Questions tagged [spherical-harmonics]

Special functions defined on the surface of a sphere, often employed in solving partial differential equations. They form a complete set of orthogonal functions and thus an orthonormal basis.

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Integrate the expression of error for photometric galaxy clustering and after compute the variance of shot noise

The error on photometric galaxy clustering under the form of covariance which is actually a standard deviation expression for a fixed multipole $\ell$ : $$ \sigma_{C, i j}^{A B}(\ell)=\Delta C_{i j}^{...
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How to normalize a linear combination of spherical harmonics?

I know the formula for normalizing individual spherical harmonics, but do not know how to normalize linear combinations of them Say I have a system $ \alpha (\theta, \phi) = aY_{l_1}^{m_1} + bY_{l_2}^...
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Clebsch-Gordan coefficients and total angular momentum basis question

Suppose I have the total angular momentum basis for two particles, formed by $\{|J,M\rangle\}$, where $J\in |j_{1}-j_{2}|\leq J\leq j_{1}+j_{2}$ and $M=m_{1}+m_{2}$ with $ -J\leq M\leq J$. There is a ...
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Legendre spherical harmonics - Does it make sense to have multipole interval lower than 1?

I introduce in the context of a physical noise $B$ a factor $\Delta\ell$ multipole appearing in the expression of the variance $\operatorname{Var}\left(B\right)$ of $B$. The formula is the following : ...
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Covariant derivative of spherical harmonics

Given is the metric $\gamma_{jk}$ for the surface of a Sphere $S^2$ with $\gamma_{22}=1,\gamma_{23}=\gamma_{32}=0$ and $\gamma_{33}=\sin^2(\theta)$. The coordinates are $x=$($t,r,\theta,\phi$) and $j$ ...
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Rotations of spherical harmonics and Wigner $D$-matrices

I seem to be having trouble understanding how Wigner D-matrices rotate spherical harmonics. I asked this question on the Maths Stack Exchange but decided to cast my net a bit wider and ask the ...
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Simplifying an integration using Legendre expansion

I'm following a derivation for the first and second moments of a conditional probability distribution $f(s; \mathbf{r}, \hat d)$ (see paper here ), where $s$ is the path length that a charged particle ...
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Dirac equation for hydrogen atom

I went over a calculation of the hydrogen wavefunction using Dirac equation (this one) and I am a bit confused by the angular part. The final result for the wavefunction based on that derivation is ...
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Converting $\psi(x,y,z)$ to $f(r)Y_l^m$ and calculating expectation value of the angular momentum operators

I've been trying to solve a particular class of problems, where I'm given a wavefunction $\psi (x,y,z)$ and I'm asked to find the expectation value or eigenvalue related to the angular momentum ...
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How could I calculate the product of two spherical harmonics

For now I have to calculate the product of two spin-weighted spherical harmonics: $ _{s}Y_{lm}\ \times\ _{s}Y_{lm}^{*}$ What’s the result of this product? Maybe it is a spin-weighted 0 spherical ...
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Relationship between 3 quantities: Density Matter power spectrum, Density Matter angular power spectrum and Temperature angular power spectrum

Summary: I would like to go deeper in the relationship between Matter power spectrum and Angular power spectrum. The starting point is the relationship between "Power spectrum $P(k)$" and ...
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What does it mean that Zernike's polynomials form a orthogonal basis on the unit disk?

I am doing a project on adaptive optics and I would like to understand a little more about Zernike's polynomials. What does it mean that they form an orthogonal basis on the unit circle? What ...
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Eigenvalue equation of orbital angular momentum operator $L^2$ and $L_z$

I am currently working on Gasiorowicz's Quantum Physics. The writer says that since $\mathbf{L}^2$ and $L_z$ commute, we have simultaneous eigenket $|l,m\rangle$, and thus we can write \begin{align} \...
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Expanding wavefunctions in terms of $L_z$ eigenkets

The angular momentum operator along the $z$-axis $L_z$ satisfies the secular equation $$ L_z|m,l\rangle = \hbar m |m,l\rangle,$$ where $l$ is the corresponding (integer-valued) eigenvalue of the ...
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Matrix representation of the $x$-component of orbital angular momentum $\hat{L}_x$

In my notes it is given that using the spherical harmonic (shown below) as basis states in this order, the matrix representing the $x$-component of orbital angular momentum $\hat{L}_x$ for a particle ...
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Looking for the experimental spherical harmonics coefficients of the CMB

Using Mathematica, I would like to plot the Cosmic Microwave Background has a spherical harmonics decomposition. As experimental data, I need to know the values of the $a_{l,m}$ coefficients in the ...
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The degeneracy of spherical harmonics

On page 336 of Shankar's 'Principles of Quantum Mechanics' the author states "The $Y_l^m$ functions are mutually orthogonal because they are nondegenerate eigenfunctions of $L^2$ and $L_z$, which ...
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Can you recover the quantum numbers from just the shape of the spherical harmonic?

