# What is Bound state: Definition and 42 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 A particle locked inside an arrangement of Dirac delta potentials

Suppose I have a 3D polyhedron with a large number of faces, and put a repulsing Dirac delta potential, ##c\delta (\mathbf{x} - \mathbf{x}_i )## with ##c>0## at each vertex point ##\mathbf{x}_i## of the polyhedron. Could this kind of an arrangement of delta potentials keep a particle such as an...
2. ### I Energies of bound state for delta function potential

Hi Let's consider a potential of the form The Schrodinger equation reads as shown in the book 'Introduction to Quantum mechanis' by D.J. Griffiths, Chaper 2 Section 5, the solution of the equation yields (only for bound state, which means E<0): My question: if i have one particle and i apply...
3. ### A Magnetic bound state in classical mechanics

Seven years ago, I wanted to share and discuss my experiments results there but it was not possible since there was no published peer review paper yet and apparently not fulfilling forum requirements. Now we have such a publication, but still not sure the subject can be discussed here. Anyway...
4. ### Can there be collision-less acceleration of electrons?

In electron cyclotron resonance of metals/solids can there be electron acceleration without them engaging in collision ? I read the last para of electron cyclotron resonance wikipedia page which stated this
5. ### Numerically find the energy of the delta-well's bound state

I'm working on an assignment where I'm required to numerically find the energy of a delta-potential's bound state. To do this, we've converted the time-independent schrödinger equation to an eigenvalue problem with E the eigen value, ψ the eigen vector and H a matrix as follows: with ##t =...
6. J

### Finding bound state and scattering matrix of Hamiltonian

Given that the Hamiltonian is H = P^2/(2m) + aδ(X − x(naught)) + bδ(X + x(naught), where x(naught) is a positive number. Find the conditions for bound states to exist and calculate their energies. Find the scattering matrix for arbitrary values of a and b. Can someone help me solve this please.
7. ### 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...
8. ### I Constraints on potential for normalizable wavefunction

We know that in one dimension if ##E>V(\infty)## or ##E>V(-\infty)## then the resulting wave function will not be normalizable. The basic argument is that if ##E>V(\infty)##, then a stationary solution to the Schrodinger equation will necessarily have a concavity with the same sign as the...
9. ### Orthonormality of Spherical Harmonics Y_1,1 and Y_2,1

Homework Statement Here is a copy of the pdf problem set {https://drive.google.com/open?id=0BwiADXXgAYUHOTNrZm16NHlibUU} the problem in question is problem number 1 which asks you to prove the orthonormality of the spherical Harmonics Y_1,1 and Y_2,1. Homework Equations Y_1,1 =...
10. ### Finding range of bound/non bound state energies of 1D finite

Homework Statement I'm currently working on a homework set for my intermediate QM class and for some reason I keep drawing a blank as to what to do on the first problem. I'm given three potentials, V(x), the first is of the form {A+Bexp(-Cx^2)}, the others I'll leave out. I'm asked to draw the...
11. ### Bound state of 3-dimensional Dirac well

Homework Statement A particle of mass ##m## is in a spherically symmetric potential ##V = -\alpha\delta(|r|-a)##. Find the minimum value of ##\alpha## such that there is at least one bound state. Homework Equations ##u = \frac{R}{r}## ##-\frac{\hbar^2}{2m} \frac{d^2u}{dr^2} + \left[V +...
12. ### Proving Bound State Exists for V(x)=-aV_0δ(x)

Homework Statement Consider V(x)=-aV_{0}δ(x). Show that it admits a bound state of energy E=-ma^2V_{0}/2\hbar^{2}.Are there any other bound states?Hint:solve Schrodinger's equation outside the potential for E<0, and keep only the solution that has the right behavior at infinity and is...
13. ### Is the Bound State of a Particle in a 1D Potential Energy Well Real or Complex?

is the bound state of a particle in a one-dimensional potential energy well real or complex?
14. ### Is the given 1-D potential an example of a bound state?

