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-1
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1answer
38 views

Orbital angular momentum eigenstates in the $|\mathbf{r}\rangle$ representation

Consider the orbital angular momentum operators $L^2$ and $L_z$. In the $|\mathbf{r}\rangle$ representation using spherical coordinates those operators actions are given by ...
0
votes
2answers
48 views

Please help me with this doubt from spherical waves

How to calculate phase difference for spherical waves? How to say whether they are in phase or out of phase? In sinusoidal we can easily say whether they are in phase or out of phase just by looking ...
0
votes
0answers
32 views

How to rewrite a wave function in terms of spheical harmonics

I'm given a wave function for a particle, in three variables (spherical coordinates): $ψ=ψ(r, θ, φ) = re^{-r/a}sin(θ)sin(φ)$. I'm tasked with rewriting $ψ$ in terms of spherical harmonics which are ...
2
votes
0answers
16 views

Conventions in defining spherical harmonics and associated Legendre polynomials

Relevant Background Spherical harmonics are defined with several different conventions: the definition used in quantum mechanics according to Wikipedia is $Y_l^{\,m}(\theta,\phi) = ...
2
votes
1answer
204 views

Analogy to Fourier transform in spherical coordinates with boundary at a certain radius

Suppose, we have a wavefuction $\phi(\vec{x})$ which is restricted in a sphere, with the spherical boundary condtion $$\phi(\vec{x}=R)=\phi_0.$$ How can I do the 'Fourier transformation' as the case ...
1
vote
1answer
41 views

Clarification about first spherical harmonic

If the quantum numer $l$ is equal to zero, each components of L has eigenvalues $m=0$. So the states that have $l=0$ are simulaneous eigenvectors of each components of L. But, my textbook says, it is ...
1
vote
1answer
52 views

Operators associated to spherical harmonic functions

$$ \newcommand{\op}[1]{\hat{#1}} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle {#1}|} $$ Let's say we have a spherically symmetric system, like a ...
2
votes
1answer
80 views

How are spherical harmonics useful outside class? [closed]

I've learned about spherical harmonics (Legendre polynomials $\longrightarrow$ Associated Legendre polynomials $\longrightarrow$ orthogonality relations $\longrightarrow$ normalization coefficient(s) ...
0
votes
3answers
87 views

Quick way to compute $\langle n^{'}l^{'}m^{'}|r^k|nlm \rangle$, $k \in I$; $|nlm\rangle$ is $H$ atom eigenfunction [closed]

I want to compute quickly (using maybe some scaling arguments) $\langle n^{'}l^{'}m^{'}|r^k|nlm\rangle$, where $k \in I$. $|nlm \rangle$ is the eigenfunction of the Hydrogen atom ($H$). Example: ...
0
votes
1answer
82 views

Confusing concepts in proof of spherical addition theorem

In http://scipp.ucsc.edu/~haber/ph116C/SphericalHarmonics_12.pdf, section 4, pages 6..9 is a proof of the spherical harmonics addition theorem. Page 8 has eq.(25), an application of Laplace series: ...
1
vote
0answers
44 views

Units of angular power spectrum

I am not sure whether this question might be better suited for crossvalidated or stackoverflow, but I will give it a try here: I have a map of the full sky in in the healpix format, units are $\rm ...
1
vote
0answers
29 views

conducting hollow sphere in magnetic monopole

if a hollow copper sphere(or any conducting hollow sphere) is connected to dc at points diametrical and a magnetic monopole is right at the center of the sphere then will there be any movement of the ...
1
vote
0answers
47 views

The product of three spherical harmonics in higher dimension

As I see, e.g., in this question, a closed formula for the following integral \begin{equation} \int_{\mathbb{S}^N} Y_{\ell_1\ldots \ell_N}\,Y_{\ell'_1\ldots\ell'_N}\,Y_{\ell''_1\ldots \ell''_N}\, d^N ...
4
votes
2answers
211 views

How can mean value of a quantity $be$ an operator?

