Spherical harmonics Definition and 126 Threads

Hello, I am watching a video about spherical harmonics, and I am at the point where the color map is being shown for various values of ##l## and ##m## My question is, what am I supposed to make of these plots? Pretty colors yes, but what do these things mean?

Hi I want to ask you if you know hot to write cos(theta)cos(phi) in terms of spherical harmonics? Thanks

Homework Statement Verify by brute force that the three functions cos(θ), sin(θ)eiφ and sin(θ)e−iφ are all eigenfunctions of L2 and Lz. Homework Equations I know that Lz = -iћ(∂/∂φ) I also know that an eigenfunction of an operator if, when the operator acts, it leaves the function unchanged...

Homework Statement Well it is not the problem itself that bothers me but the maths behind a part of it. As part of finding the coefficient I had to solve the integral of (Sin(x))^(2l+ 1). The solution given by the solution manual just pretty much jumps to the final answer...

Hello, As a science writer, I've tasked myself with acquiring a thorough theoretical and historical understanding of Quantum Mechanics. It would be interesting to know if there has ever been any experimental verification of Laplace's spherical harmonics, relating to the quantum mechanical...

Hey guys, This is my first post here, so I will apologize in advance in case I'm posting this in the wrong section. I wrote a very simple function to calculate spherical harmonics in matla, and I used this function during 3 years. Yesterday I found that the function was actually wrong, and...

Hi. http://www.nt.ntnu.no/users/jensoa/E-FY1006-31mai2012.pdf Please open the link and go to page 11, problem 3. It appears, after all, I understand nothing when it comes to the wave function of Hydrogen like atoms. So I kindly ask you to answer some questions I got: 1) "A selection of these...

If the B-mode sky plots could be Fourier transformed what would be a plot of the lowest order B-mode harmonic plotted on a sphere look like? I guess we need two functions of spherical coordinates, one function for amplitude at points on a sphere and one function for the orientation at the...

Homework Statement I want to show that <n',l',m'|\hat{z}|n,l,m> = 0 unless m=m', using the form of the spherical harmonics. Homework Equations Equations for spherical harmonics The Attempt at a Solution Not sure how to begin here since there aren't any simple eigenvalues for...

1. Problem: I have a wave function ψ(r) = (x + y + z)*f(r) and want to find the expectation values of L2 and Lz. It is suggested that I first change the wave function to spherical coordinates, then put that in terms of spherical harmonics of the form Yl,m. 2. Homework Equations ...

When solving for the spherical harmonic equations of the orbital angular momentum this textbook I'm reading.. Does this mean that there must be a max value of Lz which is denoted by |ll>? Normally the ket would look like |lm>, and since m is maxed at m=l then |ll> is the ket consisting of the...

Homework Statement For the spherical harmonics Umn; Vmn, compute the ones of orders 0,1, 2. Umn=cos(nθ)sinn(\varphi)Pmn(cos(\varphi)) Vmn=sin(nθ)sinn(\varphi)Pmn(cos(\varphi)) (b) How many non-zero spherical harmonics are there of order k? Homework Equations Equations of Umn; Vmn...

Is it possible to express (cos(\theta)sin(\theta))^2 in terms of spherical harmonics?

I am trying to show that \[ Y_{\ell}^m(0,\varphi) = \delta_{m,0}\sqrt{\frac{2\ell + 1}{4\pi}}. \] When \(m = 0\), I obtain \(\sqrt{\frac{2\ell + 1}{4\pi}}\). However, I am not getting 0 for other \(m\). Plus, to show this is true, I can't methodically go through each \(m\). How can I do this?

is there an approximation for spherical harmonics for very large l and m in closed form?

Homework Statement The question is about page 198 of Jackson's Classical Electrodynamics. The magnetic scalar potential is set to be: Phi = ∫ (dΩ' cosθ'/ |x-x'|). Using the spherical harmonics expansion of 1/|x-x'|, the book claims that only l=1 survives. I...

Hi, I'm a little confused about how to apply the real spherical harmonics when building a hydrogen wave function. I'm doing a computational project, so I want to work with a wave function which is strictly real, and I'm hoping I can do so by building the orbitals using the real spherical...

Hi guys! I was wondering if anyone knew of a particularly nice book that taught one how to solve physics problems that need the use of green's functions and/or spherical harmonics. I can't seem to find a book that actually does this other than Jackson but I'd rather not tread there (I'm guessing...

