A Non‐Standard Coupling Between Quantum Systems Originated From Their Kinetic Energy

In standard quantum mechanics, the coupling between quantum systems is described by a potential interaction term in the Hamiltonian. This type of coupling is well‐rooted in nature and shapes the universe around us, from the interactions between single photons to the attractive force between atoms that forms molecules. Quantum mechanics does not forbid other kinds of interactions to take place. In this paper, a non‐standard quantum coupling between quantum systems is proposed, originated from the kinetic energy rather than the potential interaction in the Hamiltonian. Unlike the potential‐based coupling, the proposed coupling changes the fundamental structure of quantum mechanics in the form of modified uncertainty relations that are shaped by the coupling between the particles in the system. Two prototypical examples of non‐standard systems that perform such kinetic‐based coupling are presented. In the first example, it considers a particle confined in a heterostructure, such as a quantum dot, where the coupling is between the particle and dynamic walls that determine the size of the heterostructure. The second example involves a particle in a 3D heterostructure with coupling between its position axes. It then discusses several future implications of the proposed type of non‐standard coupling.


Introduction
In quantum mechanics, coupling refers to the interaction between two or more systems, leading to correlated behavior.A standard description for the coupling between quantum systems comes through a potential interaction V appearing in the Hamiltonian of the system, e.g., for coupled quantum harmonic oscillators, we have H = p2 2 ,  > 0, and masses  1 and  2 .In materials such as heterostructures, the particles in the materials possess position-dependent effective mass (PDM) caused

Main Idea
Using the concept of PDM for particles in heterostructures, we propose to extend the concept of coupling between quantum systems based on the deformation of the kinetic terms of the Hamiltonian of the particles.Suppose we have a pair of coupled particles with positions x 1 , x 2 and with PDMs' m 1 (x 1 , x 2 ), m 2 (x 1 , x 2 ), repspectively, where m j : (x 1 , x 2 ) ∈ ℝ 2 ⇐⇒ m j (x 1 , x 2 ) ∈ ℝ > , are differentiable bivariate functions in mass units.Then, the Hamiltonian of the system is given by with the kinetic energy terms and PDMs' Here, m 0 > 0 is a constant in units of mass, and f j : (x 1 , x 2 ) ∈ ℝ 2 ⇐⇒ f j (x 1 , x 2 ) ∈ ℝ > is a unitless differentiable function.In the proposed model, , ,  are constants such that  +  +  = 2.The above model is a natural extension of the well-known von Roos model for a particle in a heterostructure, [1] where the extension is obtained by taking PDMs' that depend on both x 1 and x 2 , while in the original von Roos model, the PDM of the j − th particle only depends on the j − th position, i.e., one considers m j (x j ) instead of m j (x 1 , x 2 ) that is proposed above.Different forms of f j give rise to different features of the nonstandard quantum systems.For example, in case f j can be written as a separable function, i.e., the total kinetic energy term of the system is This means that the kinetic terms of both particles one and two are proportional to the square of the PDM of the other particle, so the deformation only modulates the kinetic terms through scaling.Unlike standard coupling that comes from the potential interaction V(x 1 , x 2 ), the kinetic-based coupling changes the foundations of the quantum system.This is followed by the change in the commutation relations, which modifies the Heisenberg uncertainty relation such that the state of one particle shapes the position-momentum uncertainty relation of the other particle.
Following (2.20) in ref. [25], we observe that By substituting the identity (6) in (2), we can rewrite the kinetic energy of the j − th particle by (7) which implies a deformed momentum operator of the form and an effective potential The commutation relation of the deformed momentum is given by which leads to the modified position-momentum uncertainty relation so the coupling between the particles changes the lower bound of the uncertainty relation.
In standard quantum mechanics, the coupling between a pair of particles with positions x 1 , x 2 only changes the energy in the system with the addition of a potential interaction V(x 1 , x 2 ).Besides that, the commutation relations of each pair of position and momentum of each particle are uncoupled to the position of the other particle.This can be seen directly through the Heisenberg uncertainty relation Δx j Δp x j ≥ ℏ 2 , j = 1, 2, which holds true independently from any interaction of the particle to other particles in the system, and so the strength of coupling between the particles does not contribute anything to the uncertainty relations.The story is different when dealing with the proposed type of interaction.The modified uncertainty relation (11) shows that unlike in standard coupling, in the kinetic-based coupling, the coupling, and its strength do affect the structure of the uncertainty relations, making it tighter than the original relations, or even less tight, where we can even get a vanished uncertainty relation for different regions in space in which ⟨f j (x 1 , x2 )⟩ ≡ 0, depending on the way the particle is coupled to the other particles in the system.This emphasizes the difference between coupling that originated from the potential interaction and the one given here, in which the uncertainty relation directly depends on the coupling in the system.