In Laundau & Lifshitz Quantum Mechanics. Non-relativistic theory in $\S29$ a problem is given: PROBLEM Average the tensor $n_in_k-\frac13\delta_{ik}$ (where $\mathbf{n}$ is a unit vector along ...
0
votes
0answers
214 views

Degeneracy of Spherical Harmonics Eigenfunctions

I'm working through Griffiths' Introduction to Quantum Mechanics (2nd edition) and I'm trying to solve problem 4.24 b). In this problem you're supposed to first find the normalized eigenfunctions to ...
4
votes
1answer
149 views

Calculating the angular power spectrum of a section of sky

To calculate the angular power spectrum $C_l$ of the whole sky, one uses the variance of the coefficients of the spherical harmonics in the temperature fluctuation field. I.e. $$C_l = ...
0
votes
1answer
119 views

Harmonics on Hyperbolic space

I would like to know if there exists an analogue for hyperbolic space of the so called spherical harmonics which play a major role in the quantum states construction in a hydrogen atom. In other words ...
0
votes
0answers
285 views

3DAnisotropic oscillator in Spherical Harmonic basis-States with $L_z=0$

I've been trying to prove a rather simple looking concept. I have a code that calculates states of a 3D anisotropic oscillator in spherical coordinates. The spherical harmonics basis used to expand ...
3
votes
1answer
120 views

My basis set isn't orthonormal?

I'm implementing a little QM calculation just for fun and to make sure I understand how it works (calculating the helium ground state energy). My problem is that my basis set doesn't seem to be ...
3
votes
1answer
82 views

State with non-zero angular momentum - cannot be described by spherical harmonic?

For a state with non-zero angular momentum, why is it that it cannot be described by the spherically symmetric spherical harmonic?
2
votes
0answers
233 views

Degeneracy, spherical harmonics

In a 3D oscillator, the energy levels are known to be $(n_x + n_y + n_z + \frac{3}{2})\hbar \omega = (n + \frac{3}{2})\hbar \omega$. Say for $n = 1$, any of the $n$'s can be $1$ and the rest are $0$. ...
1
vote
0answers
88 views

Rewriting $\langle {\bf k} \vert E,l,m \rangle$ as $\langle {\bf k} \vert ~k,l,m \rangle$ Spherical Harmonics

From Sakurai eq. 6.4.21a we have that $$\langle {\bf k} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M k}}\delta\left(E-\frac{\hbar^2 k^2 }{2M}\right) Y_l^m({\bf\hat k}),$$ where $M$ is the mass of the ...
1
vote
1answer
79 views

Multiparticle generalization of $\langle \vec k \vert E,l,m \rangle$ spherical harmonics.

From Sakurai eq. 6.4.21a we have that $$\langle {\bf k} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M k}}\delta\left(E-\frac{\hbar^2 k^2 }{2M}\right) Y_l^m({\bf\hat k}),$$ where $M$ is the mass of the ...
2
votes
0answers
96 views

One more relation with spherical spinors

Let's have the spherical spinors: $$ \mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} \\ \sqrt{l \mp m ...
0
votes
0answers
55 views

How to find an action of $(\hat {\sigma} \cdot \hat {\mathbf L} )$ on spherical spinors?

Let's have the spherical spinors $\psi_{j, m, l = j \pm \frac{1}{2}}$, $$ Y_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} ...
3
votes
1answer
220 views

A dielectric sphere in an initially uniform electric field and representation theory of SO(3)

I learned recently that the highest order spherical harmonic required to represent the spatial distribution of decay products of a particle can be used to determine its spin, by using arguments ...
0
votes
0answers
128 views

Perturbation of a Hydrogen Atom in a Quadrupole Field

Question: A hydrogen atom is located in a quadrupole field, which gives it a perturbation $$H_1=A(x^2-y^2)$$ where $A$ is some constant. Calculate the ...
0
votes
0answers
36 views

Is there a generic term for orbital groups such as $e_g$ and $t_{2g}$?

I am looking for a generic term for sets of atomic orbitals (viz. spherical harmonics) which are grouped by crystal symmetry. The most familiar examples would be $e_g$ and $t_{2g}$ (in cubic ...
2
votes
1answer
456 views

Spherical harmonics

Given the following potential: $$V(\theta,\phi)=\frac{Q}{a}\left(\sin\theta \cos\phi+\frac{1}{2}\cos^2\theta\right)$$ on the surface of a sphere of radius $a$ I am trying to solve Laplace's Equation ...
4
votes
2answers
470 views

Why does $\ell=0$ correspond to spherically symmetric solutions for the spherical harmonics?

