Questions tagged [spherical-harmonics]

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Transformation for relation between angular power spectrum and matter power spectrum

Just a precision : in one link of my previous post ( previous link ) Why does the writter say "How to write the 3D power spectrum, $P_{k}$, as an integral of the angular power spectrum, $C_{\ell}$...
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40 views

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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1answer
42 views

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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31 views

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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1answer
76 views

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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16 views

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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38 views

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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36 views

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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70 views

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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31 views

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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59 views

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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47 views

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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204 views

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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2answers
27 views

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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21 views

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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46 views

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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1answer
73 views

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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26 views

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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1answer
80 views

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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1answer
158 views

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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2answers
60 views

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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1answer
73 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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3answers
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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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1answer
36 views

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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1answer
77 views

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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2answers
106 views

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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1answer
227 views

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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37 views

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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1answer
36 views

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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1answer
157 views

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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29 views

Coordinates for coupled transition dipole moments

I am working on some calculations on coupled transition dipole moments interacting with light pulses with different polarisations. I am currently trying to find a way of writing $\theta_{\beta}$ and $\...
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0answers
46 views

Spherical harmonic expansion with radius

I'm having some trouble in computing the spherical harmonic's expansion of a general wave function $\psi(\textbf{x})$ where $\textbf{x}$ is a vector in 3D . The result on the article I'm working on is ...
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253 views

Physically unacceptable solutions for the QM angular equation

I'm reading Griffiths's Introduction to Quantum Mechanics 3rd ed textbook [1]. On p.136, the author explains: But wait! Equation 4.25 (angular equation for the $\theta$-part) is a second-order ...
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Nodes in Spherical Harmonics

In Rydberg Atoms book by Thomas F. Gallagher, Page 14, the author provides the general equation of spherical harmonics which is well-known. He states that there exist $l-m$ nodes in the $\theta$ ...
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1answer
190 views

In actuality, light can only approximate spherical waves, as it can only approximate plane waves

I am currently studying Optics, fifth edition, by Hecht. In chapter 2.9 Spherical Waves, the author says the following: The outgoing spherical wave emanating from a point source and the incoming wave ...
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1answer
27 views

The pulse has the same extent in space at any point along any radius $r$

I am currently studying Optics, fifth edition, by Hecht. In chapter 2.9 Spherical Waves, the author says the following: $$\dfrac{\partial^2}{\partial{r}^2}(r \psi) = \dfrac{1}{v^2} \dfrac{\partial^2}{...
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Why can transform the $\rm SU(2)$ spin to $S^2$ space?

Spin lies in $\rm SU(2)$ space, i.e. $S^3$ space, but when we write the spin coherent state: $$|\Omega(\theta, \phi)\rangle=e^{i S \chi} \sqrt{(2 S) !} \sum_{m} \frac{e^{i m \phi}\left(\cos \frac{\...
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Why do we care about real form spherical harmonics?

I'm studying atomic orbitals and the shape is usually represented with real form spherical harmonics, taken as an appropriate linear combination of the complex ones. If, however, the physical quantity ...
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3answers
51 views

Eigenstate of spherical harmonics

$$ \psi(\theta,\phi) = 1/(2\sqrt{3\pi}) [ \sqrt{5} \cos\theta + \sin (\theta + \phi) + \sin(\theta -\phi) $$ Calculate $\hat{L}^2 \psi$ and $\hat{L}_z \psi$. Is $\psi$ an eigenstate of $\hat{L}^2$ and ...
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Integration over all directions in spontaneous emission, complex dipole element and polarization vectors

When calculating the rate of spontaneous emission in the Weisskopf-Wigner theory the time derivative of the excited state is related to a sum of couplings to all modes (directions and polarizations) ...
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89 views

How to express a rank-2 tensor as a spherical tensor?

A common example how to write a rank-2 tensor in the spherical basis is an outer product of two vectors, $$ T_{ij} = a_i b_j $$ such that $$ T_{ij} = \frac{\textbf{a}\cdot\textbf{b}}{3}\delta_{ij} + ...
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1answer
147 views

Electric dipole field and spherical harmonics

The field of the electric dipole is $\displaystyle\vec{E}=\frac{3(\vec{p}\cdot\hat{r})\hat{r}-\vec{p}}{4\pi\epsilon_{0}r^3}$, show that it can be written as linear combination of $\displaystyle\frac{...
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1answer
114 views

Spherical tensor operators relation to spin operators

I am reading a research paper discussing theoretical calculations of electron paramagnetic resonance parameters (EPR)*. In the section about higher-order EPR parameters, it states that according to ...