In 1-D if I have an infinite potential at x<0 so the wavefunction is zero for x<0 but for x>0 the potential is zero so the wavefunction oscillates to infinity is that a bound state ? I presume this isn't bound as it can't be normalized but most definitions state that bound means the wavefunction...
15. ### Analytical solution for bound state energies of infinite well

Hi there I am trying to find bound state energies assuming infinite potential. I have been told it can be done by analytically solving Right Hand Side and Left Hand Side of an equation such as: E^1/2 tan(2ma^2E/4hbar)^1/2 = (V0-E)^1/2 If solved properly, it should give one curve (RHS), crossed...
16. ### Explore Formation of Bound State in a Delta-Function Potential Well

Homework Statement We learned in class that a particle exposed to a 1D delta-function potential well would always have a single bound state. Let us now explore this question for the case where the delta-function potential well is situated in the vicinity of the impenetrable potential wall...
17. ### Bound state negative potentials into harmonic oscillator basis

Hello readers, Given the potential V(x) = - 1/ sqrt(1+x^2) I have found numerically 12 negative energy solutions Now I want to try to solve for these using matrix mechanics I know the matrix form of the harmonic oscillator operators X_ho, P_ho. I believe I need to perform the...
18. ### Bound state of the delta function potential.

What is the physical meaning to a bound state with negative energy? As I understand it, this is the case with the delta function potential, which admits only one bound state with a negative energy. If the potential function is identically zero throughout (except at the delta function peak)...
19. ### What does l represent in the radial Schrodinger equation?

Hi, Homework Statement A particle of mass m has a potential V(r)= -Vo r<a 0 r>a Find the minimum value of Vo for which there's a bound state of energy and angular momentum are zero by solving shrodinger equation for E<0 and taking the limit E-> 0 Homework Equations The Attempt...
20. ### Bound state of a square well, no allowed bound state mean?

Homework Statement Show in the graph ,there will be no allowed bound states with odd-parity if the well depth is less than ${V_min}$ Find ${V_min}$ in terms of k and a.where a is the half of the well width. What does no allowed bound state mean? Homework Equations $cotz=-pa/z$ where p^2...
21. ### Bound state of finite square well, why do we make this statement?

Reading from http://quantummechanics.ucsd.edu/ph130a/130_notes/node150.html Again we have assumed a beam of definite momentum incident from the left and no wave incident from the right. Why is the above statement made? What does the reflected wave mean? There is now all why reflected...
22. ### Space between bound state energies in some potential.

Is there a way to know qualitative information about energy spacing of bound state energy? Infinite square well. V=0 -a/2<x<a/2 V=∞ otherwise Bound state energy E\propto n^2 space beteween succesive energies increases at higher energy (n+1)^2-n^2=2n+1 Harmonic Oscillator V\proptox^2 E\propto...
23. ### Higher Bound State: Definition & Meaning

"higher" bound state just a quick question on terminology.. if something has a higher binding energy, can it be said to be in a higher bound state? thanks
24. ### Is the Bound State Wave Function Always Real or Imaginary?

Hi :), recently I was thinking whether every bound state is only real or imaginary, not mixture? Since all bound states have degeneracy of level one, if we suppose that ψ=ψ_{r}+iψ_{i}, then ψ_{r} and ψ_{i} must be linearly dependent as in opposite case there would be a bound state with...
25. ### Bound State Problem: How can it be addressed?

Greetings. Let's say we have a bound state problem: two micro black holes in orbit around one other. Let us disregard Hawking evaporation, and solve this problem. The usual way of solving this problem is to do so quantum-mechanically by employing the Schrodinger equation, deducting the...
26. ### Bound state transitions in QFT

In non-relativistic QM, we spend a lot of time examining bound states--energy levels, spatial distributions, and all that. We can determine that an electron put into a Coulomb potential will have certain Hamiltonian eigenstates, which correspond to the orbitals of a hydrogen atom. However, in...
27. ### Could monopoles exist in a N-S bound state?

Hi, Despite decades of searching magnetic monopoles haven't been found. Could it be that they are existing as bound states of a North and South monopole? One could model such states as a Bohr atom. It seems that the ground-state binding energy would be much more negative than the...
28. ### Why doesn't a dineutron system form a bound state?

