What is Bound states: Definition and 76 Discussions
In quantum physics, a bound state is a quantum state of a particle subject to a potential such that the particle has a tendency to remain localized in one or more regions of space. The potential may be external or it may be the result of the presence of another particle; in the latter case, one can equivalently define a bound state as a state representing two or more particles whose interaction energy exceeds the total energy of each separate particle. One consequence is that, given a potential vanishing at infinity, negative-energy states must be bound. In general, the energy spectrum of the set of bound states is discrete, unlike free particles, which have a continuous spectrum.
Although not bound states in the strict sense, metastable states with a net positive interaction energy, but long decay time, are often considered unstable bound states as well and are called "quasi-bound states". Examples include certain radionuclides and electrets.In relativistic quantum field theory, a stable bound state of n particles with masses
{
m
k
}
k
=
1
n
{\displaystyle \{m_{k}\}_{k=1}^{n}}
corresponds to a pole in the S-matrix with a center-of-mass energy less than
∑
k
m
k
{\displaystyle \textstyle \sum _{k}m_{k}}
. An unstable bound state shows up as a pole with a complex center-of-mass energy.
The problem of bound states of an electron trapped in a dipole field is being studied by Alhaidari and company. (See, for example, https://arxiv.org/ftp/arxiv/papers/0707/0707.3510.pdf). It is not clear to me why the point dipole approximation is used everywhere in such calculations. Can't an...
I can't find anywhere information on how people treat bound/unbound condition for atoms with Lennard-Jones simulation. Say if I have 3 oxygen atoms flying around and two of them at some point become an O2 molecule, this means their electron shells are now fully occupied - so I am guessing the...
I am confused here. For ##x>0## particle is free and for ##x<0## particle is free. That I am not sure how we can have bond states. If particle is in the area ##x>0## why it feel ##\delta## - potential at ##x=0##. Besides that, I know how to solve problem. But I am confused about this.
If we...
I have a question from the youtube lecture
That part starts after 42 minutes and 47 seconds.
Balakrishnan said that if delta barriers are very distant (largely separated) then we have degeneracy. I do not understand how this is possible when in 1d problems there is no degeneracy for bond states.
Weinberg writes in his book on QFT Vol1 that bound states in QED are problematic because perturbation theory breaks down. consider the case of hydrogen atom, electron+proton. Weinberg explains this case and I copy from the book:
https://www.physicsforums.com/attachments/247655
what is time...
In these notes, https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes11.pdf, in the middle of page 5, it is mentioned:
We will be interested in bound states namely, energy eigenstates that are normalizable. For this the energy E of the states...
Let's suppose I have a finite potential well: $$
V(x)=
\begin{cases}
\infty,\quad x<0\\
0,\quad 0<x<a\\
V_o,\quad x>a.
\end{cases}
$$
I solved the time-independent Schrodinger equation for each region and after applying the continuity conditions of ##\Psi## and its derivative I ended up with...
I read this wiki and some of the references
https://en.wikipedia.org/wiki/Bound_state
But I can't really understand. For example the electron in hydrogen has specific energies and not general relations that the articles seem to claim.
Thanks
Before I begin, I would like to say what I am about to ask would require some sort of top-top-bottom bound state for it to function. Which (to my knowledge) has not been experimentally or theoretically predicted. Also, in case if you are wondering- no, this is not a homework question.
---
So...
Hi all - forgive me, I'd asked a series of questions in a previous post that was deemed to be circular, but I still didn't obtain a satisfactory answer to the question I was asking. In this post, I'm going to try to be very careful to use terms that are at least less 'misplaced', per se...
The question is basically find the boundary conditions when ##l=0##, for energies minor than 0.
Homework Equations
$$V(r)=\begin{cases}
& 0\text{ $r<a_0$}\\
&V_0\text{ $a_0<r<a_1$}\\
& 0\text{ $r>a_1$}\\
\end{cases}
$$
$$...
Hello. Can you shed a different light on why mass defects exist please, so that I might finally grasp it intuitively?!
I've had a look at these nuclear threads and one about GPE,
https://www.physicsforums.com/threads/why-is-there-a-mass-defect-in-the-nucleous.374443/...
Hi,
I'm trying to understand the bound states of a periodic potential well in one dimension, as the title suggests. Suppose I have the following potential, V(x) = -A*(cos(w*x)-1). I'm trying to figure out what sort of bound energy eigenstates you'd expect for a potential like this. Specifically...
