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







{\displaystyle \{m_{k}\}_{k=1}^{n}}
corresponds to a pole in the S-matrix with a center-of-mass energy less than




{\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. R

    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. K

    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. L

    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. L

    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. Jamister

    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. P

    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. A

    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. F

    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. Ian Mitchell

    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

    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

    What Are the Boundary Conditions for l=0 with Negative Energy?

    The question is basically find the boundary conditions when ##l=0##, for energies minor than 0. Homework Equations $$V(r)=\begin{cases} & 0\text{ $r<a_0$}\\ &V_0\text{ $a_0<r<a_1$}\\ & 0\text{ $r>a_1$}\\ \end{cases} $$ $$...
  12. Candidus

    I Why does a Mass Defect Exist if |PE| Increases?

    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/...
  13. C

    I Bound states of a periodic potential well in one dimension

    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...
  14. petrushkagoogol

    Behavior of charged particles in Free and Bound states

    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...
  15. H

    I Why do Hydrogen bound states span the Hilbert space?

    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.
  16. O

    Renormalization of Bound States in QFT

    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.
  17. Gvido_Anselmi

    Bound States in QFT: Learn Modern Formalism & Applications

    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...
  18. pellman

    Bound states as a solution of free particles?

    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...
  19. gfd43tg

    Bound states in finite spherical well

    Homework Statement Homework EquationsThe Attempt at a Solution for ##r \le a## and ##l = 0##, the radial equation is $$- \frac {\hbar^{2}}{2m} \frac {d^{2}u}{dr^{2}} - V_{0} = Eu $$ $$- \frac {\hbar^{2}}{2m} \frac {d^{2}u}{dr^{2}} - [V_{0} + E]u = 0$$ call ##k^{2} = \frac...
  20. sk1105

    Nuclear Shell Model - pp bound states?

    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...
  21. blue_leaf77

    Expectation value of momentum for bound states

    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...
  22. Logan Rudd

    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...
  23. Logan Rudd

    Determining bound states for delta function potential

    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...
  24. S

    Bound states of spin-dependent potential

    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...
  25. P

    Bound States of Infinite Square Well

    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...
  26. S

    Does Gaussian function give bound states for a particle?

    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.
  27. G

    Bound States, Negative Potential, Alternate Basis, Matrix Mechanics

    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...
  28. P

    Double delta function and bound states.

    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...
  29. Einj

    Variational Method and Bound States

    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...
  30. A

    Double delta function potential: two bound states vs one ?

    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...
  31. C

    Exploring Two Photon Bound States: Insights from Recent Research | Physics World

    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...
  32. C

    Can the 'mass' of bound states show up full propagator?

    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...
  33. 7

    Energies and numbers of bound states in finite potential well

    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...
  34. M

    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. P

    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. A

    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. G

    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. tom.stoer

    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. lpetrich

    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. A

    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. S

    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. E

    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. G

    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. A. Neumaier

    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. tom.stoer

    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. tom.stoer

    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. M

    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. F

    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. P

    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. D

    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...