# 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.

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1. ### A Bound states of an electron trapped in a dipole field

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...
2. ### A Lennard-Jones for bound/unbound atoms?

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...
3. ### Delta potential problem - bound states problem

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...
4. ### I Double delta potential -- Degeneracy of bound states in one dimension?

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.
5. ### A QED Bound States: Weinberg's Explanation & Breakdown

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...
6. ### I Why is e^ikx considered a plane wave?

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...
7. ### I Finite square well bound 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...
8. ### I What is the criteria for bound states

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
9. ### I Heavier hydrogen-like bound states?

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...
10. ### A Bound states and the energy-momentum relation....

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...
11. D

22. ### Understanding Scattering and Bound State Solutions in Quantum Mechanics

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...

34. ### Production of bound states of slow fermions- Peskin 5.3

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...
35. ### Bound State Wavefunctions vs Non-Bound State Wavefunctions

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...
36. ### Analytic continuation to find scattering bound states

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...
37. ### Bound states in propagator

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...
38. ### Number of bound states and index theorems in quantum mechanics?

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?
39. ### Are Standard-Model particles bound states?

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...
40. ### Could dark matter be invisible bound states of ordinary matter or ehm, aliens?

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...
41. ### What Happens When a Spherical Square Well Approaches 2mc2?

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...
42. ### In one dimension there are no degenerate bound states?

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...
43. ### Transition from bound states to continuous states

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...
44. ### Effective theory of bound states from QCD?

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?
45. ### L² Hilbert space, bound states, asymptotics of wave functions

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...
46. ### L² Hilbert space, bound states, asymptotics of wave functions

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...
47. ### Uncertainty principle and bound states?

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...
48. ### Best, -Your Quantum Mechanics Tutor

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...
49. ### Solve Bound State Problems in QFT | Identify Space of States

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...
50. ### Bound states of Yukawa potential

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...