s^{'} t h e t a^{'} o r^{'} p h i^{'}, in t er m so f t h eo t h er v a r iab l es u s in g s t an d a r d a l g e b r ai co p er a t i o n s .

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

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

main expression

Defines the angular functions used for rigid rotors and atomic orbitals.

Spherical harmonics are simultaneous eigenfunctions of

L^{2}

and Lz, so they carry the l and m quantum numbers.

Signs and relationships

positive terms: Positive terms usually represent kinetic energy, barriers, or magnitudes.
negative terms: Negative terms usually represent attractive interactions or energy lowering when present.

Free study cues

Insight

Canonical usage

Spherical harmonics are inherently dimensionless mathematical functions that describe angular distribution, often used as basis functions in quantum mechanics and other fields.

Common confusion

Students may sometimes try to assign units to spherical harmonics, forgetting they are purely mathematical functions describing angular behavior.

Dimension note

The spherical harmonics themselves are dimensionless functions. Their arguments (theta and phi) are angles, and the constants within the formula are dimensionless numerical factors.

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.

Where it shows up

Real-World Context

In atomic orbital models, the familiar p-orbital angular shapes are built from l = 1 spherical harmonics. Spherical Harmonics is used to calculate $Y_l^m(\theta, \phi)$ from the measured values. The result matters because it helps connect angular wavefunction shape to quantum numbers and orbital behavior.

Study smarter

Tips

l controls total angular shape.
m controls the z-axis projection and phi dependence.

Avoid these traps

Common Mistakes

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

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Defines the angular functions used for rigid rotors and atomic orbitals.

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

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

In atomic orbital models, the familiar p-orbital angular shapes are built from l = 1 spherical harmonics. Spherical Harmonics is used to calculate $Y_l^m(\theta, \phi)$ from the measured values. The result matters because it helps connect angular wavefunction shape to quantum numbers and orbital behavior.

l controls total angular shape. m controls the z-axis projection and phi dependence.

References

Sources

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

Overview

Derivation

Start from the model

Identify the physical pieces

Use the result carefully

Rearrangements

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Angular momentum operator

Angular momentum component commutator

Angular momentum magnitude commutator

Associated Legendre polynomial

Frequently Asked Questions

Sources