Derive / verify Legendre P (cos x)

[Sparky_]

hello,

I am trying (and failing) to verify / derive the result of the Legendre polynomial

P₁¹ (cos x) = sin x

Griffiths Quantum chapter 4 Table 4.2

I figured it would not be too bad. I have attempted this 3 or 4 times trying to be careful.

I keep getting sin(x) times some additional trig functions which even reviewing identities I cannot get them terms to go to 1

P₁¹ (cos x) = (1-cos²x)^1/2 d/dx (1/2 d/dx (cos² -1 )

= (sin x) / 2 d/dx d/dx (-sin²x))
= (-sin x) / 2 d/dx (sin 2x)
=(-sinx)(cos(2x))

= (-sinx) (cos² - sin²x)

any suggestions / help on where I am missing it, trying to get it equal to sin x

 

[BvU]

Hi,

It looks to me as if you think $d\over d(\cos x)$ is the same as $d\over dx$

Suggestion: treat $P$ as a function of $x$ instead of a function of $\cos x$

 

[Sparky_]

Thank you - that was it.

In Griffiths chap 4, the Legendre polynomial equation is introduced with "x" as the variable and I did successfully duplicate a few of the tabularized results - just an exercise in being careful with derivatives. I did read where he states that the solution to one of the angular equation is P(cos Θ) and says the result will be a polynomial in cos not x. It did not register with me.

Forest for the trees, I guess. I was in the mode of solving the derivative like a (blind) calculus problem.

Doing like you pointed out derivative with respect to cos was the issue.

(all of this is a bucket list item of mine to teach myself some quantum mechanics. I plan on stopping at the end of chapter 4: "spin")

Thank you again!

Sparky_