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4
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2answers
146 views

How can mean value of a quantity $be$ an operator?

In Laundau & Lifshitz Quantum Mechanics. Non-relativistic theory in $\S29$ a problem is given: PROBLEM Average the tensor $n_in_k-\frac13\delta_{ik}$ (where $\mathbf{n}$ is a unit vector along ...
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0answers
26 views

Convert angular power spectrum to spatial power spectrum

If we have a signal projected on a sphere, one routinely decomposes this in spherical harmonics, in analogy to a Fourier decomposition in flat space. One can then make the decomposition: ...
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0answers
59 views

Degeneracy of Spherical Harmonics Eigenfunctions

I'm working through Griffiths' Introduction to Quantum Mechanics (2nd edition) and I'm trying to solve problem 4.24 b). In this problem you're supposed to first find the normalized eigenfunctions to ...
4
votes
1answer
37 views

Calculating the angular power spectrum of a section of sky

To calculate the angular power spectrum $C_l$ of the whole sky, one uses the variance of the coefficients of the spherical harmonics in the temperature fluctuation field. I.e. $$C_l = ...
0
votes
1answer
50 views

Harmonics on Hyperbolic space

I would like to know if there exists an analogue for hyperbolic space of the so called spherical harmonics which play a major role in the quantum states construction in a hydrogen atom. In other words ...
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0answers
98 views

3DAnisotropic oscillator in Spherical Harmonic basis-States with $L_z=0$

I've been trying to prove a rather simple looking concept. I have a code that calculates states of a 3D anisotropic oscillator in spherical coordinates. The spherical harmonics basis used to expand ...
3
votes
1answer
107 views

My basis set isn't orthonormal?

I'm implementing a little QM calculation just for fun and to make sure I understand how it works (calculating the helium ground state energy). My problem is that my basis set doesn't seem to be ...
0
votes
0answers
67 views

Infinite Round Square-Well zeroes of Spherical Bessel function

Consider a potential well where the potential is zero within a spherical well of radius a centered on the origin. Taking $a$ as our length scale and $\hslash^2/(2ma^2)$ as our energy scale, we gain ...
3
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1answer
67 views

State with non-zero angular momentum - cannot be described by spherical harmonic?

For a state with non-zero angular momentum, why is it that it cannot be described by the spherically symmetric spherical harmonic?
2
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0answers
133 views

Degeneracy, spherical harmonics

In a 3D oscillator, the energy levels are known to be $(n_x + n_y + n_z + \frac{3}{2})\hbar \omega = (n + \frac{3}{2})\hbar \omega$. Say for $n = 1$, any of the $n$'s can be $1$ and the rest are $0$. ...
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vote
0answers
79 views

Rewriting $\langle {\bf k} \vert E,l,m \rangle$ as $\langle {\bf k} \vert ~k,l,m \rangle$ Spherical Harmonics

From Sakurai eq. 6.4.21a we have that $$\langle {\bf k} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M k}}\delta\left(E-\frac{\hbar^2 k^2 }{2M}\right) Y_l^m({\bf\hat k}),$$ where $M$ is the mass of the ...
1
vote
1answer
75 views

Multiparticle generalization of $\langle \vec k \vert E,l,m \rangle$ spherical harmonics.

From Sakurai eq. 6.4.21a we have that $$\langle {\bf k} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M k}}\delta\left(E-\frac{\hbar^2 k^2 }{2M}\right) Y_l^m({\bf\hat k}),$$ where $M$ is the mass of the ...
2
votes
0answers
80 views

One more relation with spherical spinors

Let's have the spherical spinors: $$ \mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} \\ \sqrt{l \mp m ...
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0answers
40 views

How to find an action of $(\hat {\sigma} \cdot \hat {\mathbf L} )$ on spherical spinors?

Let's have the spherical spinors $\psi_{j, m, l = j \pm \frac{1}{2}}$, $$ Y_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} ...
3
votes
1answer
177 views

A dielectric sphere in an initially uniform electric field and representation theory of SO(3)

I learned recently that the highest order spherical harmonic required to represent the spatial distribution of decay products of a particle can be used to determine its spin, by using arguments ...
0
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0answers
95 views

Perturbation of a Hydrogen Atom in a Quadrupole Field

Question: A hydrogen atom is located in a quadrupole field, which gives it a perturbation $$H_1=A(x^2-y^2)$$ where $A$ is some constant. Calculate the ...
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0answers
33 views

Is there a generic term for orbital groups such as $e_g$ and $t_{2g}$?

