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Questions tagged [spherical-harmonics]

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How does the hydrogen atom actually “look like”? [duplicate]

When deriving the solutions for the "simple" quantum mechanical hydrogen problem, one normally uses spherical coordinates $(r,\theta,\phi)$, since the problem has rotational symmetry. The solution has ...
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323 views

In a spherically symmetric central potential why do we look for eigenfunctions of the angular momentum operator?

In finding the solutions to the wave equation for a spherically symmetric potential $V(r)$, we look for the eigenfunctions of $\hat L^2$ and $\hat L_z$ operators. However, what is the reasoning behind ...
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Quantization of electrons' angular momentum in atoms and molecules

It is known that the Schrödinger's equation of the electron's wave function in atoms can be solved analitically only when a single electron is present (the "hydrogenlike atom"). In that case, the ...
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Systematic expansion of $e^{i\vec{k}\cdot\vec{r}}$ in atomic physics in terms of Legendre polynomials and identifying different $l$ terms

In the context of light-matter interaction one often makes the approximation $e^{i\vec{k}\cdot\vec{r}}\approx 1$. Keeping higher order terms in $e^{i\vec{k}\cdot\vec{r}}$ give magnetic dipole, ...
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Directional Eigenket?

I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it. Because the angular dependence is ...
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62 views

Linear combination of 2 spherical harmonic functions

The task is to form 2 linear combinations out of the 2 given spherical harmonic functions. I dont understand why the resultant wave function has to be multiplied with the constant $1/sqrt(2)$?
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Wigner-Eckard theorem in 3+1 Minkowski dimensions

From this source, I have: I cannot find much (if not any) information online for Wigner-Eckard in 4D, hyperboiloid harmonics etc. And there are many facts just states in this source that I would ...
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Can spherical harmonics be used in relativity equations?

I have a neutral pion of mass $m_{\pi}$, and it decays into two photons. In it's reference frame the decay is isotropic. One of the photons has a helicity of $+\hbar$ and the other $-\hbar$. In ...
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What are the boundary conditions for the Hydrogen Atom which cause the polar power series to need to terminate?

I am trying to solve the Hydrogen Atom, and I am stuck in the polar equation. To simplify, I have taken the special case in which $m=0$ to get the Legendre Equation: $$(1-x^2)P''(x)-2xP'(x)+AP(x)$$ $$(...
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38 views

matrix elements of $\hat{z}$ operator under the angular momentum basis

I found a quite challenge quantum mechanics problem in a preparation sample test for a midterm. The sample test does not have a solution, so it is bothering. The question reads as follows: ...
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How to evaluate the matrix element of coulomb repulsion term between electrons in an atom suing spherical harmonics multipole expansion?

This is a lecture notes take from the following link on numerical calculation of atomic physics:http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf I am trying to evaluate the two electron matrix element ...
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Musical Instrument that Exhibits Spherical Harmonics

A guitar string exhibits standing wave patterns when its struck, some superposition of sines and cosines, a drum head exhibits a superposition of Bessel functions when its struck. Is there any ...
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123 views

Definition of the spherical harmonics do not agree

I Griffiths' Introduction to quantum mechanics, the spherical harmonics are defined as $$Y_l^m(\theta,\phi) = \epsilon\sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi} P_l^m(\cos \theta)$$ ...
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Probability of finding a particle in the solid angle $d\Omega$ at $\theta$ and $\phi$ [closed]

For a spinless particle with the wavefunction \begin{equation} \psi(x,y,z)= K(x+y+2z)\exp(-\alpha r) \end{equation} with $r=\sqrt{x^2+y^2+z^2}$ and K and $\alpha$ are real constants. I have to ...
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What does the spherical-harmonic notation $Y^{m}_l(\hat{\textbf{r}})$ mean, and how does it relate to the usual $Y^m_l(\theta, \varphi)$?

By using the plane wave expansion, the decomposition of stationary harmonic plane wave into partial waves can be given by $$ e^{i\textbf{k}\cdot\textbf{r}} = e^{ikz} = e^{ikr\cos\theta} = \sum^{\infty}...
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Vector Spherical Harmonics and total angular momentum

In their book Akhiezer et al. give a definition of vector spherical harmonics (p.18 of Russian Edition) as $$\pmb{Y}_{j\ell m}(\pmb \Omega) = \sum_{m' \lambda} \langle \ell m' 1\lambda| jm \rangle Y_{...
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Putting the ket $|l,m_x \rangle$ in terms of the ket $|l, m_z \rangle$

Could someone guide me in my thought process of this problem? I don’t know if I’m thinking about it the right way. The problem is the following: I have a system which possible states are generated ...
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1answer
84 views

Orthonormal Basis integration and Kronecker delta

Given that this integral I'm trying to solve is $$\frac{2}{\pi}\sum^{\infty}_{l=0}\sum^{l}_{m=-l}\int_{r=0}^{\infty}\int_{k=0}^{\infty} R_{nl}(r)b_{lm}(k)j_{l}(kr)k^2 r^2 \int_{\theta = 0}^{\pi}\int_{\...
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Do relativistically-contracted electron states have the same energy and angular momentum values?