Prototypical Examples
Let us now examine two prototypical examples of non-standard quantum systems with such kinetic-based coupling between the quantum sub-systems.In our first example, the non-standard kinetic-based coupling is based on the coupling between a particle confined in a quantum dot and its barriers.Quantum dots are an important class of heterostructures, which are nanoscale semiconductor structures with dimensions on the order of nanometers.Due to quantum confinement effects, they exhibit discrete energy levels akin to artificial atoms.In dynamic quantum dots, the walls or boundaries of the quantum dot can change due to alterations in the system.As the walls move, the quantum confinement experienced by the confined charge carriers alters, causing a shift in the properties of the quantum dot.In quantum dots, electrons and holes possess PDM due to the confinement of their motion within the nanoscale dimensions of the structure.Concrete technological heterostructures that are characterized by splitting regions are, for example, SnS2/MoS2, Co9S8-NiCo2S4, and Cu(OH)2@ Ni0.66Co0.34-LDH(see, ref. [26][27][28]).
Suppose we have a particle with position x in a quantum dot with a dynamic wall in position y and a second wall that moves due to the other wall and is placed in −y (see, Figure 1), we further assume that the wall is not classical and that the particle is a quantum harmonic oscillator with a PDM.By assuming that the walls are dynamic, the Hamiltonian of the system should contain a kinetic energy term for the walls.Following the proposed non-relativistic system, the added kinetic term takes the form dy 2 where M is the constant mass of the (dynamic) wall.We note that a similar setting can be found in ref. [29], which considered the case of dynamic walls for studying the Casimir effect.Then, the Hamiltonian is given by with the potential where M is the constant mass of the (dynamic) wall, in which we assume that it is much larger than the position-dependent mass of the particle, M >> max (x,y) m, and we take m(x, y) = similar to. [30]The proposed setup is not restricted to the PDM m(x, y) = y 2 m 0 y 2 −x 2 ; however, we consider this form since a similar setup has been examined in ref. [30] for the case of static walls, and so we extend it into the case of dynamic walls which give rise to the given kinetic-based coupling.Moreover, the explicit solutions given in ref. [30] will serve us in deriving the solutions for our system (12).
For solving the Schrodinger equation that corresponds to the Hamiltonian (12), we use the Born-Oppenheimer approximation, which provides an adequate setting for the proposed model, which allows us to solve it explicitly.In the first step, we neglect the kinetic term of the wall − ℏ 2 2M d 2 dy 2 , and so the Schrodinger equation of the system is given by Then, the n − th eigen-state is given by (see, ref. [30]) where  0 = √ m 0 ∕ℏ, and C () n (x) are the Gegenbauer polynomials that can be expressed in terms of the hypergeometric function 2 F 1 , The normalizing constants are where Γ is the gamma function.The stationary states are orthogonal and the energy level that corresponds to  n is Then, following the Born-Oppenheimer approximation, the general solution to the problem is defined by and so for the full solution, we need to find the functions  p .We find  p by solving the equation Notice that the above equation is the standard Schrodinger equation with a Coulomb potential added to constant potential energy.We define the solution  p (y) to be in the form (y) = y p+1 e −y F(y) (22) for some constant , and for simplicity, we assume that M =  2 0 ℏ∕.Then, the solution to ( 21) can be expressed in the following explicit form ⋅ is the generalized Laguerre polynomial.The modified position-momentum uncertainty relation is then so the coupling of the particle with the dynamic walls shapes the precision of measuring the particle's position compared to its momentum measurement.
In the second example, we consider a single particle in three dimensions x 1 , x 2 , x 3 in a heterostructure.In this case, m = m 1 = m 2 as usual when dealing with the kinetic part of a particle in dimensions d > 1, and suppose that the heterostructure is composed of two different heterostructures, for different regions in space, where each heterostructure induces a different constant effective mass.Such constant effective mass that is due to a heterostructure has been well-studied in the literature (see, for instance, ref. [6,9]).
The PDM of the particle in the heterostructure is then given by for constants ,  ′ > 0,  ≠  ′ .Here Σ 1 , Σ 2 are some disjoint subsets of ℝ 3 occupied by the heterostructures that provide an effective mass of , while in other region the effective mass is  ′ .Σ 1∕2 have to capture the coupling between the particles, and so the value of m is determined by x 1 , x 2 , and x 3 .For the case of K different regions with different heterostructures in n spatial dimensions, we have for the given subsets Σ 1 , … , Σ K , where, as before, the sets are disjoint, and the value of the PDM is determined by the specific region in the n− dimensional system.Through all this, we assume that the entire wavefunction is continuous at the mass discontinuities, where its derivative follows the conditions where a is a point on the border between two heterostructures.
The following Figure illustrates the proposed coupling between the axes originating from the PDMs' for the 2-and 3-D set-ups.
Figure 2a illustrates a heterostructure that corresponds to the PDM for three distinguished regions Σ 1 , Σ 2 , and Σ 3 , that occupy the entire region of the heterostructure, Ω = Σ 1 ∪ Σ 2 ∪ Σ 3 , which give different (constant) effective mass to the particle ,  ′ ,  ′′ , respectively, and Figure 2b corresponds to the case of (25).Let us look more closely at the 3D case given in Figure 2b.Suppose that our particle is in three spatial dimensions x 1 , x 2 , x 3 , in an heterostructure, with PDM where Ball defines a ball set and is given by for a finite radius r 0 .Suppose now that our particle is an isotropic harmonic oscillator with the potential V(r) = 1 2 m(r) 2 r 2 , where r =

