# Zero-Energy Bound States of Neutron–Neutron or Neutron–Muon Systems

## Abstract

**:**

**r**) that are negative and go to zero as r goes to infinity, bound states may exist only for the negative total energy E. For E > 0 and for E = 0, bound states are considered to be impossible, both in classical and quantum mechanics. In the present paper we break this paradigm. Namely, we demonstrate the existence of bound states of E = 0 in neutron–neutron systems and in neutron–muon systems, specifically when the magnetic moments of the two particles in the pair are parallel to each other. As particular examples, we calculate the root-mean-square size of the bound states of these systems for the values of the lowest admissible values of the angular momentum, and show that it exceeds the neutron radius by an order of magnitude. We also estimate the average kinetic energy and demonstrate that it is nonrelativistic. The corresponding bound states of E = 0 may be called “neutronium” (for the neutron–neutron systems) and “neutron–muonic atoms” (for the neutron–muon systems). We also point out that this physical system possesses higher-than-geometric (i.e., algebraic) symmetry, leading to the approximate conservation of the square of the angular momentum, despite the geometric symmetry being axial. We use this fact for facilitating analytical and numerical calculations.

## 1. Introduction

**r**) that are negative and go to zero as r goes to infinity, bound states may exist only for the negative total energy E. For E > 0 and for E = 0, bound states are considered to be impossible, both in classical and quantum mechanics. For example, in sect. 18 of the textbook [1], it was stated that for potentials falling off at large r as −1/r

^{s}with s > 2 (which is the case for the potential analyzed in the present paper), the highest discrete energy level has some nonzero negative value E

_{max}, so that states of E > E

_{max}, including E = 0, cannot correspond to the bound states.

## 2. Analysis

**μ**

_{1}and

**μ**

_{2}, respectively. Particle 1 is electrically neutral (e.g., the neutron). Particle 2 is either charged (e.g., the muon) or electrically neutral (e.g., the neutron).

_{z}= 0, the z-axis being chosen along the common direction of the magnetic moments

**μ**

_{1}and

**μ**

_{2}. Then the spin–orbit interaction, being proportional to the scalar product of the operators

**S**and

**L**(where

**S**is the total spin) vanishes.

**μ**

_{1}

**μ**

_{2}(1 − 3 cos

^{2}θ)/r

^{3}= μ

_{1}μ

_{2}(1 − 3 cos

^{2}θ)/r

^{3},

**r**and

**μ**

_{1}

**μ**

_{2}is the scalar product (also known as the dot-product) of these two vectors.

_{z}component of the angular momentum), there is the (approximate) spherical symmetry, leading to the (approximate) conservation of the square of the angular momentum. This property of such potentials was noted and employed in celestial mechanics, e.g., while describing the motion of a satellite about the oblate Earth ([3], sect. 1.7), the motion of a circumbinary planet [4], and the motion of an interstellar interloper passing a circular binary star [5]. It was also noted and used in atomic physics while analyzing a hydrogen atom under a high-frequency, linearly polarized laser field [6,7].

_{0L}(r) Y

_{L0}(θ).

_{L0}(θ) is the spherical harmonic (corresponding to L

_{z}= 0) and R

_{0L}(r) is the radial part of the wave function (its subscript 0 indicates that it corresponds to E = 0).

_{2}(r) = −(4/7) μ

_{1}μ

_{20}/r

^{3},

_{3}(r) = −(8/15) μ

_{1}μ

_{20}/r

^{3}.

_{4}(r) = −(40/99) μ

_{1}μ

_{20}/r

^{3},

_{L}(r) is the attractive potential of the form.

_{L}(r) = −g(L) μ

_{1}μ

_{2}/r

^{3},

^{2}/(2m

_{red}) − g(L) μ

_{1}μ

_{2}/r

^{3}]ψ = 0,

_{red}is the reduced mass of the pair of the particles.

_{red}= m

_{1}m

_{2}/(m

_{1}+ m

_{2}),

_{1}and m

_{2}being the masses of particles 1 and 2, respectively.