I am looking for a generic term for sets of atomic orbitals (viz. spherical harmonics) which are grouped by crystal symmetry. The most familiar examples would be $e_g$ and $t_{2g}$ (in cubic ...
2
votes
1answer
327 views

Spherical harmonics

Given the following potential: $$V(\theta,\phi)=\frac{Q}{a}\left(\sin\theta \cos\phi+\frac{1}{2}\cos^2\theta\right)$$ on the surface of a sphere of radius $a$ I am trying to solve Laplace's Equation ...
3
votes
2answers
287 views

Why does $\ell=0$ correspond to spherically symmetric solutions for the spherical harmonics?

In quantum mechanics why do states with $\ell=0$ in the Hydrogen atom correspond to spherically symmetric spherical harmonics?
3
votes
2answers
208 views

Why 3 dipole terms in a multipole expansion?

As can be seen on this page http://en.wikipedia.org/wiki/Multipole_expansion when we take a multipole expansion without assuming azimuthal symmetry we end up with $2l+1$ coefficients for the $l^{th}$ ...
2
votes
2answers
260 views

Angular momentum of quantum system

Problem: A physical system is in the common eigenstate of $\hat{L^2}$ and $\hat{L_z}$. Calculate the following quantities: $\langle L_x\rangle,\langle L_y\rangle,\langle L_z\rangle,\langle L_x L_y + ...
1
vote
2answers
438 views

Wave function decomposition

Problem: Given the wave function $\Psi_0=A\sin^2(\theta)$ along with the Hamiltonian operator of a physical system: $H=\frac{L^2}{2I}+g B L_z$, find the eigenvalues and eigenfunctions of $\hat{H}$ ...
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0answers
140 views

Total angular momentum in multielectron atoms

I have some confusion about orbitals in multielectron atoms. Let's say we consider an atom (Lithium, for example, $1s^2\, 2p^1$) and that the state of the last electron is [n=2, l=1, ml=0, s=1/2, ...
4
votes
1answer
157 views

What is an anapolar moment?

I just read this: Anapolar Dark Matter I'm not sure i've heard the term 'anapolar' before, so i google and i found this: http://en.wikipedia.org/wiki/Toroidal_moment This confuses me, my ...
2
votes
1answer
116 views

One body harmonic oscillator states expressed in terms of creation operators

I am reading trough chapter one of Moshinsky's "The harmonic Oscillator in Modern Physics". However i am having some trouble with the mathematics in section 8 of chapter 1. I will sketch a summary of ...
3
votes
1answer
1k views

Hydrogen wave function in momentum space

We can seperate the wave function of an hydrogen atom in a radial and an angle part: $$ \phi_{n,l,m} (\mathbf{r}) = R_{n,l,m}(r) Y_{l,m}(\vartheta,\varphi) \, , $$ where $Y_{l,m}$ are the spherical ...
5
votes
2answers
587 views

Why must the angular part of the Schrodinger Equation be an eigenfunction of L^2?

I was reading about the solution to the Schrodinger Equation in spherical coordinates with a radially symmetric potential, $V(r)$, and the book split the wavefunction into two parts: an angular part ...
1
vote
1answer
475 views

How to use Legendre polynomials in order to determine the (an)isotropy of an on-lattice cluster aggregate?

I am currently testing various models of on-lattice (square lattice in two dimensions) cluster growth for anisotropy. I end up with a cluster, the boundary of which, in case of a truly isotropic ...
0
votes
1answer
358 views

Expansion in solid spherical harmonics on the lattice

I'm interested in calculating scattering processes (e.g. Coulomb scattering of an electron beam by a single ion) in the context of lattice quantum field theory, and wonder if there is something like ...
1
vote
1answer
108 views

Acoustic wave equation for a closed sphere

I am looking to model the nodal surfaces in a resonating closed sphere. The sound source is external. What sort of wave equation will reveal the spherical harmonics depending on the frequency, speed ...
3
votes
1answer
60 views

A nice overview (and maybe derivation) of the Poincaré transformations of the Vector Spherical Harmonics

With $Y_{lm}(\vartheta,\varphi)$ being the Spherical Harmonics and $z_l^{(j)}(r)$ being the Spherical Bessel functions ($j=1$), Neumann functions ($j=2$) or Hankel functions ($j=3,4$) defining ...