Spherical harmonics Definition and 120 Threads

Homework Statement The angular part of a system’s wavefunction is <\theta, \phi | \psi>\propto (\sqrt{2}\cos\theta + \sin{\theta}e^{−i\psi} - \sin{\theta}e^{i\psi} ). What are the possible results of measurement of (a) L^2 , and (b) L_z , and their probabilities? What is the...

Homework Statement Hi! I need help with this. I have to calculate the expression of a function using spherical harmonics. The relevant equations are given below. An example of a function can be f(theta,phi) = sin(theta)... Can you help me to calculate de coeficients and to express the...

Whats a good book to learn about these topics?

I would appreciate some input about how to program spherical harmonics in Matlab. http://mathworld.wolfram.com/SphericalHarmonic.html I want to program a double summation that looks like this. G(\Omega_{1},t_{1}|\Omega_{0}) = \sum_{l=0}^\infty \sum_{m=-l}^l...

expansion of polarized plane waves into spherical harmonics, please help! Hi all, I would like to get some guidance in how to expand a polarized (i.e. linear polarization) plane wave into a series of spherical harmonics. I am aware of the formula applying to scalar plane waves (please see...

I wonder if you can recommend a good book treating "Spherical Harmonics" in some details. Thanks for help

Homework Statement Why is is the for physical applications of the spherical harmonics |m| must be less than or equal to l, with both being integers? Homework Equations Y(m,l)=exp(im phi)P{m,l}(cos theta) Hopefully my notation is clear, if not please say. The Attempt at a Solution Well...

Homework Statement At t=0, a given wavefunction is: \left\langle\theta,\phi|\psi(0)\right\rangle = \frac{\imath}{\sqrt{2}}(Y_{1,1}+Y_{1,-1}) Find \left\langle\theta,\phi|\psi(t)\right\rangle. Homework Equations \hat{U}(t)\left|\psi(0)\right\rangle =...

Homework Statement I want to expand sin(theta) in spherical harmonics. Well, actually I want (3cos^2(theta+45°)-1)*exp(i*(psi+45°)) but I think I could find my mistake by the above simple example. Homework Equations http://en.wikipedia.org/wiki/Spherical_harmonics The Attempt at...

This isn't exactly a part of any problem, but a part of a generic principle. I don't understand the use of raising and lowering operators. L_{^+_-}=\hbar e^{^+_- i l \phi}({^+_-}\frac{\partial}{\partial \theta}+ i cot \theta \frac{\partial}{\partial \phi}) So how does one use L_{^+_-}Y_l^m...

Homework Statement I want to expand 1+sin(phi)sin(theta) in the spherical harmonics. I am not sure if this will be an infinite series or not? If it were infinite that would seem rather difficult because the spherical harmonics get really complicated when l > 3. Also, all of the sine terms in...

Homework Statement The spherical harmonics Y^m_l with l=1 are given by Y^{-1}_1 = \sqrt{\frac{3}{8\pi}}\frac{x-iy}{r}, Y^0_1 = \sqrt{\frac{3}{4\pi}}\frac{z}{r}, Y^1_1 = -\sqrt{\frac{3}{8\pi}}\frac{x+iy}{r} and they are functions of L^2 and L_z where L is the angular momentum. i) From...

Hi, I'm trying to get the Y_l^l spherical harmonic and I'm running into problems evaluating the following expression: \frac{d^{2l}(\cos^2(\theta) - 1)^l}{d\cos(\theta)^{2l}} The 2lth derivative with respect to cos theta of cos squared theta - 1 to the lth power it just seems like I'm going...

Happy New Year all! i have a question regarding the addition theorem for spherical harmonics. In JD Jackson book pg 110 for e.g. the addition theorem is given as: P_{L}(cos(\gamma))=\frac{4\pi}{2L+1}\sum_{m=-L}^{L}Y^{*}_{Lm}(\theta',\phi')Y_{Lm}(\theta,\phi) where...

I'm calculating the zz Component for the quadruple tensor. Q_{zz} = 3cos^2\theta-1 (r=1 in this case), and the Y_{lm}(\theta,\phi) would be l=2, m=0. I would like to calculate the result in either maple or mathematica - I have not used either very much - I want to check the result using...

In solving the 3D hydrogen atom, we obtain a spherical harmonic, Y such that, Y_{lm}(\theta,\phi) = \epsilon\sqrt{\frac{(2l+1)}{(4\pi)}}\sqrt{\frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi}P^m_l(cos \theta) where \epsilon = (-1)^m for m \geq 0 and \epsilon = 1 for m \leq 0 . In quantum, m = -l...

There is this excersise in Griffith's QM text that I can't seem to solve. It's about the calculation of the normalization factor of the spherical harmonic functions using the angular momentum step up operator. These definitions/results are given: Y_l^m = B_l^m e^{im\phi} P_l^m (\cos\theta...

Hello, My question is fairly simple. My instructor solved in class today Laplace's equation in spherical coordinates which resulted in spherical harmonics. I have not taken any quantum mechanics yet so this is my first exposure to spherical harmonics. What do the "l" and "m" terms in the...

Greetings, I´m calculating cuadratic dispersion of some quantum systems. I need to expand x^2 in terms of spherical harmonics (using Clebsch-Gordan coefficients, or threeJ as well) in order to be able to use Gaunt espression in the integral solving. I start from the expansion of x as...

Viva! I wonder if anyone could explain me the difference between real spherical harmonics (SH) and complex SH. What's the difference in doing an expansion in either situations? And what are the orthogonality relations for each case? Any help would be great...( websites, books..)...