Boundary Conditions for the DKP Particle in the One-Dimensional Box

1 Introduction

For each quantum mechanical system, one defines a Hilbert space ℋ. Every measurable quantity is called an “observable” (e.g. energy, momentum, angular momentum) and has to be represented by a self-adjoint operator acting on ℋ. In particular, the Hamiltonian H is a very special observable because it generates the time evolution of the states, and its spectrum represents the energy of the system.

To define the Hamiltonian properly, besides the formal expression as a differential operator, its domain, in particular the boundary conditions (“BCs”), must be specified. The canonical example in non-relativistic quantum mechanics is the “free” particle in a one-dimensional box, where a vanishing normal component of the probability current is a sufficient condition to obtain an impenetrable boundary surface and this might be accomplished by imposing Dirichlet, Neumann, or mixed BCs upon the wavefunction. The particle will then be considered as “free” (i.e. at the box, but not confined to the box) if the domain of the Hamiltonian operator consists of functions satisfying the BCs. Meanwhile, one cannot extrapolate those BCs to the relativistic case without proving beforehand that the relativistic Hamiltonian will be self-adjoint for them. For example, for the relativistic Dirac particle, the wavefunction is a spinor of four complex components, which are coupled into a system of first-order differential equations, so imposing the Dirichlet condition at the boundaries leads to incompatibility in the relativistic scattering as well as in the energy eigenvalues problem. A detailed study of the possible BCs for this particle in a box has been considered in [1–4] using different approaches. However, to the best of our knowledge, the problem of the BCs that may be imposed for a “free” Duffin–Kemmer–Pétiau (“DKP”) particle in a box has not been yet considered in the literature.

The interest in the DKP equation is twofold. Firstly, it is not a priory obvious if the DKP equation including interactions is equivalent to the KG equation (in the free case, both the theories are equivalent). We have considered this question in the case of time-like Lorentz vector interaction [5–7]. Indeed, we have demonstrated the equivalence between the two theories based on the Feschbach–Villars (FV) analogy, and we have obtained the DKP Green function in terms of the KG one. Notice that although the KG-based approach has been widely adopted for the meson–nuclear interactions, it was demonstrated in [8] that the quark model interpretation of the pion and the smoothness of the pion propagator support the DKP equation over the KG one, and that when assuming a Lorentz scalar interaction, the important modified Kisslinger term in the traditional pion-nucleus potential arises naturally from the DKP equation. Secondly, since the DKP equation looks formally like the Dirac equation, it is useful for practical purposes to see how far one can stretch the analogy to the Dirac equation.

Before spreading on the subject, let us discuss about the motivation of the DKP theory, or rather, why this renewed interest, which goes back to 1970s [9]. In fact, before people only conceived of one meson (π), so the KG and the DKP equations were considered to be equivalent since they yielded the same energy levels for the π mesic atom, and the same results for processes like π–e scattering, e⁺–e→π⁺–π⁻, and actually everywhere one checked, especially for quantum electrodynamics. When mesons of different mass were discovered (such as the K), it was necessary to see whether equivalent proofs for equal initial and final masses could be generalised to the case of unequal masses, such as for K₀→π⁻e⁻ν̅. In 1970, a collaboration began investigating this question and found that when there is symmetry breaking in meson current processes, the description of the process using the DKP formulation no longer yields the same results as the standard KG description. This plus an interest in the relativistic properties of wave equations has renewed the interest in the DKP equation and its corresponding algebra. This generated several articles enriching the literature, particularly during the past two decades, although many of those papers were contested. Indeed, starting from the fact that the correct use of nonminimal vectorial interaction is required due to wide application in the description of elastic meson–nucleus scattering, it was reported in [10] that in many recent papers and for the three-dimensional DKP equation with Coulomb and harmonic oscillator potentials, the nonminimal vector coupling has been used improperly, and that in some papers the space component of the nonminimal vector potential is absorbed into the spinor. To correct this misconception, it is demonstrated that the solution of the problem can be found by solving the Schrödinger-like equations. The absence of Klein’s paradox being attributed to the fact that the nonminimal vectorial interaction refers to a kind of charge conjugate invariant coupling that behaves like a vector under a Lorentz transformation. The invariance of the nonminimal vector potential under charge conjugation means that it does not distinguish particles from antiparticles. Furthermore, in [11], attention is drawn to the misunderstanding by many authors of two points relative to minimal electromagnetic coupling in the DKP theory. Indeed, it is demonstrated that the reported apparent difference between the interaction terms for scalar bosons DKP theory and KG theory is caused by the incorrect use of an expression for the physical form of DKP field that is valid only in the free-field case, not in the minimally coupled case, and in agreement with [12], that the apparent anomalous term in second order and Hamiltonian forms of the DKP equation disappears when the physical components of DKP field are selected so that this term has no physical meaning. This is independent of a specific choice of the β matrices of the DKP algebra, nor the degree of the representation, being quite general.

