16
votes
Accepted
Why do we need the Condon-Shortley phase in spherical harmonics?
You don't need it: it's a sign convention and the only thing you need to do with it is to be consistent. (In particular, this means always checking that the sign and normalization conventions for $Y_{...
- 135k
10
votes
Accepted
Why in QM the solution to Laguerre equations are ONLY Laguerre polynomials?
This is related to the fact that the solutions in the textbook are written only for the discrete spectrum of energies. We are interested in the normalized solutions, i.e. such that
\begin{equation*}
\...
- 8,739
5
votes
Accepted
Green Function expressed in terms of Hankel function (of the second kind)
There is a useful class of dummy integration variable trick, which applies also later when calculating Feynman diagrams. You can make the following observation:
$$
\frac{1}{x} = \int_0^{+\infty}dt e^{-...
- 15.1k
3
votes
Accepted
Spherical Bessel Equation has different forms?
To arrive at this, I begin manipulating the Radial equation:
$$\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)-\frac{2mr^2}{\hbar^2}(V(r)-E)R=l(l+1)R$$
You forgot some factors of $r$ while manipulating ...
- 180
3
votes
What does it mean that Zernike's polynomials form a orthogonal basis on the unit disk?
It means that any function (sensible) $f(\rho, \phi)$ defined on the unit circle can be expanded into a linear combination of Zernike polynomials:
$$ f(\rho, \phi) = \sum_{m,n}{a_{m,n}R_n^m(\rho)\cos{...
- 39.6k
3
votes
Addition theorem for Spherical Bessel function
The identity you seek is
$$j_0(|\vec{p}-\vec{p}\,'|r)=\sum_{n=0}^{\infty}(2n+1)j_n(pr)j_n(p'r)P_n(\cos\theta).$$
It follows from (10.60.2) here.
- 52.2k
3
votes
Accepted
Spherical Harmonics Sum Identity
In the process of proving the sum rule
\begin{equation}
\sum_{m=-l}^l |Y_l^m(\theta,\phi)|^2 = \frac{2l+1}{4\pi} \, ,
\end{equation}
you often start from the more general addition theorem for the ...
- 1,621
3
votes
Accepted
References regarding Green's function on a square domain in 2D
Thanks to @Nephente advice, I'm now able to answer my question.
Finding the Green's function for the Laplacian in a 2D square can be considered as a particular case of the more general problem of ...
3
votes
What physically determines Bessel functions' orders?
The "$n$" in the $J_n(\kappa r)$ refers to the number of nodes in the angular direction. A complete set of eigenfunctions of $-\nabla^2$ in ${\mathbb R}^2$ are
$$
\psi_{n,\kappa}(r,\theta)= e^{in\...
- 56.6k
2
votes
Is there a generating function for Hermite polynomials of 2n?
Of course. Parity.
Recall
$$
H_{2n+1}(x) + H_{2n+1}(-x)=0, \qquad H_{2n}(x) + H_{2n}(-x)= 2 H_{2n}(x),
$$
so that
$$
e^{2xt -t^2} + e^{-2xt -t^2} = \sum_{n=0}^{\infty} (H_{n}(x)+H_n(-x) ) \frac{...
- 66.3k
2
votes
Accepted
Why are some associated Legendre functions not orthogonal to each other?
Because the orthogonality is supplied by the azimuthal direction when the two $z$-components of the angular momentum differ. This is why we usually bundle them together into the spherical harmonics ...
- 22.8k
2
votes
Accepted
How fast should you rotate a chain hoop so that it doesn't tilt?
Segments of chain in the loop are subject to tension forces from each side. The resultant of these must have a radial component which provides the centripetal acceleration, and a vertical component ...
- 12.2k
2
votes
Accepted
Deriving recurrence of the Hermite polynomials
$$ g(0,t)= e^{-t^2}= \sum_m {(-t^2)^m \over m!}= \sum_n H_n(0) {t^n\over n!} \leadsto \\
H_n(0) = \begin{cases}
0 & \text{for odd }n, \\
(-2)^\frac{n}{2} (n-1)!! & \text{for even }n.
\...
- 66.3k
2
votes
Manipulating Hypergeometric functions
I think you want the connection formulae that link solutions about the three regular singular points to each other. There is a brief discussion starting on page 408 of my lecture notes:
http://...
