Simultaneous eigenstates and quantization of angular momentum

[@MariaPereira-ho2ks]

this teacher is amazing!

[@李愚-f7j]

@19:40 How could we know that this Legendre differential equation is solved by a series solution when simplified as m = 0? Thanks ^^ and I did not follow the calculate trick when plugging in the series solution in equation 3.16 and get the recursion relations between a(k+2) and a(k)

[@blackberry596]

Necroposting, but I believe the way to find this is by using the Frobenius method for solving differential equations which includes guessing at a solution of the form of a power series and then solving for the recursion relations to get explicit expressions for the coefficient a_n of the power series.

[@neeteshmudgal1209]

at 10:00 why prof, says that " phi -2pi will lead to inconsistency".. please explain anybody..

[@李愚-f7j]

@Hamish Blair THANKS^^

[@zphuo]

@6:00, I don't understand why Lamda=l(l+1). Can somebody give me a explain ? thanks.

[@geovanemarquez1531]

ZiPan Huo did you figure it out brother

[@zphuo]

No. I still don't know why we need it..

[@geovanemarquez1531]

ZiPan Huo did you use Griffiths book too?

[@zphuo]

yes.

[@m.s.6449]

At that point, it's just a deliberately weird way of writing a positive real number (you can always find l so that l(l+1) equals any given positive real number). In the end, it will make formulas simpler.

[@kumailhaider7933]

8:30 can someone explain how we get the Legendre polynomial here in solution of differential equation

[@berketozlu]

He just wrote it as an ansatz since the differential equation should solve for phi but there might be a term of theta too. So he just wrote it as Legendre polynomial without knowing what is that theta dependent term of the psi.

[@andrewmanti235]

can anyone please explain the substituion of pl at 20:24,and how he got (k+1)(k+2)ak+2...? please

[@立成王-n3i]

subtitute the P(x) Polynomial to the equation and let the coefficient of x^k(k=0,1,2...) equal to zero，just like the step in the harmonic oscillatior

[@rodolfogonzalez9983]

it comes from solving the differential equation with power series. And it´s called recurrence relation

[@cafe-tomate]

Why would the series diverge at X=±1 ?

[@cordi-fm9tb]

I had the same question. I think it's because the nth-term divergence test of the series. like if lim k goes to inf, the last term goes to 0 when abs(x)

[@李愚-f7j]

also in ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes20_21.pdf note of page 9 when we pick l = m terminate the series but( equ3.19) the recursion relations becomes zero, nothing exists how could we make p_sub _k(x) a k degree polynomial, also below is not in the lecture video (maybe too simple for them)the (3.21)Rodriguez formula and generating function (3.22)I searched Wikipedia en.wikipedia.org/wiki/Rodrigues%27_rotation_formula I still lost. Could anyone please enlighten me?

[@jiaqigan6398]

When we choose l=k as the highest term in the solution, terms which's power is smaller than k still remain there. To write a complete solution, you just need to set up the value of coefficient a_k, and then, go backwards, derive the value of a_(k-2),a_(k-4).... (You can find those steps in many good mathematical physics books). For the

[@jiaqigan6398]

For the Rodriguez formula, apply binomial theorem on the left hand side of the equation, then do the derivation, it's not hard. For the generating function of the

[@jiaqigan6398]

Legendre polynomials,it’s very similar to the generating function of the hermit polynomials.You can find some clear statements in Schiff’s book.

[@李愚-f7j]

@@jiaqigan6398 THANK YOU VERY MUCH, I found the related contents in Schiff's book you mentioned , thanks!

[@jiaqigan6398]

@@李愚-f7j 哈哈，不客气

[@davalentinalopez]

2:34 23:37

[@wondererasl]

if m ==0, then p to the power of m is 1, right ?

[@not_amanullah]

Thanks 🤍❤️

[@not_amanullah]

This is helpful ❤️🤍