Because of the singularity of integer spin algebras, the construction of the Hamiltonian for the DKP equation which has more than 2(2s+1) components (for particle–antiparticle) is fraught with difficulties, which can, however, be avoided if one connects the FV theory [13] with the well-established equivalence between both the KG and DKP theories for time-like Lorentz vector interactions, by using the Sakata and Taketani (ST) decomposition [14]. This decomposition separates out the “physical components” of the wavefunction solution of the DKP equation, from the “subsidiary components”. In this way, one obtains the DKP–Hamiltonian in terms of the FV one, for the physical components, in case of barrier and well potentials of Woods–Saxon (WS) [6, 7, 15]. One will not fail to remember that the Hamiltonian equation for the subsidiary components was derived for the first time in [14], and it was shown that its solution is an identity in terms of the particle components. The ST “particle components” are commonly called the FV equation, since they were by themselves derived in another manner from the KG formulation by Feshbach and Villars [13] obtaining 2(2s+1) component matrix equations of the same form. As it is mentioned in [16], DKP-spin 0 equation manifests the covariance, whereas the ST-spin 0 equation manifests the causal properties. In particular, the presence of noncausal wave propagation when there is coupling to a second-rank tensor field is apparent from the form of the ST equation, in which the coefficients of all the space derivatives depend on the external field. Furthermore, there were many other attempts in the literature to obtain the DKP-Hamiltonian, and in [17], it is shown that the Hamiltonian version of the DKP theory with electromagnetic coupling brings about a source term at the current that disappears from the scenario if one uses the correct physical form for the DKP field, regardless of the choice for representing the DKP matrices. In fact, it is argued that as the DKP spinor has an excess of components, the theory has to be supplemented by an equation that allows one to eliminate the redundant components. That constraint equation is obtained by multiplying the DKP equation by 1–β⁰β⁰. Then, considering only the minimal vector interaction, the DKP theory has its Hamiltonian version, H being Hermitian with respect to η⁰ – not in the usual sense ([η⁰(β⁰H)]⁺=η⁰(β⁰H))⁻.

One will recall briefly that the DKP equation [18–21] is a first-order relativistic wave equation for spin 0 and 1 bosons. In case of interaction with an electromagnetic field, it will be written as (ℏ=c=1):

where the matrices β^μ satisfy the DKP algebra:

The convention for the metric tensor is here g^μν =diag(1, –1, –1, –1). The algebra (2) has three distinct representations: the trivial one with β^μ =0, a five-dimensional representation describing scalar particles, and a ten-dimensional one for the vectorial particles. We choose the representation in which β^k+=–β^k (k=1, 2, 3) and β⁰⁺=β⁰, so we write for the spin 0:

with

The ρ_T denotes the transpose matrix of ρ, and 0 denotes the zero matrix. For the spin 1, the β^μ are given by

with

where 0 and 1 denote, respectively, the zero matrix and the unity matrix, and s_i being the standard nonrelativistic (3×3) spin 1 matrices:

The aim of this article is to characterise a “free” DKP particle in a one-dimensional box, that is, we want to obtain the Hamiltonian operator that answers to the conditions for a “free” DKP particle in a box, i.e. which BCs define its domain. Recall that a necessary condition in order to have a particle enclosed inside a box [a, b] is that the probability current density j vanishes at the walls: j(a)=j(b)=0. Likewise, a necessary condition to have a “free” particle in a box is that the probability current density must satisfy j(a)=j(b)≠0, which would permit us to say that the walls are transparent to the particle.

This article is organised as follows: in Section 2, a formal Hamiltonian for the DKP equation is constructed based on the FV analogy, and the BCs are then chosen in the way that the property of the pseudo-self-adjointness of H will be fulfilled. Concluding remarks are given in Section 3.

