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.
322 questions
2 votes
3 answers
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Observing $L_x$ of a particle which is present in an eigenstate of $L_z,L^2$
I wonder what will be the probability of finding the particle in a particular $L_x,L^2$ state after we know its $L_z,L^2$. Definitely,the $L^2$ value will not change. Here is what I tried: First of ...
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2 votes
1 answer
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Associated Legendre equation solution for the angular equation
So in Griffith's, after doing separation of variables on the angular equation $$ \sin\theta \frac{\partial}{\partial\theta}(\sin\theta \frac{\partial Y}{\partial\theta}) + \frac{\partial^2 Y}{\partial ...
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2 votes
2 answers
269 views
Why are there two different types of visual depictions of electron orbitals?
I was refreshing myself on electron orbitals and quantum numbers. I’m seeing two different looking depictions of some of these orbitals. the second diagram aligns more with what i’m reading on the ...
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Why doesn’t ΛCDM predict any $ℓ=5,10,15…$ resonances in the CMB?
The CMB power spectrum is nearly scale-invariant, yet some models predict spikes at $ℓ=5n$. Is there a fundamental reason ΛCDM lacks such features, or is it just unparameterized?
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Tensor products and simultaneous eigenstates
In A Modern Approach to Quantum Mechanics, Townsend writes: One of the most evident features of the position-space representations (9.117), (9.127), and (9.128) of the angular momentum operators is ...
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1 answer
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What is $s$-wave sector in the context of black holes?
Can someone explain the terminology $s$-wave sector that appears here? Let us begin by reviewing the derivation of Hawking radiation in the $s$-wave sector,.....
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1 vote
1 answer
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Hermicity not preserved in Hamiltonian [closed]
My question is in the context of a matrix element in a diatomic molecule. I will rephrase it as well as possible to remove any unnecessary complexity. I denote the spherical harmonic as $Y_m^l = |m,l\...
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Direct product basis and product of spherical harmonics [duplicate]
I am recently reading about the Wigner-Eckart and Clebsch-Gordon sections in Sakurai, Modern Quantum Mechanics, 2nd Ed (2014). On p. 234, Eq. (8.72), I found that he derived the expression for the ...
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10 votes
4 answers
2k views
What determines the shape of electron suborbitals?
The azimuthal quantum number determines the shape of electron distribution around a nucleus ($s$ orbital has spherical distribution, $p$ orbital has dumbell-like distribution and so on). But what ...
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1 vote
1 answer
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The concept of spin direction in real space
We can expand the real space eigenvectors of positions in terms of angular momentum states as: $$|x,y,z\rangle =\sum_{l,m} \alpha_{l,m}|l,m\rangle$$ so it seems reasonable to have direction for ...
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Can a solution to Laplace's equation for prolate ellipsoid be understood in spherical coordinates?
First posted to Math StackExchange, then thought this forum would be better. I have been through the details of finding solutions to Laplace's Equation in spherical coordinates for a magnetized sphere,...
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Sign issue with the boundary integral of Green's Function
I feel it crucial to start by writing out the derivation of the solution to Poisson's equation using Green's formula: $$\nabla \cdot \phi \nabla \psi = \phi\nabla^2\psi + \nabla\phi\cdot\nabla\psi$$ ...
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1 answer
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If $\sigma^+$ light couples states $|ℓ,m_ℓ\rangle$, $|ℓ',m_ℓ+1\rangle$ does it also couple states $|ℓ',m_ℓ+1\rangle$, $|ℓ,m_ℓ+2\rangle$?
Consider atomic states $|\ell,m\rangle$, where each state is $(2\ell + 1)$-fold degenerate. Descriptions of optical pumping often involve arguments where a polarized light source, say $\sigma^+$, ...
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1 vote
1 answer
216 views
Expansion of electric field into Spherical Harmonics
A solution of of the Laplace equation in spherical coordinates that is regular at origin can according to Zangwill can be written as $$ \varphi(r,\theta,\phi) = \sum_{lm} A_{lm} r^l Y_{lm}(\theta,\phi)...
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The use of Dirac notation to derive identities involving a product of spherical harmonics
In quantum mechanics we can identify the "Angular momentum operator" $\textbf{L}$ via: $$ \langle { \theta,\varphi } | \textbf{L} | \psi\rangle = - i (\textbf{r} \times \nabla) \psi(\theta,\...