Homework Statement Consider 2 conductor spherical shells of radii a and b (where a>b). The inner shell is at zero potential and the outer shell is at a potential given by ##V(\theta, \phi )=V_0 \sin \theta \cos \phi ## where ##V_0## is constant and theta and phi are the usual spherical...

$$ Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi} $$ For ##\ell = m = 1##, we have $$ \sqrt{\frac{(2 + 1)(0)!}{4\pi(2)!}}P^1_{1}(\cos\theta)e^{i\varphi} = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta $$ But Mathematica...

$$ Y_{\ell}^m = \sqrt{\frac{(2\ell + 1)(\ell - m)!}{4\pi(\ell + m)!}}P^m_{\ell}(\cos\theta)e^{im\varphi} $$ For $\ell = m = 1$, we have $$ \sqrt{\frac{(2 + 1)(0)!}{4\pi(2)!}}P^1_{1}(\cos\theta)e^{i\varphi} = \frac{1}{2}\sqrt{\frac{3}{2\pi}}e^{i\varphi}\sin \theta $$ But Mathematica is telling...

Homework Statement An electron in a hydrogen atom is in a state described by the wave function: ψ(r,θ,φ)=R(r)[cos(θ)+eiφ(1+cos(θ))] What is the probability that measurement of L2 will give 6ℏ2 and measurement of Lz will give ℏ? Homework Equations The spherical harmonics The...

Hi, I am looking to use the definition from WGS84 to calculate Earth's gravitational potential using spherical harmonics, however I am having some difficulty finding the definition of one of the variables. Gravitational potential is given as the following: V = \frac{GM}{r}\left [ 1 +...

Homework Statement http://img109.imageshack.us/img109/1065/87070684.png Homework Equations 1) L_{\pm}=\pm\hbar e^{\pm i \phi}(\frac{\partial}{\partial\theta}\pm i cot\theta \frac{\partial}{\partial\phi}) 2) L_{\pm}Y^m_l = \hbar\sqrt{(l \mp m)(l \pm m+1)}Y^{m \pm 1}_{l} 3)Answer...

What's the difference in the representation of spherical harmonics and the orbitals themselves? they look exactly the same to me... unlike the radial part of the wavefunction though.

Homework Statement Prove that {Y_{L}^{M}\left ( 0,\varphi \right )=\left ( \frac{2L+1}{4\pi } \right )^{1/2}\delta _{M,0}Homework Equations Y_{L}^{M}\left ( \theta,\varphi \right )=\left ( \frac{(2L+1)(L-M)!}{4\pi(L+M)! } \right )^{1/2}P_{L}^{M}(cos\theta )e^{im\varphi } \int_{\varphi...

Homework Statement Evaluate the integral ∫∫dΩ V(Ω)Yml(Ω) for V(Ω) = +V0 for 0<θ<π/2 ; -V0 for π/2<θ<π Homework Equations I was hoping to apply the orthonormality properties of the spherical harmonics but this is a little more difficult since the integral breaks into two integrals over...

Homework Statement I'm trying to solve I_l = \int^{\pi}_{0} d \theta \sin (\theta) (\sin (\theta))^{2l} Homework Equations the book suggest: I_l = \int^{+1}_{-1} du (1 - u^2)^l The Attempt at a Solution I think it's something related to Legendre polynomials P_l (u) =...

I didn't get any bites in the Calculus section a few days ago so I'm hoping since this is likely a pretty basic part of spherical harmonics that someone here can aid me. Also hoping reposting in a new section after a few days is allowed. Thank you in advance for your assistance! Homework...

Okay, so I'm working on using spherical harmonics to fit a model to some data. The thing is, everything can apparently be described as a "linear combination of spherical harmonics" but nobody is explaining in plain English what that means, at least to me! :D I see lots of double sum...

I'm having a hard time grasping the logical flow from orbital angular momentum to spherical harmonics. It feels like it's just sort of been sprung out of nowhere from both my lecture notes and the textbook. Can anyone help fill in the gaps that clearly must link them somehow? How did I get from...

Hey guys I am trying to understand a code for a Fortran 77 subroutine which calculates spherical harmonics using the CERN library RASLGF for legendre functions. The code looks like this subroutine harmonics(max,theta,phi,Yr,Yi) implicit none integer max,k,nn,n,grens...

Homework Statement I want to understand spherical harmonics. I want to really grok them deeply. I want to be able to visualize them and understand them. I'm the sort who can't take anything on faith, especially where quantum mechanics is concerned. So I want to understand angular...

Hi folks, I'm looking for a derivation of the following statement (formula 76) http://img845.imageshack.us/img845/1550/screenshot4op.png Do you know any reference, where I can find a bit more detailed description? I reckon, you can find it in Jackson's electrodynamic book, but I couldn't find...