2 The DKP Particle in the One-Dimensional Box

The axioms of quantum theory were built to explain the phenomenon of energy quantisation in atoms and molecules. A simple example of the application of these rules is the study of a particle in a one-dimensional box, i.e. an infinite one-dimensional square well. This has a great pedagogical value, since in many introductory courses of physics or chemistry, the study of electronic wavefunctions in atoms and molecules is made by analogy with a particle in a box, without having to solve the more involved Schrödinger equation for systems ruled by the Coulomb potential.

The DKP particle is in the one-dimensional box (see Fig. 1), which is defined by

so we can choose for ψ(z, t) the form e^−iEtκ(z), then get the following eigenvalue equation [6, 7]:

where κ(z)^T=(φ, A, B, C) with A, B, and C being, respectively, the vectors of components A_i, B_i, and C_i; i=1, 2, 3. According to the equations they satisfy, we gather the components of κ(z) in this way:

with

and

Ψ(z) = φ(z)V,

where

Figure 1:

The one-dimensional box.

Thus, we denote by φ(z)^T=(Ψ, Φ, Θ) the solution of (9). We set A, B, and C being complex constants, OKG = d2dz2 + [E2 − m2] is the free KG operator, V^T=(1, 1, 1) is the vector related to the three directions of the spin 1, and p²=E²–m² where p is real (|E|>m).

The Hamiltonian for a wave equation is that operator H which satisfies the eigenvalue equation:

where (13) is to be obtained from the fundamental wave equation (1). The procedure is withal complicated for the DKP equation, and this comes about because β matrices are not invertible. To overcome this difficulty, we proceed by separating out the 2(2s+1) physical components of φ(z), which express the charge symmetry existing in the problem, that is, Ψ and Φ. These components play a similar role to that of the sum and the difference of the two components of the wavefunction ξ^T=(ξ₁, ξ₂) of FV equation, so we write [6, 7]

B being the matrix defined by

We get then the eigenvalue equation:

where H is a formal free Hamiltonian for the DKP equation – not in the ordinary sense – obtained in terms of the FV Hamiltonian [6, 7]:

τ′1 = τi ⊗ I3,τ_i, i=1, 2, 3 being the isospin Pauli matrices. We write

and this is equivalent to the following coupled equations:

Notice that H satisfies the relation:

with

so we get the condition:

We recall for the following that while having similar results, one can pass from the case of vectorial particle (s=1) to the case of scalar particle (s=0) – and vice versa – only by replacing in (9) the 10×10 matrices β⁰ and β³ by the 5×5 ones, and in (17) the τ′1 matrices by τ_i themselves. Therefore, ω will be of dimension 2×1 instead of 6×1.

Along each direction of the spin 1, the element ω has the form:

where the vectors V_j are related to the three directions of the spin 1:

V1T = (1, 0, 0), V2T = (0, 1, 0), V3T = (0, 0, 1),

and φ is defined by (12). We define the norm by [6, 7]

with ω¯j = 12ωj+τ′1, ωj+ being the adjoint of ω_j. This norm is not positive definite, it is equal to (+1) for the particles, and (–1) for the antiparticles. We will say then that ℋ is “pseudo-Hilbertian”.

In order to characterise φ(z), we eliminate Φ from (19) and (20), so we obtain the equation:

for which a general solution is given by

with k²=E²–m², A and B are complex constants, and φ⁽¹⁾ and φ⁽²⁾ particular independent solutions satisfying

From the continuity relation ∂_μJ^μ =0, we derive the charge density:

and the current density:

Wavefunctions ω(z) and (Hω)(z) belong to a proper subset of the “pseudo-Hilbertian” space H, where there exists a basis in which one may expand every ω with a scalar product denoted by 〈,〉.

Our goal is to know whether the DKP particle will be confined or not in the box. So, using the formal Hamiltonian H (17), and following the idea that for each quantum mechanical system, the Hamiltonian has to be self-adjoint (pseudo-self-adjoint in our case), we will look for the BCs which ensure that. If for those BCs the current density vanishes at the walls, the particle is said confined inside the box. However, if it is different from 0, the walls are permeable for it, and one will say that the particle is “free”. We recall that for obtaining quantisation, it is not so much the type of differential equation that must be solved, but the BC that must be obeyed by the solution: a particle confined to a finite region of space does have discrete energy levels.