So I was wondering, in quantum physics beautiful graphs are introduced displaying spherical harmonics relying on the quantum numbers of $m$ and $l$. But is it possible to recover these quantum numbers ...
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Angular part solutions of Schrodinger's equation

Hello Physics Community i'm trying to solve the angular part of Schrodinger's equation, specifically the $\theta$ part , $$ \left[\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d}{d \theta}\...
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coupled vectors in spherical coordinates

I have asked a similar question before (Coordinates for coupled transition dipole moments). What I am trying to do here is to write the $X$ component of $\vec{\mu}_{\beta}$, which is $\sin(\theta_{Z\...
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Quantum Mechanics Griffiths Problem [closed]

I was doing problem no. $4.4$ from Griffiths Third Edition. I cannot understand one thing related to the solution of the normalization of $Y_2^1$. $$Y_2^1 = -\sqrt{\frac{15}{8\pi}}e^{i\phi}sin\theta ...
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What ranges of $l$ and $m$ give the spherical harmonics for disk features?

The spherical harmonics represent a spherically symmetric basis and correspond to the eigenfunction solutions of the three-dimensional Laplacian. They take the form $$ Y_l^m(\theta,\phi) = \sqrt{\frac{...
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How do I fit a Fermi surface using cylindrical harmonics?

I am trying to fit a Fermi surface to cylindrical harmonics as done in this paper: https://journals.aps.org/prb/pdf/10.1103/PhysRevB.96.075163 (Table 1) And its supplementary material: https://...
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Uniqueness of representation of a quantum system in polar coordinates

Are spherical harmonics the only possible Hilbert basis for a polar description of a quantum system? In other words, is the following representation for a wave function $\Psi(\theta, \phi):$ $$\Psi (\...
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Simplest possible solution to the Klein-Gordon field equation has a (KG) norm which is not constant in time

It is a fact that the Klein-Gordon inner product must be constant for all $t>0$, where the Klein-Gordon product is defined by $$ \langle f, g \rangle \ := \ i \int d^3x \; \left[ f^{\ast}(t,\mathbf{...
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Geometric interpretation of orbital angular momentum eigenvalues

The eigenvalues of the square of the orbital angular momentum are defined as: $$ \hat{L}^2 | \phi \rangle = \hbar^2 l(l+1) | \phi \rangle $$ $$ \hat{L}_z | \phi \rangle = \hbar m | \phi \rangle, ...
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How can real $d$ orbitals be computed from complex orbitals?

I recently completed MIT's 8.04 quantum mechanics course on edX and have been writing python code to compute hydrogen-like electron orbitals, basically just for fun. My program computes the ...
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What exactly azimuthal symmetry means?

In quantum mechanics, when discussing scattering by a potential, it is written that we are assuming the potential is spherically symmetric so the function cannot depend on $φ$. Azimuthal symmetry ...
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Span of eigenstates of angular momentum

Do the eigenstates of the angular momentum operator span the space of all possible wave functions even if the potential is not central?
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Can you expand a real scalar field $\phi(t,\mathbf{x})$ in terms of spherical harmonics?

A massless real scalar admits the expansion $$ \phi(t,\mathbf{x}) = \int \frac{d^3\mathbf{p}}{(2\pi)^{3/2} \sqrt{2|\mathbf{p}|}} \bigg( e^{ - i |\mathbf{p}| t + i \mathbf{p} \cdot \mathbf{x} } a_{\...
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Is it better to think of electron shells as depending on the value of $n$, or depending on the energy difference between sub-shells?

Often electron shells are defined as 'states with the same principal quantum number $n$' which would suggest that $3s$, $3p$, $3d$ sub-shells are all in the same shell. Conversely, it is also often ...
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What significance can we assign to 'breathing' of $s$-shell orbitals?

In my atomic lecture notes it says We can visualise an $l=0$ $s$-state as a spherical cloud expanding and contracting - breathing, as the the electron moves in space. Similarly in the video enter ...
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Rotational wavefunction for diatomics in the ground state

Frozen diatomic molecule looks like a cylinder. When it rotates, since it rotates fast, its wavefunction will have some symmetry, depending on the rotational state (which are quantized). In the ground ...
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Where is my blunder in this angular momentum "identity"?