Hi, this may seem like something I should ask in the math forums but, as I came into this problem in atomic physics I'm confident that this is a question more appropriate here than in the math forums. So far I've been only able to find the common integral of a product of three spherical...

Homework Statement I have to construct 3, 3X3 matrices for Lz, Lx, Ly for the spherical harmonics Y(l,m) given l=1 and m = 1,0,-1 So I can determine the relevant harmonics for these values of l and m. I act with Lz on Y to get L Y(1,0) = 0 L Y(1,1) = hbar Y(1,1) L Y(1,-1) =...

Homework Statement The function cos(theta)*cos(phi) in spherical coordinates cannot be expanded to a series of spherical harmonics. Explain why. Homework Equations As far as I can recall, the spherical harmonics are a complete set over a sphere, meaning every function which is SI over a...

Hi, I need expand a spherical function(real function not a complex function) in terms of spherical harmonics. Expansion coefficients are complex numbers. If i need to observe physics quantities that are represented by the spherical harmonics coefficients which part should i look at- real part...

u(r,\theta,\phi)=R(r) Y(\theta,\phi) Where Y is the spherical harmonics \frac{\partial^2 Y}{\partial \theta^2} + cot\theta \frac{\partial Y}{\partial \theta} + csc^2 \frac{\partial^2 Y}{\partial \phi^2} + \mu Y = 0 The book said this equation has nontrivial solutions when \mu =...

When solving the time-independent Schrodinger equation for a spherically symmetric potential, using the separation of variables, we find that solutions of the form \psi =R(r)Y_l^m(\theta ,\phi) where the Y_l^m are the spherical harmonics. We apply this to the (idealized) electron in a Hydrogen...

Homework Statement The complete wavefunction for a particular state an atom, is Si(r,theta,phi)=Ne^(-Zr/a_0)(Z/a_0)^3/2sqrt(1/4pi). What is the eigenvalue Lz for this state?Homework Equations see above The Attempt at a Solution This is the last one that I'm having trouble with. I have no...

Hello everyone, I desperately need some help in understanding spherical harmonics and I would be really grateful if someone could help me understand them intuitively. So, as I understand SH are another way to represent a function as a linear combination of some basis functions but the...

Hello, I had posted this in the 'General math' section and did not get any response. Maybe it belongs in this group as it is more related to function decomposition. I hope I am not breaking any forum rules and it is not my intention to cross-post. Just reading an essay about spherical...

Let's define operator P: P \phi(\vec{r})=\phi(-\vec{r}) Does anyone know simple and elegant prove that P|lm\rangle = (-1)^l |lm\rangle (|lm\rangle is spherical harmonic).

Homework Statement I am trying to calculate the angular momenta for \psi(x,y,z) = A(ar^2 + bz^2) A is given as a constant. Homework Equations The Attempt at a Solution I know that z=r\sqrt{4\pi/3} * Y_0^1 What I have so far is:- \psi(x,y,z) = r^2Aa +...

Homework Statement What is the result of raising momentum ladder operator (L+) acting on spherical harmonics Y04 (\theta,\phi) Homework Equations The Attempt at a Solution I was expecting Y14 (\theta,\phi) I applied L+ on Y04 (\theta,\phi) and ended up with Y14...

Lets consider the equation: \nabla^2 f=0 I know that in spherical coordinates this equation may be decomposed into two equations, first which depends only on r, and the second one which has the form of spherical harmonics equation except that the l(l+1) is an arbitrary constant, let's say C...

Homework Statement Find the speherical harmonics (Y_1)^1, (Y_1)^0, (Y_1)^-1 as functions of the polar angles \theta and \psi and as functions of the cartesian coordinates x, y , and z. Homework Equations \(phi_l)^l= sin^l(\theta)*e^il\psi L_\(phi_l)^l=(d/(d\theta))*\phi_l^l-l...

Hello, everyone! I'm working on parametrizing a magnetic field using spherical harmonics. The equations Yc n,m (theta, phi) = (R/R0)^n * Pn,m(cos(theta)) * cos(m*phi) Ys n,m (theta, phi) = (R/R0)^n * Pn,m(cos(theta)) * sin(m*phi) where Pn,m is a Legendre polynomial where n is degree and m...

I am trying to write the term "Sin^2 theta * Sin 2 phi" in terms of spherical Harmonics (they form a combination of Y(2,-2) and Y(2,2)) but the term I get contains the imaginary number 'i'. Am I doing something wrong.. In fact this term is a part of a Hamiltonian and when I get the eigenvalues I...