The machinery of Von Neumann’s theory for self-adjoint extensions of operators, which we will follow, deals with symmetric operators, that is, operators with dense domains. Recall that in a normed space 𝒜, a subspace 𝒰 is said dense in 𝒜, if ∀ξ∈𝒜, ∃ (ξ_n)_n∈ℕ∈𝒰, such that, ξ_n converges to ξ for the topology defined by the norm: limn → ∞||ξn − ξ|| = 0. However, we cannot speak about the density of the domain of H, since the norm is not positive definite. We introduce in this case the notion of weak density.

Definition 1In a pseudo-normed space 𝒜, a subspace 𝒰 is said weakly dense in 𝒜, if ∀ξ∈𝒜, ∃(ξ_n)_n∈ℕ∈𝒰, such that, ξ_nconverges weakly to ξ for the weak topology defined by the scalar product: ∀g ∈ A, limn → ∞〈ξn, g〉 = 〈ξ, g〉.

We recall that a linear densely defined operator A acting in a Krein space (ℋ, [., .]_J) with a fundamental symmetry J (i.e. J=J⁺ and J²=I) and an indefinite metric [., .]_J=(J., .) is called J-self-adjoint – or pseudo-Hermitian – if A⁺J=JA, see [22]. Thus, in the sense of (23), the linear weakly dense operator H is τ′1-self-adjoint on the Hilbert space ℋ endowed with the indefinite inner product:

which is called a Krein space (ℋ, [., .]τ′1).

Thereafter, it is necessary that H be symmetric, meaning that every state in the Krein space (ℋ, [., .]τ′1) could be arbitrarily well approximated by states in the domain of H denoted 𝒟(H), and that (using integration by parts):

for all ξ, η∈𝒟(H). Also, since its “pseudo-adjoint” H̅ defined by the same formal operator (22) verifies 𝒟(H)⊆𝒟(H̅); it is necessary that

If we choose for the domain of H,

where ω′ = dωdz, then H is symmetric. However, H will not be τ′1-self-adjoint since the domain of H̅, which is defined by

with

for all ξ∈𝒟(H) and π∈𝒟(H̅), is larger than the domain of H, that is, H̅ is defined on a manifold of wavefunctions taking arbitrary values at the end points of the interval Ω. The problem will consist then in choosing a sufficiently general set of BCs for which 𝒟(H)=𝒟(H̅). Widening the initial domain of H, we achieve that both domains coincide, in which case H will be τ′1-self-adjoint. That is to say, once 𝒟(H) is fixed, H̅ will be the pseudo-adjoint of H, if its maximal domain 𝒟(H̅) is consistent with the vanishing of [ξ+(1 + τ′3)dπdz − dξ+dz(1 + τ′3)π]0L for all ξ∈𝒟(H). Furthermore, when taking ω(0)=0 and ω′(0)=0, i.e. φ(0)=0 and φ′(0)=0, one will get the homogeneous system:

where determinant φ(1)dφ(2)dz|z = 0 − φ(2)dφ(1)dz|z = 0 is different from zero due to (28). So, A=B=0, that is, the only solution is the trivial one, i.e. ω=0. The same results will be obtained if one sets ω(L)=0 and ω′(L)=0.

The non-existence of non-trivial solutions for the BCs (34) is certainly a consequence of the fact that (26) is an elliptic equation so that there are no non-trivial solutions if the function φ and its derivative φ′ have to vanish simultaneously at the boundaries of the interval Ω.

By imposing upon the wavefunction, the Dirichlet BC

i.e. φ(0)=φ(L)=0, we obtain

ωj = 2iAsinknz(1Enm) ⊗ Vj

with

the discrete momentum values, and

the positive eigenvalues of the Hamiltonian, corresponding to the particle states with positive norm (+1). The symmetry condition ρ(0)=ρ(L) is verified:

also the impenetrability condition

However, even though in non-relativistic quantum mechanics, a vanishing wavefunction at the boundaries is one among the self-adjoint extensions of the free Hamiltonian, in relativistic quantum mechanics, the vanishing of the entire relativistic wavefunction at the beginning of an impenetrable box is not admissible [1–3]. Then we cannot consider that the Hamiltonian has the BC (38) as one of its τ′1-self-adjoint extensions.