I was searching for ways to reduce the following expression $$ A = \sum_{m_3=-l_3}^{l_3}C_{l_1m_1,l_3m_3}^{l_2m_2} Y_{l_3m_3}(\hat u) $$ where $\hat u$ is a unit vector, $Y_{lm}(\hat u)$ is a ...
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Integral of the product of 4 spherical harmonics

Recently, I saw a closed formula for the integral of the product of three spherical harmonics in two dimensions here Integral of the product of three spherical harmonics and I was wondering if someone ...
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Probability of finding certain value of the $z$-component of the angular momentum when the wave function contains multiple $l$-values [closed]

Suppose that $$ \psi = \frac12 Y_{00}+\frac1{\sqrt 3}Y_{11}+\frac 12 Y_{1,-1}+\frac1{\sqrt6}Y_{22}.$$ This wave function is not an eigenstate of $\hat{L}_z$. If a measurement of the $z$-component of ...
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Non-Periodic Boundary conditions for wave equation

I already know that when we have a problem with spherical symmetry and periodic boundary conditions in both angles the Helmholtz equation (like the equations that we find in physics when we have wave ...
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Why is the solution of the radial Schrödinger equation valid at $r=0$?

The Schrödinger equation for a particle in a central potential is $$\left[\frac{p_r^2}{2m}+\frac{\ell(\ell+1)}{2mr^2}+V(r)\right]\psi(r,\theta,\varphi)=E\psi(r,\theta,\varphi).$$ This gives solutions ...
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Spherical Harmonics Sum Identity

I'm taking a course in Quantum Mechanics and this problem is causing me some struggles. Can anyone help me to prove this identity? $$\sum_{m = -l}^l m^2 |Y_{l}^{m}(\theta, \phi)|^2 = \frac{l(l+1)(2l+1)...
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Spin spherical functions in hydrogen-like solution to Dirac equation

I saw a video where the following equality is wrote for the angular part of the solution of the Dirac equation for hydrogen-like atoms: \begin{align} \Omega^{j=l\pm 1/2}_{lm}&= \frac{1}{\sqrt{2l+1}...
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197 views

Completeness relation of spherical harmonics

In spherical coordinates, the resolution of the identity can be written as $$ 1=\int_0^{2\pi}d\phi\int_0^{\pi}\sin\theta\, d\theta\, |\theta,\phi\rangle\langle\theta,\phi| \equiv \int d\Omega |\Omega\...
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Properly understanding the concept of Wave Function and its versatility

Let's start from the beginning: The state of a quantum particle is represented by a vector $|\psi\rangle$ in the Hilbert space, observables are represented by Hermitian operators which eigenvalues ...
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Why does Jackson's book use different method to calculate the vectorial multiple expansion coefficients in chapter 10.3 and 9.7?

In chapter 9.7: $$Z_0a_E(l,m)f_l(kr)=-\frac{k}{\sqrt{l(l+1)}}\int{Y^*_{lm}\mathbf{r\cdot E}d\Omega}\tag{9.123}$$ In chapter 10.3 $$a_\pm(l,m)j_l(kr)=\int{\mathbf X^*_{lm}\mathbf\cdot\mathbf E(\mathbf ...
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How should we think about Spherical Harmonics?

Studying Quantum Mechanics I only thought about Spherical Harmonics $Y_{l,m}(\theta , \phi)$: $$Y_{l,m}(\theta , \phi)=N_{l,m}P_{l,m}(\theta)e^{im\phi}$$ as the simultaneous eigenfunctions of $L_z$ ...
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What is the best way to solve an half spin particle exercise?

This question is strongly related to this other question regarding the definition of spinor. Let's take for example the following exercise: Given a particle with mass $m$ and spin $1/2$, described by ...
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Arguments of the eigenfunctions of angular momentum

Once we have defines the angular momentum operators $L_z,L_y,L_x,L^2$ ($L^2=L_z^2+L_y^2+L_x^2$) suppose we focus on the eigenstates $|l \ m \rangle$ common to both $L_z$ and $L^2$: $$L_z|l \ m\rangle =...
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Expansion of Green's Function in Spherical Coordinates in Jackson

So I was reading about the expansion of the Green function in Spherical coordinates from Classical Electromagnetism by J.D. Jackson and I'm really confused about a subtle step that he makes to go from ...
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Understanding vector spherical harmonics

I am studying this classical paper by Regge and Wheeler Stability of a Schwarzschild singularity. In the second page they introduce their formalism with spherical harmonics and generalization thereof. ...
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Legendre Polynomials Integrals for Laplace Solutions

I know how to normalize Legendre polynomials, but I have a sphere with 0 to pi/3 boundaries where the potential is $V$, otherwise zero. For normalization it is -1 to 1, what changes with different ...
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Interpreting spherical harmonics angular momentum eigenstates

I'm trying to get some intuition behind the spherical harmonics being the angular momentum eigenstates in quantum mechanics. Firstly, am I correct in saying that we can imagine the angular momentum ($|...

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