^{2}/(2m

_{red}r

^{2})] [−r

^{2}(d

^{2}R/dr

^{2}) − 2r(dR/dr) + L(L + 1)R] − [g(L) μ

_{1}μ

_{2}/r

^{3}] R = 0,

^{2}R/dr

^{2}+ (2/r) (dR/dr) − L(L + 1)R/r

^{2}+2g(L)m

_{red}μ

_{1}μ

_{2}R/(ħ

^{2}r

^{3}) = 0.

^{2}χ/dr

^{2}− [L(L + 1)/r

^{2}]χ + B/r

^{3}= 0,

_{red}μ

_{1}μ

_{2}χ/(ħ

^{2}).

^{2}χ/dr

^{2}≈ −Bχ/r

^{3}.

^{b}).

^{b+2}] sin(a/r

^{b}) − [a

^{2}b

^{2}/r

^{2b+2}]χ ≈ − (a

^{2}b

^{2}/r

^{2b+2})χ ≈ −(B/r

^{3})χ.

^{1/2},

^{1/2}/r

^{1/2}).

^{2}χ/dr

^{2}≈ [L(L + 1)/r

^{2}]χ.

^{q}.

^{q+2}≈ [L(L + 1)/r

^{q+2}.

^{L}.

_{rms}exist only for L ≥ 2).

_{rms}= 8.3 × 10

^{−13}cm, which exceeds the neutron radius by an order of magnitude. We also estimate the average kinetic energy <K>—to make sure that the values of <K> do not contradict the nonrelativistic treatment of this physical system. (Here and below, the symbol <…> means the “average value”). Specifically, we estimate K by using the uncertainty relation p ~ ħ/(2r), so that

^{2}/(8m

_{red})] <1/r

^{2}>,

_{n}= 0.87 × 10

^{−13}cm for avoiding the divergence of the integral at small r. The estimated average kinetic energy is <K>~8.6 Mev, so that the nonrelativistic treatment is justified since the rest energy of the muon is about 106 Mev.

^{L}(see the textbook [1]). Since the two particles are identical, in the state of the parallel spins, and thus parallel magnetic moment (which is the state we are interested in), the coordinate wave function must be antisymmetric (see the textbook [1]), so that the parity P = −1. Therefore, only the states of odd values of L are admissible. Together with the above restriction L ≥ 2, this means that the lowest admissible value of the angular momentum for this system is L = 3.

_{z}, the situation for the neutron–neutron systems is significantly less restrictive than for the neutron–muon systems. Indeed, for the latter systems, we required L

_{z}= 0 to “kill” the spin–orbit interaction. For the neutron–neutron systems, there is no such restriction because for these systems the spin–orbit interaction does not exist.

_{rms}= 7.2 × 10

^{−13}cm, which exceeds the neutron radius by an order of magnitude. The average kinetic energy in this state is estimated as <K>~1.6 MeV, so that the nonrelativistic treatment is justified since the rest energy of the neutron is about 939 Mev.

## 3. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The normalized wave function of the bound E = 0 state of the neutron–muon system for L = 2 at a relatively small r.

**Figure 2.**The normalized wave function of the bound E = 0 state of the neutron–muon system for L = 2 at a relatively large r.

**Figure 3.**The normalized wave function of the bound E = 0 state of the neutron–neutron system for L = 3 at a relatively small r.

**Figure 4.**The normalized wave function of the bound E = 0 state of the neutron–neutron system for L = 3 at a relatively large r.

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**MDPI and ACS Style**

Oks, E.
Zero-Energy Bound States of Neutron–Neutron or Neutron–Muon Systems. *Foundations* **2023**, *3*, 65-71.
https://doi.org/10.3390/foundations3010007

**AMA Style**

Oks E.
Zero-Energy Bound States of Neutron–Neutron or Neutron–Muon Systems. *Foundations*. 2023; 3(1):65-71.
https://doi.org/10.3390/foundations3010007

**Chicago/Turabian Style**

Oks, Eugene.
2023. "Zero-Energy Bound States of Neutron–Neutron or Neutron–Muon Systems" *Foundations* 3, no. 1: 65-71.
https://doi.org/10.3390/foundations3010007