Acta Polytechnica https://doi.org/10.14311/AP.2025.65.0546 Acta Polytechnica 65(5):546–553, 2025 © 2025 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague DIRAC FERMION IN A TIME-DEPENDENT SPHERICAL BOX Kuvonchbek Matchonova, Davron Matrasulovb,c, Jaroslav Dittrichd,∗ a National University of Uzbekistan, 4 University Str., 100174 Tashkent, Uzbekistan b Turin Polytechnic University in Tashkent, 17 Niyazov Str., 100095 Tashkent, Uzbekistan c Tashkent University of Architecture and Civil Engineering, Yangishahar Str. 9A, 100095 Tashkent, Uzbekistan d Czech Academy of Sciences, Nuclear Physics Institute, 25068 Řež, Czech Republic ∗ corresponding author: dittrich@ujf.cas.cz Abstract. We consider a Dirac particle in a spherical box with a time-dependent radius. Analytical and numerical solutions of the time-dependent Dirac equation with time-dependent boundary (Dirichlet) conditions are obtained. Using the obtained solutions, physically observable characteristics of the dynamical confinement, such as the average kinetic energy (as a function of time) and average quantum force acting on the particle by the moving wall, are calculated. The trembling motion is analysed by computing the average coordinate of the Dirac particle as a function of time. The absence of the geometric phase is shown by direct calculation. Keywords: Dirac equation, ball with time-dependent radius, dynamical confinement, Dirichlet boundary condition. 1. Introduction Dirac fermions appear in various topics of the con- temporary physics. In early stages of quantum me- chanics, they have been considered in the context of high energy physics and quantum field theory, how- ever, during past two decades, quasiparticles described in terms of the Dirac equation attracted significant attention in condensed matter physics [1–5]. This happened due to the fact that in materials, such as graphene, topological insulators and Weyl semimetals, quasiparticles (electrons and holes), can “mimic” rela- tivistic behaviour being described in terms of Dirac equation. Therefore, these materials are unified under the general term, “Dirac materials” [1–5]. Besides the Dirac materials, relativistic quasiparticles described in terms of the Dirac equation appear in some optical systems [6, 7]. In most of the cases, Dirac fermions appear in confined form and the boundary of the con- fining domain is not always static, i.e. changes in time. Such changes/fluctuations of the confinement boundary can affect the physical properties of sys- tem under consideration, as they cause changes in the boundary conditions for the quantum mechan- ical wave (e.g. Schrödinger, Dirac, etc.) equations. Such an effect leads to the important mathematical problem, solving the quantum evolution equations with time-dependent boundary conditions. In physics, this problem is called “dynamical quantum confine- ment”. Earlier dynamical confinement attracted much attention in the context of nonrelativistic quantum mechanics described by Schrödinger equation (see, for example, the [8, 9], for review). Despite the consider- able progress made in the study of dynamical confine- ment in nonrelativistic quantum mechanics, relativis- tic cases, especially the case of relativistic spin-half particles, are still not the focus of many studies. We note that dynamical confinement for relativistic scalar particles have been studied in the context of dynami- cal Casimir effect [10]. In this paper, we address the problem of dynamical confinement for spin-half Dirac fermions by considering a spherical confining domain, where the time dependence of the boundary does not break the spherical symmetry, so that the system is described in terms of the radial Dirac equation. We note that recently, an one-dimensional counterpart of such a problem was considered in the [9]. 2. Dirac particle in a spherical box The Dirac equation on confined domains has been studied in [11–16]. [11] presents a pioneering study of the Dirac equation in a hard-wall box, where physically relevant self-adjoint general boundary conditions were derived. One-dimensional Dirac equation in a quan- tum box has been studied in details in [12, 13]. The one-dimensional Dirac equation in time-varying do- mains was studied in [14], by considering special cases of the dynamical confinement and deriving an analyt- ical solution to the problem. Practical applications of the Dirac equation to graphene quantum dots were considered in [16]. The dynamics of a Dirac particle under an one-dimensional dynamical confinement was studied in a recent paper [9]. Here, we consider