If there are 3 positive charges of +1, +3, +5 coulombs equidistant from a negative charge of 1 coulomb what positive charge will this negative charge be attracted to ?
Is the result different if the charges exist in a “bound” state (resulting in electrovalent compounds) where a positive charge...
As the title says, why does the set of hydrogen bound states form an orthonormal basis? This is clearly not true in general since some potentials (such as the finite square well and reversed gaussian) only admit a finite number of bound states.
Hi, I am about to work on the problem of trying to find a renormalization program for bound states in QFT. Any suggestions/advice on where to start would be much appreciated.
Hello everybody.
I'm interested in some problems of bound states in external fields in QFT (especially QED).
I wonder are there any lectures/books or reviews which provide modern treatment of this subject?
I would like to learn more about general formalism and applications in QED (I allready...
It came to me just now that because we can always take the Fourier transform of a well-behaved function, this means we can think of any such state as a superposition of free-particle momentum eigenstates. E.g., the Hermite polynomial eigenfunctions of the harmonic oscillator. They have a...
I have looked around for help with this, including on existing threads, but I can't quite find what I'm looking for.
I know that in the nuclear shell model we fill the shells in the same way as with electrons, so 2 protons in the first and 6 in the second etc, with the same being true for...
Homework Statement
I'm curious in proving that expectation value of momentum for any bound state is zero. So the problem is how to prove this.Homework Equations
$$ \langle \mathbf{p_n} \rangle \propto \int \psi^*(\mathbf{r_1}, \dots ,\mathbf{r_N}) \nabla_n \psi(\mathbf{r_1}, \dots...
1)So from my understanding, as long as ##E>0## you will have scattering states and these scattering states will always result in an imaginary ##\psi##, but bound states can also have an imaginary ##\psi##? Is this correct and or is there a better way of looking at this maybe more conceptually...
I'm working on a problem out of Griffith's Intro to QM 2nd Ed. and it's asking to find the bound states for for the potential ##V(x)=-\alpha[\delta(x+a)+\delta(x-a)]## This is what I'm doing so far:
$$
\mbox{for $x\lt-a$:}\hspace{1cm}\psi=Ae^{\kappa a}\\
\mbox{for $-a\lt x\lt...
Homework Statement
Hi! My issue here is that I need to find the bound states (if any) of the potential:
U(r)=-C\frac{s_1\cdot \hat{r}\, s_2\cdot \hat{r}-s_1\cdot s_2}{r}.
Here s_1 and s_2 are the spins of the two spin-one particles involved in this interaction. The two particles have...
Hi all,
So I was recently set straight on the fact that bound state does *not* necessarily mean E<0 but rather is the statement that E<V(+/- infinity). So how do we apply this definition to the infinite square well where the potential at +/- infinity vanishes, and yet the bound states have...
Hello everyone. I was yesterday asked in an interview to draw a gaussian curve. I drew. And then they asked in what region would this give rise to bound states? I am really confused how to conclude if a function gives bound state or not. Please help.
Thanks.
Homework Statement
Given the potential
V(x) = - 1/ sqrt(1+x^2)
Consider this in a 50x50 matrix representation of the hamiltonian in the basis of a one dimensional harmonic oscillator. Determine the eigenvalues and eigenvecotrs, the optimal parameter for the basis, and cop ate the...
Homework Statement
Given the delta function -α[δ(x+a) + δ(x-a)] where α and a are real positive constants.
How many bound states does it possess? Find allowed energies for \frac{hbar2}{ma} and \frac{hbar2}{4ma} and sketch the wave functions.
Homework Equations
I know there are three parts of...
Homework Statement
Consider a potential function V(x) such that:
$$
\begin{cases}
V(x)\leq 0\text{ for }x\in[-x_0,x_0] \\
V(x)=0 \text{ for }x\not\in[-x_0,x_0]
\end{cases}
$$
Show, using the variational method that:
(a) In the 1-dimensional case \lambda^2V(x) always possesses at...
In the double delta function potential well, where one delta function ( -αδ(x) ) is at -a and one at +a, if the energy is less than zero, there can be either one or two bound states, depending on the magnitude of α...if α is large enough, there can be two bound states, but if α is small, there...
With great interest I read an article about a paper where scientists were able to create two photon bound states ("molecules of light").
http://physicsworld.com/cws/article/news/2013/sep/26/physicists-create-molecules-of-light
I was quite astonished since light normally does not...
The result of the Kallen-Lehmann spectral representation is that the two point correlation (and thus also the full propagator) has a pole in the physical mass of the particle. In Peskin and Schroeder it is also argued that multiparticle states show up as a cut, but bound states can also show up...
