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50 views

Imaginary argument in bessel function for a wavefunction

I am solving for the continuum model of haldane model with one of the site being a potential well. The Dirac equation for a topologically non trivial case gives a solution for the states in the band ...
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0answers
22 views

Equivalence between two different ways of calculating phase shifts in quantum scattering

Quantum scattering problem from a potential of the form $ V={{a} \over {r^2}}, $ regarding the calculation of phase shifts. One standard way to solve the problem is to equate in the limit where r ...
2
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1answer
86 views

How are spherical harmonics useful outside class? [closed]

I've learned about spherical harmonics (Legendre polynomials $\longrightarrow$ Associated Legendre polynomials $\longrightarrow$ orthogonality relations $\longrightarrow$ normalization coefficient(s) ...
7
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1answer
89 views

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, ...
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1answer
133 views

Book on gamma functions with applications in physics

I have heard that in my next semester, our quantum mechanics teacher will be giving a great emphasis on difficult integrals with the most of them having to do with gamma functions. Does anybody know ...
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0answers
53 views

Energy Oscillations in a One Dimensional Crystal?

Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially interested in ...
2
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0answers
142 views

Why does regularization work in this Bessel function integral?

I encountered some days before an integral representation for a modified Bessel function and should differentiate it. But in this representation : $$K(\omega,a)=\int_0^{\infty} \frac{ds}{s} ...
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1answer
105 views

I want to solve Mathieu Equation $y''(x)+(a−2q \cos(2x))y(x)=0$. How to solve it using Floquet solution?

I want to solve Mathieu Equation $$y''(x)+(a−2q \cos(2x)) \, y(x)=0.$$ How to solve it using Floquet solution? In Floquet solution for integer order of $v$ and $π$ periodicity We have Solution ...
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1answer
474 views

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 ...
7
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1answer
208 views

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 $\psi$ ...
12
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2answers
949 views

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 ...
2
votes
3answers
641 views

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 to: $$ \left[ ...
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1answer
449 views

Shift operator (integral calculus involving Hermite polynomials) [closed]

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 ...
2
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1answer
196 views

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 ...