- 56.6k
2
votes
Commutator of raising operator in angular momentum with partial derivative wrt z
You are aiming to show
$$
[p_z,L_x+iL_y]= p_x+ip_y.
$$
Have set $\hbar=1$ for simplicity.
This near duplicate answer works out
$$
[p_i, L_j] =
i \epsilon_{ijm}p_m,
$$
yielding your answer.
As @...
- 66.3k
1
vote
What is the form of general expression (one expression) for the eigenfunctions of discrete and continuous spectra of motion in the Coulomb potential?
There is no unified expression, because we simply have two different ordinary differential equations, each with its own set of solutions, according to whether a certain parameter (eventually found to ...
- 4,421
1
vote
What does a star on a spherical harmonic mean?
It has nothing to do with spherical harmonics in particular, an asterisk is one of the common notations for the complex conjugate.
- 129k
1
vote
Accepted
Orthonormalization condition for $L^2$ operator
The differential solid angle (in spherical polar coordinate ) is $$d\Omega=\sin\theta \ d\theta \ d\phi$$
$$\int_0^{2\pi}\int_0^\pi d\Omega \ Y^*_{l'm'}(\theta,\phi) \ Y_{lm}(\theta,\phi)=\int_0^{2\...
- 12.1k
1
vote
What is the general solution of the Associated Legendre differential equation when $A$ does not equal $\ell(\ell+1)$?
What is the general solution of the Associated Legendre differential equation when $A$ does not equal $\ell(\ell+1)$?
When you allow $\ell$ and $m$ to be arbitrary complex numbers $\lambda$ and $\mu$,...
- 52.2k
1
vote
To what extent can one recover plane waves from the Airy eigenfunctions of a linear potential as the field is turned off?
Let me add a few aspects being somewhat complementary to the already existing answers.
The crucial point of your interesting problem is the fact that the spectrum $\sigma(H_{F_0})$ of the operator $$...
- 7,298
1
vote
Integral of Hermite functions
I found in the work of Groenwold, 1946, a closed form for the solution of
\begin{equation}
k_{mn}(\zeta_1,\zeta_2)=\frac{1}{\sqrt{\pi 2^{m+n} m!n!}}\int_{\mathbb R} \mathrm d \zeta \; e^{-\zeta^2} H_{...
- 385
1
vote
Accepted
Integral of Hermite functions
By experimenting in Mathematica I found that
$$\int_{-\infty}^\infty d\zeta e^{-\zeta^2}H_n(\zeta+\zeta_1)H_{n-1}(\zeta+\zeta_2)=
\sqrt{\pi}2^n(n-1)!\zeta_1L_{n-1}^1(-2\zeta_1\zeta_2)$$
holds for $n=...
- 52.2k
1
vote
Is there a generating function for Hermite polynomials of 2n?
....A far more convenient formulation is actually
available, which expresses the $\,H_n\,$ in terms of a generating function $\,S\left(\xi,s\right)$.
\begin{equation}
S\left(\xi,s\right)\...
- 16.4k
1
vote
Different levels of physical model solvability and why reality doesn't care
If reality follows the same rules, why is it so "easy" for her to move the planets in the solar system according to GR and produce atoms with many electrons, hyperfine structure and whatnot, while we ...
- 13.4k
1
vote
Expanding the Green's function in spherical harmonics
It is both the symmetry and the reality of the Green's function that implies this. For example, if I know $A\left(\theta\right)Y\left(\theta^{\prime}\right)$ is both symmetric and real, then \begin{...
- 263
1
vote
Accepted
On fusion transformation in Liouville CFT
First of all let me point out that Ponsot-Teschner has been superseded by Teschner-Vartanov https://arxiv.org/abs/1202.4698 , whose formulas are more symmetric (although not less complicated).
Then ...
- 2,792
1
vote
Resources for special functions
I absolutely recommend the text by James Seaborn Hypergeometric functions and their applications. This text is completely approachable and contains all the special functions required at the ...
1
vote
Accepted
Manipulating Hypergeometric functions
With a substitution $x\equiv r/R$ one can transform the ODE into a hypergeometric equation:
$$x(1-x)\phi''(x)-\phi'(x)+l(l+1)\phi(x)=0.$$
This corresponds to $(a,b,c)=(-1-l,l,-1)$ or equivalently $(l,-...
- 3,071
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