All Questions
Tagged with scattering hilbert-space
62
questions
2
votes
0
answers
60
views
Asymptotic states and physical states in QFT scattering theory
Context
In the scattering theory of QFT, one may impose the asymptotic conditions on the field:
\begin{align}
\lim_{t\to\pm\infty} \langle \alpha | \hat{\phi}(t,\mathbf{x}) | \beta \rangle = \sqrt{Z} \...
- 108
2
votes
0
answers
91
views
Is Sakurai's derivation of the Lippmann-Schwinger equation correct?
I am using Sakurai's Modern Quantum Mechanics 3rd ed. The following is from the beginning of chapter 6.
The defining equation for the $T$-matrix is
$$\langle \vec{k}' \lvert U_I(t, t_0) \lvert \vec{k} ...
- 2,676
2
votes
0
answers
56
views
Does Sakurai's definition of $S$-matrix assume a particular type of scattering?
I am using Sakurai's Modern Quantum Mechanics 3ed.
In chapter 6, Sakurai defines the $T$-matrix via the equation
$$\langle \vec{k}' \lvert U_I(t, t_0) \lvert \vec{k} \rangle = \delta_{k'k} - \frac{i}{\...
- 2,676
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
1
vote
1
answer
75
views
Difference between stationary states, collision states, scattering states, and bound states
A few weeks ago, I was presented one-dimensional systems in my QM class, and of course one-dimensional potentials too. Nonetheless, I'm still a bit unclear about the terminology my professor uses. ...
- 1,616
3
votes
0
answers
64
views
Deriving a contradiction from the LSZ condition
I'm reading the LSZ reduction formula in the wikipedia:
https://en.wikipedia.org/wiki/LSZ_reduction_formula
To make the argument simple, let $$\mathcal{L}=\frac{1}{2}(\partial \varphi)^2 - \frac{1}{2}...
- 61
0
votes
0
answers
51
views
Question about In-States in S.Weinberg Lectures On Quantum Mechanics
At the beginning of chapter 7 Potential Scattering of Lectures on Quantum Mechanics by S. Weinberg, he has introduced the in states (in Heisenberg picture) $\Psi^{\text{in}}_{\pmb{k}}$, which ...
- 529
1
vote
0
answers
76
views
Closed form expression for scattering states in Hydrogen atom
The self-adjointness of the Hamiltonian of the Hydrogen atom follows from the Kato-Rellich theorem, as is mentioned e.g. in Brian Hall's book. The eigenstates of $H$ corresponding to $E<0$, i.e. ...
- 689
5
votes
1
answer
244
views
Details in the derivation of the Lippmann-Schwinger equation
So the argument goes that for a slightly perturbed Hamiltonian
$$
H = H_0 + V,
$$
there will be some exactly known states, $\left|\phi\right>$, solving
$$
H_0\left|\phi\right> = E\left|\phi\...
- 525
3
votes
1
answer
445
views
Asymptotic states in the Heisenberg and Schrödinger pictures
One can show that, in the interacting theory, the operators that create single-particle energy-momentum eigenstates from the vacuum are
\begin{align}
(a_p^{\pm\infty})^\dagger=\lim_{t\to\pm\infty}(...
- 1,060
4
votes
1
answer
184
views
Why is the $S$-matrix calculated using the free vacuum state and not the full interacting vacuum state?
Let $H = H_0 + H_I$ be a Hamiltonian that is the sum of a free Hamiltonian and an interacting Hamiltonian. Denote the free vacuum state by $| 0 \rangle$ and the full vacuums state by $|\Omega \rangle$....
- 3,350
0
votes
0
answers
73
views
How to use Born approximation with degeneracy?
According to the scattering theory, the scattering state wave function $|\psi_l\rangle$ follows the Lippmann-Schwinger equation. If we take the first order Born approximation,then
$$
|\psi_l\rangle=|l\...
- 1
4
votes
2
answers
338
views
When do we exclude non-normalizable solutions and when not?
I'm a bit confused on when we should keep any non-normalizable solutions and when not. What do I mean?
Let's say that we have the free particle system. The energy eigenstates are not normalizable - ...
- 369
2
votes
0
answers
193
views
Vacuum loop in $\phi^4$ scattering
$\newcommand{\Ket}[1]{\left|#1\right>}$
$\newcommand{\Bra}[1]{\left<#1 \right| }$
I am studying QFT through Peskin & Schroeder's book and currently I am doing the part of scattering in $\phi^...
- 21
0
votes
1
answer
87
views
Question about asymptotic assumption in LSZ reduction formula derivation
I have a silly question in derivation of LSZ reduction formular, I can go directly with the derivation until I found a assumption that I can't convince myself.
In the book Quantum Field Theory and the ...
