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# Questions tagged [spherical-harmonics]

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### 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-...
390 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 ...
978 views

### 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 ...
1k views

416 views

### Can a standing wave exist on a spherical surface?

I've often seen the DeBroglie wave illustrated by a two dimensional surface as a standing wave, but then the 'electron cloud' surrounding an atom is hardly two dimensional and furthermore held to the ...
166 views

### Is there an intuitive interpretation of the shape of the angular momentum eigenstate?

I was watching a MIT lecture video on angular momentum eigenstate. Toward the end of the lecture, the professor had shown some plots of the first few spherical harmonics, in an attempt to explain ...
47 views

### Angular power spectrum: Calculating bias from N weighted events

I'm interested in calculating the angular power spectrum $C_{l,N,\omega}$ of $N$ weighted (weight $\omega_i$ for event $i$) events from a full sky map with distribution $C_l$? Interesting quantities ...
600 views

### How to calculate the angular momentum states of isotropic quantum harmonic oscillator?

While trying to calculate the angular momentum states for the first non trivial even and odd states ($N=2$ and $N=3$). When $N=n_x + n_y + n_z$ By solving the radial problem one can see that there 6 ...
1k views

### Klein Gordon equation in Schwarzschild spacetime (spherical harmonic mode expansion)

My Question: In his GR text, Robert Wald claims that solutions, $\phi$, to the Klein-Gordon equation, $\nabla_a\nabla^a \phi = m^2 \phi$, in Schwarzschild spacetime can be expanded in modes of the ...
176 views

### EM Field Quantization in Spherical polars

Is it possible to quantize the electromagnetic field in spherical polar coordinates instead of cartesian ones? Such that creation and annihilation operators correspond to harmonic oscillator modes ...
862 views

### Which direction do electrons orbit around/near the nucleus differ in aligned magnetic atoms?

Atoms can be aligned in magnets to create magnetic fields. Does that alignment give an atom a north and south pole or certain atoms have a unique electron orbitals giving an atom a north and south ...
1k views

### Why do we need the Condon-Shortley phase in spherical harmonics?

I'm confused with different definitions of spherical harmonics: $$Y_{lm}(\theta,\phi) = (-1)^m \left( \frac{(2l+1)(l-m)!}{4\pi(1+m)!} \right)^{1/2} P_{lm} (\cos\theta) e^{im\phi}$$ For example here ...
179 views

### Why is $\langle x \rangle =0$ for the ground state hydrogen atom?
From Griffiths, Introduction to Quantum Mechanics, 2nd ed: I found $\langle r \rangle =\frac{3a}{2}$ and $\langle r^2 \rangle =3a^2$. Now I need to find the expectation value of x. However, I don't ...