Hermite Differential Equation and Hermite Polynomials

[@ErwinSchrodinger64]

Stationary harmonic oscillators... in terms of partial differential equations... Hermite polynomials will be involved. Anyone that's going to study Quantum Mechanics, Hermite polynomials will come into play when V(x)=harmonic oscillator or V(x,y,z... N)=N-dimensional harmonic oscillator. Thank you Faculty of Khan. My students will thank you soon. I will thank you when I teach Quantum Theory this fall.

[@chiranjibimahapatra708]

Yo bro Im here for this reason ...QM

[@of_discourse]

Derivative of the sum is the sum of derivatives is only generally true when the sum is finite, but I think you can do this for any Taylor series, assuming it converges in a neighborhood of that region.

[@FacultyofKhan]

Good catch, and thanks for clarifying! I'll put an edit in the description accordingly!

[@canyadigit6274]

New pfp nice

[@mohamedjahimti8618]

Can you make videos on Generating Functions for Legendre, Bessel and Hermite Polynomials. It would be very helpful if you can do it ASAP

[@holyshit922]

Exponentital generating function for Hermite polynomials is easier to get starting from recurrence relation H_{n+1}(x)=2xH_{n}(x)-2nH_{n-1}(x) H_{0}(x) = 1 H_{1}(x) = 2x H_{n+2}(x)=2xH_{n+1}(x)-2(n+1)H_{n}(x) H_{0}(x) = 1 H_{1}(x) = 2x E(x,t) = sum(H_{n}(x)*t^n/n!,n=0..infinity) After some calculations you will get following initial value problem B''(t) + 2(t -x)B'(t)+2B(t)=0 B(0) = 1 B'(0) = 2x Particular solution to this equation is easier to guess after reduction to Riccati Reduction to Riccati y'' + p(x)y'+q(x)y = 0 y'' = - p(x)y' - q(x)y | :y y''/y = -p(x)*(y'/y) - q(x) | -(y'/y)^2 y''/y - (y'/y)^2 = -(y'/y)^2 - p(x)*(y'/y) - q(x) y''/y - y'*y'/y^2 = -(y'/y)^2 - p(x)*(y'/y) - q(x) (y''*y-y'*y')/y^2 = -(y'/y)^2 - p(x)*(y'/y) - q(x) (y'/y)' = -(y'/y)^2 - p(x)*(y'/y) - q(x) Let y'/y = z z' = -z^2 -p(x)z - q(x) y' = yz Reduction Riccati to Bernoulli assuming you somehow guessed particular solution Let z_{1} be the particular solution to Riccati equation z' = p(x)z^2 + q(x)z + r(x) (1) z_{1}' = p(x)z_{1}^2 + q(x)z_{1} + r(x) (2) Lets subtract eq no 2 from eq no 1 z' - z_{1}' = p(x)(z^2-z_{1}^2)+q(x)(z-z_{1}) z' - z_{1}' = p(x)(z - z_{1})(z + z_{1}) + q(x)(z-z_{1}) z' - z_{1}' = p(x)(z - z_{1})(z - z_{1} + 2z_{1}) + q(x)(z-z_{1}) z' - z_{1}' = p(x)(z - z_{1})(z - z_{1}) + 2z_{1}p(x)(z-z_{1}) + q(x)(z-z_{1}) z' - z_{1}' = p(x)(z - z_{1})^2 + (2z_{1}p(x) + q(x))(z-z_{1}) (z-z_{1})' - (2z_{1}p(x) + q(x))(z-z_{1}) = p(x)(z - z_{1})^2 w = z-z_{1} w' - (2z_{1}p(x) + q(x))w = p(x)w^2

[@vivekpanchal3338]

For physicist Hermite polynomials, why we set a0 nd a1 diffrent for each order of Hermite polynomials? I am confused whether it is done for making the wave function normalised or there is any other reason?

[@andreasatakan6183]

King!

[@cicerohitzschky8855]

Excellent class, teacher! What software do you use to write on the screen?

[@mehmoodsaleem6749]

Great Sir💗💗💗

[@Ryze107]

Could you please explain where the ODE comes in the first place? How do you derive the ODE for the processes it is meant to describe?

[@mingmiao364]

He said in the beginning of the video: quantum mechanical harmonic oscillator. For details of the derivation, see: opentextbc.ca/universityphysicsv3openstax/chapter/the-quantum-harmonic-oscillator/ On the webpage above, search for keywords “time-independent Schrödinger equation”. Hope it helps.

[@lichifang632]

Would you consider making a video on Mathieu function (differential equation)? Almost no video about that is available currently on youtube

[@eulefranz944]

That a good idea!

[@anaeem86]

I am making videoes on that this week. Maybe u can check my channel up by then. Thanks

[@anaeem86]

M planning to do hermite polynomial, airy's, laguerre, laplacian and mathieu....the whole shabang

[@anaeem86]

Can u plz share as to which video editor u use

[@manjuriroy9351]

Thankyou sir❤️

[@nitayweksler3051]

nice vid, thanks!!!!

[@morethanjustasloth5528]

"This equation will really help us in quantum mechanics." Man I just wanna figure out what how to draw a line in GLSL.

[@ahkypc9171]

Great

[@cpmontanapromoblackheartta3886]

❤

[@ricardoraymond9037]

Your subtitles are larger than your black board written characters. Very distracting......

[@enriqueniemannconcha7267]

Did you try turning them off??

[@seanziewonzie]

Pronounced "air meet" not "her might"

[@didierfavre2356]

It looks simple. ????????????????? Let's try.

[@sidneynatzukajr6099]

first!!

[@abublahinocuckbloho4539]

mate you have a way with words, you make a mountain out of a mole hill. something that can be explained in a sentence, you take a paragraph. consider "instead of having an infinite series containing the sum of a bunch of polynomial terms" how about "instead of an infinite series" instead. you might want to put your script through grammarly to avoid a lot of unnecessary waffling like you are finding words to make sure you reach a word limit. there are a bunch of other examples in this video and every other fucken video you made