√
x 2 1 + x 2 2 + x 2 3 , and consider the spherical coordinates (r, , ).Then, the PDM is defined by At the region r ≤ r 0 , the solution is given by the 3D harmonic oscillator with mass where with the normalizing constant Similar to the first example, L (⋅) ⋅ is the generalized Laguerre polynomial.The solution for the region r > r 0 is then Φ  kl (r, , ).To get a physical solution of the wavefunction, we require boundary conditions.Let Ψ  ′ ≤ and Ψ  ′ > be the wavefunction solutions for the regions r ≤ r 0 and r > r 0 , respectively.The proposed boundary conditions are then for some known A 0 ≠ 0, where A 1 , A 2 are such that ‖A‖ 2 2 = 1, A = (A 0 , A 1 , A 3 ) T .Then, the boundary conditions are satisfied for suitable amplitudes A 1 , A 2 that satisfy the pair of equations and is the radial part of the associated wavefunction, i.e., Φ  ′ kl (r, , ) = R  ′ kl (r) ⋅ Y lm (, ).This allows us to get the explicit form of the wavefunction of the system.
We can rewrite m(r) (31) using indicator functions one ⋅ , in particular m(r) = 1 r>r 0 +  ′ 1 r≤r 0 .This allows us to find the corresponding function f associated with the PDM, where  ′ =  ′ ∕m 0 and  = ∕m 0 are unitless constants.
By substituting (40) in ( 11), the uncertainty relations of the particle in the x j axis is then given by

Conclusion
At the heart of the coupling between the particles in nature lies the concept of potential energy and its associated potential interaction terms.This framework, deeply rooted in the realm of quantum mechanics, guides the behavior of particles at the most fundamental level, giving rise to the diverse phenomena in the microscopic world, manifesting in the macroscopic one.In the paper, we have proposed a new way to achieve coupling based on the kinetic energy rather than the potential one.Unlike the coupling originating from potential energy, in the presented case, the coupling modifies the position momentum commutation relations, which implies a modified version of the position momentum uncertainty relations where the lower bound of the relations is based on the strength of the coupling in the system.Besides the proposed two prototypical examples shown in this paper, such examples can be obtained using different forms of PDM and different settings of the experiments.The kinetic-based coupling shares a similar feature of deformation of the momentum.In future research, we propose studying the analogies between particles in curved space and kinetic-based coupling better to understand the relations between deformed momentum due to kineticbased coupling and such a deformation due to the curvature of space and spacetime.Another future research that has potential implications for deepening our understanding of quantum foundations is to explore and compare the results with those related to the Higgs field, which gives the mechanism of mass in quantum field theory.

Figure 1 .
Figure 1.The experimental setup for the particle in the quantum dot with the dynamic walls.

Figure 2 .
Figure 2. a) The kinetic-based coupling arises in the case of a 2D heterostructure with three distinguished regions that include different structures.b) The 3D case with two different regions that occupy different structures.