I've been reading that electron bound states are defined by four quantum numbers, $n$, $l$, $m_l$, and $m_s$, respectively the principal quantum number, the azimuthal quantum number, the magnetic ...
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Why are there three $p$-orbitals?

This question is specifically about Schrödinger quantum mechanics, but if an answer in some other mode would illuminate it could be acceptable, as demonstrating a physical or mathematical reason for ...
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Spherical harmonic about normal mode

Since spherical harmonic could describe the normal mode on spheroid(like Earth), I want to ask why the order m must between degree -ℓ ~ ℓ? I can understand it by the math during the proof process, it ...
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What are spin-2 spherical harmonics and why are they needed?

A function $f(\theta,\phi)$ (with $\theta,\phi\in \mathbb{S}^2$) can be expanded in terms of spherical harmonics $Y_{l.m}(\theta,\phi)$. Recently, in this Particle Data Group review titled Cosmic ...
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Simplifying CMB correlation function with spherical harmonics

I originally asked this on the physics Stack Exchange site, but perhaps it could be more easily answered here. Given the definition of the correlation function for CMB temperature fluctuations as $$ ...
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2answers
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Expectation Value $\langle \frac{1}{r^2} \rangle$ using Hellmann–Feynman theorem

Suppose we have the hydrogen atom$$ H ~=~ \frac{p_r ^2}{2m} + \frac{L^2}{2mr^2} -\frac{e^2}{r} \,.$$And have solved the Schrodinger equation finding $$E_n = - \frac{me^4}{2 \hbar ^2 n^2} $$ and $$ Ψ_{...
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Hydrogen Atom, polar equation eigenfunctions

In my textbook, Quantum Mechanics by David McIntyre on page 235, the solutions to the polar equation resulting from the separation of variables to the hydrogen atom are the eigenstates: The book ...
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What is a force multipole?

In a recent talk about physics and mechanics inside the cell, I heard such terms as 'force monopole' and 'force dipole'. What do such terms mean? Are they talking about the angular distributions, ...
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82 views

Increase in energy with increasing orbital quantum number

If we consider the problem in a spherically symmetric potential, we can see that with an increase in the orbital quantum number, the energy state in the spectrum increases. This is observed both in ...
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97 views

Spherical Harmonics: Physical Meaning behind the Mathematics: Dirac

Hello Physics Exchange, I have a series of concept questions. Mostly to prove or disprove my understanding. This will be my first post, so if you have any suggestions to improve my formatting - I ...
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Writing electric and magnetic directional derivatives?

I have just decomposed a massless Maxwell field using the basis of vector spherical harmonics given by: $$\overline{Y}_{lm}^r=Y_{lm} \hat{r}$$ $$\overline{Y}_{lm}^e=r \nabla Y_{lm}$$ $$\overline{Y}_{...
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What is the geometry of DeBroglie standing waves?

I asked a similar question here . But never received a complete answer. So I've made the question more specific to DeBroglie waves. So from what I've read DeBroglie waves are indeed standing waves ...
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Poisson equation for a point charge in spherical coordinates

While learning for an exam, I came across an exercise in electrostatics. Although I think I understand what the goal of the exercise is, I'm confused about notation and the hint in the exercise. It is ...
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Integral of spherical harmonics and their derivatives

What is the Integral of the product of spherical harmonics and derivatives of spherical harmonics? More precisely, I am looking for $$\int_\Omega d\Omega\, Y_{l}^m Y_{l^\prime}^{m^\prime} \partial_\...
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315 views

Compact Formula for Wigner D-matrix

I have done an elementary calculation that appears to be giving me a simple (intuitive) formula for an arbitrary Wigner D-matrix. I can't seem to find this formula anywhere. In the following, $\...
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311 views

Why spherical harmonics are related to certain rotations (and not others)?

