Questions tagged [special-functions]
57 questions
16
votes
4 answers
To what extent can one recover plane waves from the Airy eigenfunctions of a linear potential as the field is turned off?
Consider a single massive particle in one dimension under the action of a static linear potential, with the hamiltonian
$$
\hat H=\frac{\hat p^2}{2}+\hat{x}F_0.
$$
The eigenstate at energy $E$ is, with this normalization, given by
$$
\langle…
Emilio Pisanty
- 137,480
13
votes
2 answers
Why can the Euler beta function be interpreted as a scattering amplitude?
The Wikipedia article on the Veneziano Amplitude claims that the Euler beta function can be interpretted as a scattering amplitude. Why is this?
In another word, when the Euler beta function is interpreted as a scattering amplitude, what features…
Achmed
- 1,139
12
votes
1 answer
Why do we need the Condon-Shortley phase in spherical harmonics?
I'm confused with different definitions of spherical harmonics:
$$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}$$
For example here they claim, that one can decide whether to include…
ilciavo
- 243
10
votes
1 answer
What's the relation between the Euler $\psi$ function, the digamma function, and the hypergeometric function?
Can somebody help me out with the intermediate details of eqn. (2.5) in this paper?
Generalized gravitational entropy. Aitor Lewkowycz and Juan Maldacena. arXiv:1304.4926.
Is the Euler $\psi$ function appearing in the above equation the same as…
user29126
- 131
5
votes
1 answer
Why in QM the solution to Laguerre equations are ONLY Laguerre polynomials?
I am studying eigenfunction methods to solve Fokker-Planck equations and I got stuck with a calculation that is related to some typical problems in QM. In particular, the radial part of an hydrogen atom can be characterized through a Laguerre…
Javi
- 1,305
4
votes
0 answers
Derivation of the Bessel function representation of the Green function of the inhomogeneous Klein-Gordon equation
I will link the following question, as it is partly related to the problem I am trying to deal with.
Green's function for the inhomogenous Klein-Gordon equation
As you can read from this User´s question, they are trying to solve the Inhomogenous…
Lorenzo
- 41
4
votes
3 answers
Complex energy eigenstates of the harmonic oscillator
Given the Hamiltonian for the the harmonic oscillator (HO) as
$$
\hat H=\frac{\hat P^2}{2m}+\frac{m}{2}\omega^2\hat x^2\,,
$$
the Schroedinger equation can be reduced…
Wizzerad
- 437
3
votes
1 answer
Green Function expressed in terms of Hankel function (of the second kind)
I am reading Davies' and Birrell's book on QFT in Curved Spacetimes. In Chapter 2 and specifically in the subchapter 2.7, the authors derive an expression for the Feynman propagator
$$G_F(x,x')=\frac{-i\pi}{(4\pi…
schris38
- 4,409
3
votes
1 answer
Shift operator (integral calculus involving Hermite polynomials)
I didn't know whether to pose this question on Physics.stackexchange or Math.stackexchange. But since this is the last step of a development involving the eigenfunctions of an Harmonic oscillator and a shift operator matrix, I thought it'd be better…
mwoua
- 243
3
votes
1 answer
Calculation of spherical Bessel functions - meaning of $\left(\frac{1}{x}\frac{d}{dx}\right)^{l}$
I'm trying to understand the calculation of spherical Bessel functions in chapter four of Griffiths' Introduction to Quantum Mechanics (2nd ed, p142). He gives…
Peter4075
- 3,109
3
votes
1 answer
References regarding Green's function on a square domain in 2D
Premise: I know this question would be better suited to MathSE, but since I endeavour to solve a free CQFT on a bounded domain, I'm confident I'll find a more exhaustive answer here.
I'm trying to obtain the Green's function on a planar square…
3
votes
1 answer
On fusion transformation in Liouville CFT
It is known in Liouville CFT from the crossing symmetry that the four points $s$-channel and 4t$-channel conformal blocks are related to each other via an integral transformation
$$\mathcal{F}\left[\begin{matrix}\theta_1,\theta_{t}\\…
Gropillon
- 67
3
votes
1 answer
Solving Special Function Equations Using Lie Symmetries
The lie group + representation theory approach to special functions & how they solve the ode's arising in physics is absolutely amazing. I've given an example of it's power below on Bessel's equation.
Kaufman's article describes algebraic methods…
bolbteppa
- 4,168
2
votes
0 answers
Expansion of interaction potential in terms of spherical harmonics in unconventional superconductivity
I am currently reading the book Introduction to Unconventional Superconductivity by V. P. Mineev and K. V. Samokhin. In Equation (1.6), the interaction potential $V$ is expanded in terms of spherical harmonics as…
caz
- 21
2
votes
0 answers
What is the relation between Chebyshev polynomials and coupled oscillators?
I have been told that Chebyshev polynomials are key for finding the normal modes of oscillations of a linear chain of coupled oscillators, since they are the eigenmodes of the system. However, I cannot see why is the case nor can I find any textbook…
Fernando
- 106