Spherical Harmonics Y(l,m) Calculator

Compute the real part, imaginary part, magnitude, and normalization of the spherical harmonic Y(l,m) from degree l, order m, and the angles theta and phi.

Input

Enter the degree l, order m, and the angles theta (colatitude) and phi (azimuth) to compute the real part, imaginary part, and magnitude of the spherical harmonic Y(l,m).

Degree l

Non-negative integer

Order m

Integer with -l ≤ m ≤ l

Colatitude theta

Angle from the pole (0 to pi)

Azimuth phi

Angle in the equatorial plane (0 to 2 pi)

Angle unit

Pick a common orbital

Selecting one fills in l and m

Result

Real part of Y(l=2, m=1)

-0.2365436739

Unit is 1 / sqrt(steradian)

Imaginary part

-0.23654367

Magnitude

0.33452327

Squared magnitude

0.11190582

Normalization

0.25751613

Associated Legendre

-1.29903811

Argument (radians)

-2.35619449

Magnitude of Y(l=2, m=1) versus theta

Horizontal axis is theta from 0 to pi, vertical axis is magnitude. Peak value is about 0.41717614.

Magnitude for each order m at l=2

Magnitudes for each order at the same theta. These do not depend on phi.

Order m	Magnitude
-2	0.28970565
-1	0.33452327
0	0.07884789
1	0.33452327
2	0.28970565

How it works

The spherical harmonic is defined as Y(l,m)(theta, phi) = N(l,m) times P(l,m)(cos theta) times exp(i m phi), where N(l,m) is the normalization factor and P(l,m) is the associated Legendre function.
The normalization factor is N(l,m) = sqrt( (2l+1)/(4 pi) times (l-m)!/(l+m)! ), using the physics convention that includes the Condon-Shortley phase.
The associated Legendre function P(l,m) is evaluated by a three-term recurrence starting from P(m,m) = (-1)^m (2m-1)!! (1 - z^2)^(m/2), where z = cos theta.
The order m must be an integer between -l and l inclusive. An out of range m or a negative degree l produces an error.
Negative orders use the relation Y(l,-m) = (-1)^m times conj( Y(l,m) ), where conj denotes the complex conjugate.
Angles can be entered in degrees or radians. When degrees are selected they are converted to radians by multiplying by pi/180.
The primary result is the real part. The imaginary part, magnitude, squared magnitude (the probability density), normalization factor, and argument are also shown.
Because the computation uses floating point, rounding error can appear for large degree l or for angles near singular points.

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