When considering the Neumann BC:

i.e. φ′(0)=φ′(L)=0, we obtain

with

kn = nπL, n = 0, 1, 2..,En = [m2 + (nπL)2]1/2,

respectively, the discrete momentum values and the positive eigenvalues of the Hamiltonian. The current density verifies at the walls the impenetrability condition:

as well as the charge density given by

verifies the symmetry condition:

Figure 2 shows that in this case, there is no particle for z<0 and z<L, in other words, that the particle is constrained by the potential to stay inside the box Ω. The charge density at the walls is not zero, and this could be interpreted physically by saying that the particle is bouncing between the walls of the box, that is, the particle reaches the walls but cannot cross them since j=0, so it turns back. The situation is comparable to the motion of charged particles in a material medium. Fulfilling the Lorentz equation j→ = ρv→,v→ being the velocity of the particles, the current j→ equals to zero means the vanishing of their velocity. The vanishing of the relativistic current density at the walls of the box has been used in the MIT bag model of hadrons [23], in which an infinite spherical well (bag) confines the otherwise free constituent quarks inside the bag.

Figure 2:

Plot of ρ/ρ_max in the one-dimensional box for the Neumann BC. The charge density is maximal at the walls, ρmax = 4Enm|A|2.

The BC (43) is acceptable physically for being a τ′1-self-adjoint extension of the Hamiltonian H and can be used if we consider the walls of the box to be impenetrable barriers, i.e. R=1, T=0, R and T being, respectively, the coefficients of reflection and transmission.

The mixed periodic or anti-periodic BCs

yield

with the discrete momentum values

for the periodic conditions, and

for the antiperiodic conditions. The positive eigenvalues of the Hamiltonian are, respectively,

and

It can be seen that the ground energy level of (52) in the subspace of positive energies is not degenerate and corresponds to n=0 state:

However, the ground energy level of (53) in the subspace of positive energies is degenerate and corresponds to n=0 and n=–1 states:

The charge and the current densities verify at the walls of the box:

and

With the BCs (48), H is a τ′1-self-adjoint operator. The not null current at the box walls must be interpreted physically. One may say that the box walls are transparent to the particle or that the “free” particle is travelling through the box in a condition of resonance [1–3], i.e.

R = 0, T = 1.

The negative eigenvalues of the Hamiltonian En = −kn2 + m2 are related to the antiparticles that will be created near the box walls. Recall that the phenomena of the DKP particle crossing a potential V(z), which admits an infinite jump at a point z=z₀:

is connected to the creation of particle–antiparticle pairs, since the sign of the charge density near z=z₀ changes as ρ(z0+) = E − eV2E − eV1ρ(z0−) [5]. This being the Klein paradox which can be explained with the charged DKP field since it is a question of bosons. The paradox for fermions was resolved using the hole theory by Dirac. It was a generic for the Dirac equation that was resolved very soon after its inception. The KG equation also shows the paradox where it is an artefact of its not being a proper quantum mechanical equation with a positive-definite probability density.

3 Conclusion

We have analysed the relativistic DKP particle in a one-dimensional box by following the machinery of Von Neumann’s theory for self-adjoint extensions of operators and have proposed some appropriate BCs to the problem. We have first reduced the DKP equation to the physical components of the wavefunction. The relativistic wavefunction at the boundaries of a non-permitted region cannot vanish entirely, so we did not accept the Dirichlet BC as a τ′1-self-adjoint extension for the Hamiltonian. Furthermore, we have accepted the Neumann BC, and the walls of the box are then impenetrable barriers between which the particle will be enclosed. However, with the mixed periodic or antiperiodic BCs (except in the case of the periodic condition when n=0, the particle is then enclosed inside the box), the walls are transparent for the DKP particle which is “freely” travelling through the box. This resonance phenomenon is closely related to the creation of particle–antiparticle pairs [15] and is common to the DKP and KG particles [24] when interacting with the one-dimensional potential well of Woods–Saxon, which is able to bind particles. Such resonances do not exist for subcritical potentials.

Since the DKP equation is of Dirac type, we examine some analogies and differences from the Dirac equation and we notice that the DKP and Dirac particles behave likely in a one-dimensional box under periodic or antiperiodic conditions. Mention that among the infinite BCs for which the Dirac “free” Hamiltonian is self-adjoint, the CPT symmetry is not spontaneously broken only if the domain of H consists of functions satisfying the periodic BC [1–3]. Otherwise, in contrast to self-adjoint operators in Hilbert spaces, which necessarily have a purely real spectrum, self-adjoint operators in Krein spaces, in general, have a spectrum that is only symmetric with respect to the real axis. Pairwise complex conjugate eigenvalues, as part of the discrete spectrum, are connected with spontaneously broken PT symmetry [22]. It would be stimulating to complete our study by specifying which BCs (periodic or antiperiodic) is invariant under the CPT transformation.