Real Spherical Harmonics – Removing the Imaginary Terms

▶️ Watch on 3Speak - Telegram - YouTube - Summary - Hive Notes - Sections playlist - MES Links

In this video, I derive the real spherical harmonics by excluding the imaginary complex i terms from the complex spherical harmonics. The real spherical harmonics are what are typically plotted when graphing atomic orbitals or magnetic fields, since they are easier to visualize in 3D and still inherit the important properties of complex spherical harmonics. I derive the proportion forms (thus excluding the proportionality constants) of the real spherical harmonics from their typical definition, and make use of complex conjugation and Euler's formula to obtain a compact formula for the cases where m is positive or negative.

#math #sphericalharmonics #calculus #quantumphysics #3d

7 Spherical Harmonics.png

Timestamps

Real spherical harmonics are defined to remove the complex terms – 0:00
Complex spherical harmonics are chosen using the basis solutions rather than the linear combinations – 2:24
Real spherical harmonics in their proportional forms, ignoring the normalization constants – 3:49
Complex conjugate in polar form flips the sign of the exponential power via Euler's formula – 4:43
The Associated Legendre equation depends on m², so -m gives a solution that is proportional to that using +m, via the Rodrigues' formula – 11:23
Case where m is greater than 0 – 18:23
Case where m = 0 – 26:39
- Note that this yields just the ordinary Legendre polynomial – 30:20
Case where m is less than 0 – 32:20
Putting it all together to get our real spherical harmonics in proportional form – 45:29
- Writing it in more compact form – 47:53

Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join

DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate

SUBSCRIBE via EMAIL: https://mes.fm/subscribe

MES Links: https://mes.fm/links

MES Truth: https://mes.fm/truth
Official Website: https://MES.fm
Hive: https://peakd.com/@mes

Email me: [email protected]

Free Calculators: https://mes.fm/calculators

BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm

Free Online Tools: https://mes.fm/tools

iPhone and Android Apps: https://mes.fm/mobile-apps

▶️ 3Speak

Congratulations!

You have obtained a vote from CHESS BROTHERS PROJECT

✅ Good job. Your post has been appreciated and has received support from CHESS BROTHERS ♔ 💪

♟ We invite you to use our hashtag #chessbrothers and learn more about us.

♟♟ You can also reach us on our Discord server and promote your posts there.

♟♟♟ Consider joining our curation trail so we work as a team and you get rewards automatically.

♞♟ Check out our @chessbrotherspro account to learn about the curation process carried out daily by our team.

🥇 If you want to earn profits with your HP delegation and support our project, we invite you to join the Master Investor plan. Here you can learn how to do it.

Kindly

The CHESS BROTHERS team