Let's take a direction eigenket $|{\bf\hat{n}}\rangle$ in 3-dimensional space oriented with angles $\theta\in\left[0,\pi\right]$ and $\phi\in\left[0,2\pi\right]$ in spherical coordinates. Next take ...
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339 views

Laplace equation azimuthal and non azimuthal symmetry

In general we can say that if the potential is specified in the surface of a sphere with azimuthal symmetry $(m=0)$ has a solution: $$ \Phi(r,\theta)= \sum_{l=0}^\infty \left[ A_lr^l + B_lr^{-(l+1)}\...
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Is there a relation between the Legendre generator function and Spherical Harmonics for a Potential?

Recently I had to solve a simple problem in which I had a sphere of radius $R$ with a constant potential (but with different sign), on both of the hemispheres, and I was asked to get the electrostatic ...
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298 views

How do multipole moments relate to a Taylor expansion, with regards to Newtonian potential?

Given the Newtonian gravitational potential, $$ \phi(\mathbf{x}) = - \int \frac{G \rho(\mathbf{x'})}{|\mathbf{x} - \mathbf{x'}|}$$ One can construct a 'multipole expansion' by using the Taylor ...
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Can you write the Einstein equation as an eigenvalue equation for analytical solutions (to local problems) on non-flat spacetimes

In reading about various local solutions to Einstein's field equation it is easy to forget that they almost all assume a flat background spacetime (at least asymptotically). Considering this made me ...
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Expanding Schwarzschild solution using spherical harmonics?

The well known Schwarzschild solution is given by \begin{equation} (ds)^2 = \left( 1 - \frac{2GM}{c^2r} \right) dt^2 - \left( 1 - \frac{2GM}{c^2r} \right)^{-1} dr^2 - r^2 d\Omega^2. \end{equation} ...
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221 views

Queries of Proof of Wigner-Eckart Theorem

With regard to the Wigner-Eckart Theorem the following is stated: The following is an outline of the proof in a text I am using: "Consider the action of a tensor-operator component on an angular-...
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1answer
208 views

What does it mean to normalize a combination of Spherical Harmonics?

Using the following as an example: Show that the combinations $$-\frac{1}{\sqrt{2}}\left(Y_{11}-Y_{1-1}\right)\quad\text{&}\quad\frac{i}{\sqrt{2}}\left(Y_{11}+Y_{1-1}\right)$$ are real and ...
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Irreducible form of Spherical tensor operators

In the section on spherical tensors in Sakurai, he introduces the idea of going from Cartesian tensors to irreducible spherical tensors. He states the following: A spherical harmonic can be written ...
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Position operator in spherical basis

In a set of notes it states that we can define spherical basis vectors in terms of the Cartesian basis vectors $\hat{x}, \hat{y}$ and $\hat{z}$ by $$\hat{e}_{\pm 1} := \mp \frac{1}{\sqrt{2}}(\hat{x} \...
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Spherical Formulation of Quantum Mechanics

I always wondered, during my QM courses, if we don't explore enough of the freedom that the Lagrangian and Hamiltonian Classical Dynamics give us. Classically, we can always make canonical ...
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The dipole radiation pattern and spherical harmonics $Y_{10}$

I am studying the multipole expansion of electromagnetic wave radiation pattern, and it is said that any fields can be decomposed into the spherical harmonics $Y_{lm}$. However, for $l=1$, which ...
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Is it a coincidence or a convention that the spherical harmonics $Y_{\ell m}(\theta, \phi)$ are eigenstates of $L_z$?

The $z$-component of the angular momentum $L_z$ is an eigenstate of the spherical harmonics $Y_{\ell m}(\theta, \phi)$. The spherical harmonics $Y_{\ell m}(\theta, \phi)$ are also the eigenstates of ...
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Monopole Spherical Harmonics

I was following Yakov Shnir his book about magnetic monopoles, there they derive the monopole spherical harmonics. I will sketch the derivation briefly, The starting equation is the eigenvalue ...
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357 views

Spherical harmonics and angular momentum

I'm studying the relation between rotation matrices and spherical harmonics $\langle\hat{r}|l,m\rangle = Y_{lm}(\theta,\phi)$. In one demonstration it is stated that if $|\hat{r}\rangle=|\hat{e}_z\...
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Why are circular unit vectors often defined as $\hat{\mathbf e}_\pm = \mp (\hat{\mathbf e}_x \pm i \hat{\mathbf e}_y)/\sqrt{2}$

When dealing with circular polarizations, spherical harmonics, and generally with any vector-valued rotationally-invariant quantity, it is often a requirement to define complex-valued unit vectors of ...
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Meaning of different Stokes' coefficients in harmonic expansion of gravitational potential

I know that geophysical quantities, such as gravitational potential, can be written in terms of spherical harmonics: $$W(r,\phi,\lambda;t) = \frac{G m_e}{r} + \frac{G m_e}{r} \sum_{l=2}^{N_{max}}\left(...