A function 𝑓(𝐫) is called "harmonic" if it satisfies Laplace's equation.pic.twitter.com/nzyulWuuRW
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The 𝑺𝒑𝒉𝒆𝒓𝒊𝒄𝒂𝒍 𝑯𝒂𝒓𝒎𝒐𝒏𝒊𝒄𝒔 𝑌(θ,φ) satisfy the angular part of Laplace's Equation.
They are the basis functions for a "Fourier Expansion on a sphere"
The 𝑃_lm are the associated Legendre Functionspic.twitter.com/yfQgnoeH8I
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The spherical harmonics 𝑌(θ,φ) are the reason why the electron orbitals have the shape they do.
Go ahead, plot them for different quantum numbers (ℓ,𝑚) !
Here I've plotted the probability density for (3,1)pic.twitter.com/JH4S4b8QzR
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I kind of learned to like the Rodriguez formula and legendre polynomials by the end of last semester... they are soothing..
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I saw this beast in modern physics last semester.
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