Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,054 questions , 2,207 unanswered
5,347 answers , 22,726 comments
1,470 users with positive rep
818 active unimported users
More ...

  Special question about particle-particle oscillations

+ 2 like - 0 dislike
547 views

Suppose we have two scalar fields $\varphi, \kappa$. Next, suppose there is a region in space where they are mix with each other, i.e., we have a lagrangian
$$
\tag 1
L_{\text{int}} = A \varphi \kappa
$$
By taking into account their kinetic term, we have following EOMS:
$$
\left(\omega^{2} + \partial_{\mathbf{r}}^2 - \begin{pmatrix}0 & A \\ A & 0\end{pmatrix}\right)\begin{pmatrix}\varphi\\ \kappa\end{pmatrix} = 0
$$
It gives rise to particle oscillations. 

Next, suppose we have a beam of $\varphi$ particles propagating along $z$ axis. After entering the domain (say, at $z=0$) in which there is the interaction $(1)$ it begins to oscillate into $\kappa$ particle. I want to calculate the probability of oscillation at $z>0$. It turns out that it is proportional to
$$
P_{\varphi\to\kappa}\sim |e^{-ik_{+}z}-e^{-ik_{-}z}|, \quad k_{\pm} = \sqrt{\omega^2 \mp A} 
$$ 
It turns out that for $|A|>\omega$ one of the momenta $k_{+}$, $k_{-}$ becomes imaginary, and the probability doesn't behave as oscillating function, but instead is exponentially amplified or damped.

What is the physical reason for this?

asked May 26, 2017 in Theoretical Physics by NAME_XXX (1,060 points) [ revision history ]
edited May 26, 2017 by NAME_XXX

1 Answer

+ 1 like - 0 dislike

I haven't done any calculations and I do not know if your equations reflect some physics, but I reasoned in the following way: both fields are "twins" - the same frequency, the same interaction. We may consider them as components of a non relativistic spin 1/2 field, $\omega^2$ being energy $E$. Then the interaction constant $A$ may be considered roughly as a "potential barrier". When the component $\varphi$ collides with it, the component $\kappa$ may appear due to interaction with the barrier. The difference $\omega^2 - A$ may cause propagating waves inside the barrier (real valued wave vector $\bf{k}$), or reflecting waves (evanescent ones inside the barrier)). (I guess the reflecting waves exist in any case, you must admit them.) The physical solutions is chosen by the physically dictated setup - no amplified wave.

answered Feb 2, 2023 by Vladimir Kalitvianski (102 points) [ revision history ]
edited Feb 3, 2023 by Vladimir Kalitvianski

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOver$\varnothing$low
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...