All Questions
Tagged with mathematical-physics scattering
23
questions
7
votes
1
answer
364
views
Determining Bound States from Møller Operator
Hello I came across an interesting property of the Møller operator, which I summarize below:
The Møller operator $\Omega^{(+)}$ maps in-states that belong to the continuum spectrum of the free ...
- 199
2
votes
0
answers
154
views
Mathematical explanation of the extinction paradox
I am trying to learn properly about scattering. For this I was pointed to Wave Propagation and Scattering in Random Media by Ishimaru.
I got a bit stuck in section 2-2 General properties of the Cross ...
- 91
1
vote
0
answers
76
views
Closed form expression for scattering states in Hydrogen atom
The self-adjointness of the Hamiltonian of the Hydrogen atom follows from the Kato-Rellich theorem, as is mentioned e.g. in Brian Hall's book. The eigenstates of $H$ corresponding to $E<0$, i.e. ...
- 689
4
votes
1
answer
441
views
Why are scattering states orthogonal in general?
A similar question was asked here. Unfortunately, I don't think the question was well-formulated/explained enough for people to actually care about the answer, so let me provide some reason to why I'm ...
- 2,123
26
votes
1
answer
531
views
Is the converse of Weinberg's statement on the cluster decomposition principle true?
In Weinberg's "The Quantum Theory of Fields, Vol. 1", Section 4.4, page 182, the author says:
We now ask, what sort of Hamiltonian will yield an $S$-matrix that satisfies the cluster ...
- 36.4k
0
votes
0
answers
111
views
How to properly understand the residue in the LSZ theorem?
The LSZ theorem for a scalar field reads
$$
\mathcal M=\lim_{p^2\to m^2}\left[\prod_{i=1}^n(p^2-m^2)\right]\tilde G(p_1,\dots,p_n)
$$
where $G$ is the $n$-point function, to wit,
$$
G(x_1,\dots,x_n)=\...
0
votes
1
answer
80
views
Simple substitution of partial waves [closed]
(...)
$$
\psi({\bf r}) \approx \sqrt{n}\sum_l C_l \left[\frac{e^{i(kr - l\pi/2 + \delta_l)} - e^{-i(kr - l\pi/2 + \delta_l)}}{2ikr}\right]P_l(\cos\theta) \tag{1302}
$$
which contains both incoming ...
- 39
1
vote
0
answers
414
views
Lippmann-Schwinger equation and time dependence
Consider the Lippmann-Schwinger equation (LSE)
$$ |\psi\rangle = |\phi\rangle + \hat{G}_0(\epsilon) \hat{V} |\psi\rangle \tag{1}$$
where $\hat{G}_0(\epsilon) = \frac{1}{\epsilon - \hat{H}_0 + i\eta}$...
- 1,001
0
votes
1
answer
378
views
Lippmann-Schwinger equation and $T$ expansion
Lippmann-Schwinger equation, in operator form, is: $$
T=V+V\frac{1} {E-H_0+i \hbar \varepsilon} T=:V+V\Theta_0T,
$$
where $H_{tot}=H_0+{V}$ is the hamiltonian ($H_0$ is the free particle hamiltonian ...
user78618
5
votes
2
answers
746
views
Dirac delta function definition in scattering theory
I'm studying scattering theory from Sakurai's book.
In the first pages he gets to the following expression:
$$\langle n|U_I(t, t_0)|i\rangle=\delta_{ni}-\frac{i}{\hbar}\langle n|V|i\rangle\int_{t_0}^...
user78618
6
votes
1
answer
1k
views
Complex scaling method for solving resonance states
I am now reading about the complex scaling method for solving resonance states. As far as I understand, the procedure goes like this:
Let us take the 1d potential $V(x) = A e^{-x^2} x^2 $ as an ...
- 1,179
6
votes
0
answers
182
views
What is the physical interpretation of the automorphism on bounded operators induced by an S matrix?
In a QFT, the S-matrix $S$ is a unitary operator, that fixes the vacuum and commutes with the unitary operators implementing the action of the Poincare group on an appropriate Hilbert space $H$.
...
- 1,151
0
votes
2
answers
134
views
A conjecture about the Møller operator
Consider the Møller operator
$$ \Omega_+ = \lim_{t \rightarrow -\infty } e^{i H t } e^{- i H_0 t } , $$
Now, suppose a state $\psi $ is located far away from the potential $V = H- H_0$. I feel that $...
- 4,122
1
vote
0
answers
149
views
Deriving general boundary conditions from first principles for elastodynamic scattering
It seems that most of the relevant books only give the linear case and the rest say something along the lines of "here are common examples of boundary conditions."
What are the most general boundary ...
- 41
3
votes
0
answers
997
views
Derivation of the Lippmann-Schwinger equation
I was trying to understand the derivation of the Lippmann-Schwinger equation in Sakurai's Modern Quantum Mechanics, Section 6.1. Our teacher presented a much simpler derivation, similar to that on ...
- 131