hmparticle9's latest activity
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Hhmparticle9 replied to the thread I Derive the orthonormality condition for Legendre polynomials.I have started with $$\int_{-1}^1 D^{2n}(x^2-1)^{2n} = \int_{-1}^1 D^{2n-1}(x^2-1)^{2n-1}2x$$ I don't think that is the correct way. I...
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Hhmparticle9 replied to the thread I Derive the orthonormality condition for Legendre polynomials.According to my book (in the question statement), ##||P_n||^2 = \frac{2}{2n+1}##. Okay cool I will think about it for a bit
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Hhmparticle9 reacted to fresh_42's post in the thread I Derive the orthonormality condition for Legendre polynomials with
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Yes, and the proof for ##\|P_n\|=1## is along similar lines. We have an integrand of the form ##\left(D^n(x^2+1)^n\right)^2## and... -
Hhmparticle9 replied to the thread I Derive the orthonormality condition for Legendre polynomials.First and foremost. Thanks for persevering with me :) I understand everything you said in the above post. We proceed as you have...
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Hhmparticle9 replied to the thread I Derive the orthonormality condition for Legendre polynomials.My problem with what you are saying is that there are monomials ##x^k## with ##k \in [m,2m]## in the expansion of ##(x^2-1)^m## which...
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Hhmparticle9 replied to the thread I Derive the orthonormality condition for Legendre polynomials.Are we also assuming that ##m \geq n##? Also, this does not conclude the proof? I can see that ##(x^2-1)^m## is a sum of monomials...
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Hhmparticle9 replied to the thread I Derive the orthonormality condition for Legendre polynomials.Okay I think I have made a breakthrough. Since the derivative acts on everything to the right of it we can say $$\int_{-1}^{1} x^n...
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Hhmparticle9 reacted to fresh_42's post in the thread I Derive the orthonormality condition for Legendre polynomials with Like.
Yes, but my hint is still valid. Prove it for the summands of ##P_l## instead of the entire polynomial.
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Hhmparticle9 replied to the thread I Derive the orthonormality condition for Legendre polynomials.I am sorry for the inconvenience. Could we stick to the definition given in my text book? (my original post). I know I am sort of...
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Hhmparticle9 replied to the thread I Derive the orthonormality condition for Legendre polynomials.For example: $$\int_{-1}^{1} \bigg( \frac{d}{dx}\bigg)^m(x^2-1)^m \bigg( \frac{d}{dx}\bigg)^l(x^2-1)^l \text{ d}x = \int_{-1}^{1}...
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Hhmparticle9 replied to the thread I Derive the orthonormality condition for Legendre polynomials.It is not that I did not want to use integration by parts, it is just that I am uncomfortable and want to become comfortable with it. :)...
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Hhmparticle9 posted the thread I Derive the orthonormality condition for Legendre polynomials in Calculus.I am stuck at the gate with this one. $$\int_{-1}^{1} P_m P_l \text{ d}x = \frac{1}{2^m m!} \frac{1}{2^l l!} \int_{-1}^{1} \bigg(...
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Hhmparticle9 replied to the thread Understanding how to "tack on" the time wiggle factor.Yes I did :) After your post it clicked immediately: The Hamiltonian determines how a state evolves in time. I was treating the...
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Hhmparticle9 replied to the thread Spherical conducting shell enclosing a non-conducting core.Sorry ! :) $$\int_S \mathbf{E} \cdot \text{d} \mathbf{S} = E(r) 4 \pi r^2 = \frac{Q - \hat{Q}}{\epsilon_0} \implies E(r) = \frac{Q -...
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Hhmparticle9 replied to the thread Spherical conducting shell enclosing a non-conducting core.Surely the charge ##\hat{Q} < Q## would accumulate at ##r = b##. Gauss's law would say $$\int_S \mathbf{E} \cdot d \mathbf{S} = \frac{Q...