Hello I understand how to approach finite potential well. However i am disturbed by equation which describes number of states ##N## for a finite potential well (##d## is a width of a well and ##W_p## is potential):
$$
N \approx \dfrac{\sqrt{2m W_p}d}{\hbar \pi}
$$
I am sure it has something to...
Hi all,
I've been reading section 5.3 of Peskin and Schroeder, in which the authors discuss the production of a bound state of a muon-antimuon pair close to threshold in electron-positron collisions.
Here \xi,\xi' are the Weyl spinors used to construct the Dirac spinors for the muon and...
Bound vs "not"bound states
Homework Statement
Hi, I do not understand how two bound state wavefunctions differ from not bound state
wavefunctions.
To be more precise I m thinking about the graphical representation.
[b]ons[/b2. Relevant equati
The Attempt at a Solution
I speculate that bound...
Hello,
I am trying to understand the idea of using analytic continuation to find bound states in a scattering problem. What do the poles of the reflection coefficent have to do with bound states? In a problem that my quantum professor did in class (from a previous final), we looked at the 1D...
Why must it be true that a system that has a bound state must have its scattering amplitude have a pole in the upper half of the complex wave-number plane?
For example, if the scattering amplitude as a function of the initial wave number magnitude |k| is:
A=\frac{1}{|k|-iB}
with B>0, then...
Just an idea: is there an index theorem for an n-dimensional Hamiltonian
H = -\triangle^{(n)} + V(x)
which "counts" the bound states
(H - E) \,u_E(x) = 0
i.e. eigenfunctions and eigenvalues in the discrete spectrum of H?
So far, we've discovered this compositeness hierarchy:
Atoms - bound states of electrons, nuclei, photons
Nuclei - bound states of nucleons and other hadrons
Hadrons - bound states of quarks and gluons
So are any Standard-Model particles bound states of any other particles?
The...
I've thought about dark matter and I'm wondering if it could possible be made up invisible bouond states of ordinary matter? Wikipedia says "According to consensus among cosmologists, dark matter is composed primarily of a new, not yet characterized, type of subatomic particle." But why a...
Homework Statement
I'm dealing with a dirac particle in an attractive spherical square well. I've solved for the transcendental equation to find energy, found the normalized wave function, and now I'm trying to explain what happens when the well becomes very deep, when V0 ≥ 2mc2. If I plug...
Hi.
In the book I'm reading I've come to a question regarding degenerate states in one dimension. It says that in one dimension there are no degenerate bound states.
But say I have a stationary state with some energy E, and assume that it is normalizable. You can easily show that the complex...
Transition from bound states to "continuous" states
If I have the Hamiltonian for the Hydrogen atom and a perturbation given by a classical electric field (the kind of problems you get in an ordinary course about QM, no QFT involved), can I have a transition from a bound state (I intend a...
Effective theory of bound states from QCD??
Do you know any work that actually succeeds in producing the action of an effective
field theory for nucleons and mesons, starting from the QCD action?
Hi,
I asked this question in the quantum physics forum https://www.physicsforums.com/showthread.php?t=406171 but (afaics) we could not figure out a proof. Let me start with a description of the problem in quantum mechanical terms and then try to translate it into a more rigorous mathematical...
Hi,
I discussed this with some friends but we could not figure out a proof.
Usually when considering bound states of the Schrödinger equation of a given potential V(x) one assumes that the wave function converges to zero for large x.
One could argue that this is due to the requirement...
i have two questions that i am struggling with and i have tried all i can think of with them and i am still not getting the answers correct.
1)Estimate, using the Uncertainty Principle, the kinetic energy of an electron if it were bound in the nucleus.
Answer: ∼ 200 MeV for R ∼ 1 fm...
Hey Guys,
Am working through Relativistic Quantum Mechanics: Wave equations by W.Greiner and have a simple question about the Klein-Gordon equation: is it fair to say that bound states only occur between -m<=E<=m? (c=1). There are a few problems where they show that you can get pair...
How does one solve bound state problems in QFT(like an electron positron atom)? How does one identify the space of states. The Fock space seems to lose it definition when a bound state problem is discussed. There is also no meaning to wave functions or potentials that are used in standard...
Say you have a Yukawa potential (a.k.a. screened coulomb potential) V(r) = -\frac{e^2}{r}e^{-rq} where q is the inverse screening length, how would you find the critical q for having bound states? I'm working on reproducing N.F. Mott's argument about the critical spacing of a lattice of...