In quantum mechanics why do states with $\ell=0$ in the Hydrogen atom correspond to spherically symmetric spherical harmonics?
3
votes
2answers
253 views

Why 3 dipole terms in a multipole expansion?

As can be seen on this page http://en.wikipedia.org/wiki/Multipole_expansion when we take a multipole expansion without assuming azimuthal symmetry we end up with $2l+1$ coefficients for the $l^{th}$ ...
2
votes
2answers
289 views

Angular momentum of quantum system

Problem: A physical system is in the common eigenstate of $\hat{L^2}$ and $\hat{L_z}$. Calculate the following quantities: $\langle L_x\rangle,\langle L_y\rangle,\langle L_z\rangle,\langle L_x L_y + ...
1
vote
2answers
614 views

Wave function decomposition

Problem: Given the wave function $\Psi_0=A\sin^2(\theta)$ along with the Hamiltonian operator of a physical system: $H=\frac{L^2}{2I}+g B L_z$, find the eigenvalues and eigenfunctions of $\hat{H}$ ...
1
vote
0answers
198 views

Total angular momentum in multielectron atoms

I have some confusion about orbitals in multielectron atoms. Let's say we consider an atom (Lithium, for example, $1s^2\, 2p^1$) and that the state of the last electron is [n=2, l=1, ml=0, s=1/2, ...
4
votes
1answer
174 views

What is an anapolar moment?

I just read this: Anapolar Dark Matter I'm not sure i've heard the term 'anapolar' before, so i google and i found this: http://en.wikipedia.org/wiki/Toroidal_moment This confuses me, my ...
2
votes
1answer
124 views

One body harmonic oscillator states expressed in terms of creation operators

I am reading trough chapter one of Moshinsky's "The harmonic Oscillator in Modern Physics". However i am having some trouble with the mathematics in section 8 of chapter 1. I will sketch a summary of ...
3
votes
1answer
2k views

Hydrogen wave function in momentum space

We can seperate the wave function of an hydrogen atom in a radial and an angle part: $$ \phi_{n,l,m} (\mathbf{r}) = R_{n,l,m}(r) Y_{l,m}(\vartheta,\varphi) \, , $$ where $Y_{l,m}$ are the spherical ...
5
votes
2answers
786 views

Why must the angular part of the Schrodinger Equation be an eigenfunction of L^2?

I was reading about the solution to the Schrodinger Equation in spherical coordinates with a radially symmetric potential, $V(r)$, and the book split the wavefunction into two parts: an angular part ...
1
vote
1answer
677 views

How to use Legendre polynomials in order to determine the (an)isotropy of an on-lattice cluster aggregate?

I am currently testing various models of on-lattice (square lattice in two dimensions) cluster growth for anisotropy. I end up with a cluster, the boundary of which, in case of a truly isotropic ...
0
votes
1answer
419 views

Expansion in solid spherical harmonics on the lattice

I'm interested in calculating scattering processes (e.g. Coulomb scattering of an electron beam by a single ion) in the context of lattice quantum field theory, and wonder if there is something like ...
1
vote
1answer
119 views

Acoustic wave equation for a closed sphere

I am looking to model the nodal surfaces in a resonating closed sphere. The sound source is external. What sort of wave equation will reveal the spherical harmonics depending on the frequency, speed ...
3
votes
1answer
68 views

A nice overview (and maybe derivation) of the Poincaré transformations of the Vector Spherical Harmonics

With $Y_{lm}(\vartheta,\varphi)$ being the Spherical Harmonics and $z_l^{(j)}(r)$ being the Spherical Bessel functions ($j=1$), Neumann functions ($j=2$) or Hankel functions ($j=3,4$) defining ...
10
votes
3answers
4k views

Integral of the product of three spherical harmonics

Does anyone know how to derive the following identity for the integral of the product of three spherical harmonics?: \begin{align}\int_0^{2\pi}\int_0^\pi ...