Why doesn't a dineutron system form a bound state? Why doesn't 2 neutrons with one spin up and the other spin down form a bound state but a neutron and proton with both spin up or down form a bound state
29. ### Delta Function Bound State

Homework Statement A particle moves in one dimension in the delta function potential V= αδ(x). (where that is an 'alpha' ... not 'a') An initial wave function is given \Psi = A(a^2-x^2) for x between -a and a and Psi=0 anywhere else What is the probability that an energy measurement will...
30. ### Quantum Mechanics: Choose an acceptable bound state function

1. Which of the following is an allowed wave function for a particle in a bound state? N is a constant and α, β>0. 1) Ψ=N e-α r 2) Ψ=N(1-e-α r) 3) Ψ=Ne-α x e-β(x2+y2+z2) 4) Ψ=Non-zero constant if r<R , Ψ=0 if r>R Only one is correct. 2. What are the criteria for...
31. ### Exploring the Relationship Between Gravity and Bound States in Hydrogen Atom

In hydrogen atom the electron and the proton come very close to each other statistically(their wavefunctions even merge), so why we do not see the effect of gravity which should be on the order of other forces at Planck distance. Otherwise, compton to compton wavelength distance is too high for QG.
32. ### Average electronic momentum in bound state: please see this

Someone please tell me if I am thinking right: Let's consider an unperturbed electronic state of an atom/molecule. If we denote it by [a>, then the average electronic momentum in state [a> is, <p> = <a]p[a> = (<a]p<a])* (because p is hermitian) = (<a]*p*[a>*)...
33. ### Delta well + infinite barrier -> bound state

Hi, I'm trying to understand the quantum mechanical solution to this potential: V(x) = \left\{\begin{array}{cc}\infty & \mbox{ for } x < 0,\\-\lambda\delta(x-d) & \mbox { for } x > 0\end{array}\right. A particle of mass m is constrained to move on the half straight line \{x \in \mathbb{R}: x...
34. ### Bound state of a spin 3/2 particle in a potential

Hello, Up until now I was certain that a bound state is a state with energy below the minimum of the potential at infinity. However, in this question I don't know at all how to proceed. Homework Statement A spin 3/2 particle moves in a potential V=V_0(r)+\frac{V_1}{r^3}L\cdot S and V0 > 0. We...
35. ### A plane wave vs the bound state of Hydrogen atom: orthogonal?

These days I met one problem and asked a professor for help. But I can not understand his answer. Can you help me explain his answer? My question is that whether we can assume that a plane wave is orthogonal to the bound state of Hydrogen atom when t->\infty? Professor answers...
36. ### Calculating Expectation Value of Kinetic Energy in 3D Bound State

ok. this is an easy enough thing to prove in one dimension but my question seems to be 3 dimensional and it's causing me some hassle: show the expectation value of the kinetic energy in a bound state described by the spherically symmetric wavefunction \psi_T(r) may be written \langle...
37. ### Exploring Bound State Calculations in Quantum Field Theory

Can anybody recommend a good review article (or a book) for bound state calculations in QFT? I have never seen anything along these lines, other than brief sections or paragraphs in various textbooks about the connection to the Schrodinger equation in the non-relativistic limit for two particle...
38. ### Bound State Condition: Definition Explained

What is the precise definition of the bound state condition? Thanks in advance.
39. ### Matter + Antimatter Bound State Mathematics

Well, it has been ~ four years ago now I request help with this question in another thread, long dead, so I thought I would bring it to forum again in updated form: So, my question is: Does anyone know the mathematics that would explain the quantum dynamics of how a matter helium-3...
40. ### Purely imaginary bound state wave functions?

Hello all. I’m researching rotational motion with a nearly harmonic potential using the basis of the particle on a ring eigenstates e(n*i*theta) defined from theta=0 to theta=2*pi. The total systems wave functions (eigenfunctions of the full Hamiltonian (KE+PE)) are then linear combinations of...
41. ### Bound state for a Dirac delta function potential

Homework Statement Find the bound state energy for a particle in a Dirac delta function potential. Homework Equations \newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } } - \frac{\hbar^2}{2 m} \ \pd{\psi}{x}{2} - \alpha \delta (x) \psi (x) = E\psi (x) where \alpha >...
42. ### Understanding the Impossibility of a Bound State of Two Identical Nucleons

Hello, Can someone explain to me exactly why a bound state of two identical nucleons is not possible? I have a feeling its something to do with antisymmetric wavefunction, but haven't found a satisfactory explanation in any book. Cheers.