dynam- ical confinement caused by a spherical box with a time- varying radius, described in terms of the radial Dirac equation with time-dependent boundary conditions. We choose standard representation of the Dirac ma- trices according to [17] and follow the conventions for the spherical spinors used there. For the spherically symmetric situations, the Hamiltonian can be decom- 546 https://doi.org/10.14311/AP.2025.65.0546 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 65 no. 5/2025 Dirac fermion in a time-dependent spherical box posed into a direct sum of parts with a given value of the total momentum j, its third component j3, and parity (−1)l ([17], see, e.g., [18] for more details). Let us remind here only the basic formula: Ψ(t, x⃗) = ( rf(t, r)Ωjlj3(n⃗) (−1) 1+l−l′ 2 rg(t, r)Ωjl′j3(n⃗) ) , r = |x⃗|, n⃗ = x⃗ r . Further, notations l = j + 1 2µ, l′ = j − 1 2µ, µ = ±1, κ = µ(j + 1 2 ), ν = l + 1 2 , ν′ = l′ + 1 2 are used. Notice, that κ = ±1,±2, . . . . We consider the Dirac operator on the three- dimensional sphere of time dependent radius R0(t), let us assume that R0 is C2-smooth. To assure a unitary evolution of the system, we replace the time derivative with the modified one given as (cf. [19]): ∇t = ∂t + Ṙ0(t) 2R0(t) (r∂r + ∂rr) . (1) The modified radial Dirac equation now reads as follows: i∇tΨ = ( −iα∂r + α1 κ r + βm ) Ψ, (2) where: α = ( 0 −i i 0 ) , α1 = ( 0 1 1 0 ) , β = ( 1 0 0 −1 ) . It is easy to check that Equation (2) provides the conservation of probability: d dt ∫ R0(t) 0 Ψ(r, t)†Ψ(r, t)dr = 0. We use the following transformations of distance, time and the wave function: y = r R0(t) , τ = ∫ t 0 1 R0(ξ) dξ, ψ(r, t) = 1√ R0(t) φ(y, τ(t)), (3) with φ = ( φ1 φ2 ) ∈ L2((0, 1),C2). Then, from the Equa- tion (2), we obtain: i∂τφ(y, τ) = ( −iα∂y + α1 κ y +mR0(t(τ))β ) φ(y, τ), (4) where y ∈ (0, 1) and φ1(0, τ)=φ2(0, τ) = 0. Thus, we “mapped” the problem onto the Dirac equation on a fixed interval but with a time-dependent mass M(τ) = mR0(t(τ)). 2.1. Radial operator domain The self-adjointness and the basic spectral properties of the considered Dirac operator in three dimensions were established in [20]. A simple example given there shows that the operator domain is not contained in the Sobolev space H1. However, the domains of the corresponding radial operators, and consequently those on the right hand side of Equation (4), are contained in H1. This was observed already in [18] on the basis that the full Dirac operator has the domain inside H1 loc(R3,C4) and the radial operators contain only terms regular at the bag surface. We give here an independent proof based on the form of the radial operator itself only, without the reference to its origin. In Equation (4), the formal differential operator: T = −iα∂y + α1 κ y +Mβ appears. The maximal operator Tmax domain con- sists of functions φ ∈ L2((0, 1),C2) such that Tφ ∈ L2((0, 1),C2). Proposition 1. For κ = ±1,±2, . . . , the domain: D(Tmax) = {φ ∈ H1((0, 1),C2) |φ(0) = 0}, (5) where φ(0) is the trace of φ at 0. Proof. Let φ ∈ D(Tmax). Evidently, φ ∈ H1((ε, 1)) for any 0 < ε < 1 as α is constant and invertible and (α1 κ y +Mβ)φ ∈ L2((ε, 1)). Let f be the upper/lower component of φ. Then: f ′(y) ± κ y f(y) = F (y), (6) with a function F ∈ L2((0, 1)) according to the form of T . Consider first the case of upper sign and κ ≥ 1. The above equation has the solution: f(y) = y−κ ∫ y 0 ξκF (ξ) dξ + cy−κ, (7) with a constant c provided that the integral is conver- gent. However:∣∣∣∣∫ y 0 ξκF (ξ) dξ ∣∣∣∣ ≤ 1√ 2κ+ 1 yκ+ 1 2 ∥F∥L2 , by the Schwarz inequality, so c = 0, and f(0) = 0. Let us now consider the case of lower sign and κ ≥ 1. The solution is now: f(y) = −yκ ∫ 1 y ξ−κF (ξ) dξ + c1y κ, (8) with a constant c1. As:∣∣∣∣yκ ∫ 1 y ξ−κF (ξ) dξ ∣∣∣∣ ≤ 1√ 2κ− 1 √ y − y2κ∥F∥L2 , we obtain, again, f(0) = 0. The cases of κ ≤ −1 are also included due to the different signs in the equation above. We need further to refine our estimates to prove that f ∈ H1. Let us start with Equation (7) (c = 0). Here: f ′(y) = −κy−κ−1 ∫ y 0 ξκF (ξ) dξ + F (y). 547 K. Matchonov, D. Matrasulov, J. Dittrich Acta Polytechnica As F ∈ L2, we have to prove the square integrability of the first term. Let us calculate:∫ 1 0 ∣∣∣∣y−κ−1 ∫ y 0 ξκF (ξ) dξ ∣∣∣∣2dy ≤ ∫ 1 0 y−2κ−2 (∫ y 0 ξκ|F (ξ)|dξ )2 dy = ∫ 1 0 y−2κ−2 (∫ y 0 ξκ|F (ξ)|dξ )(∫ y 0 ηκ|F (η)|dη ) dy = 2 ∫ 0<ξ<η 0 for n > 0 and εn < 0 for n < 0. Let us denote kn = |εn|. The solution of Equation (12) can be written as: ψn(y) = An ( sgn(n)√yJν(kny) µ √ yJν′(kny) ) , (ψn, ψk) = δn,k, (13) where An = [ 1 2 (Jν′(kn)2 − Jν+1(kn)Jν−1(kn)) ]− 1 2 . Using the boundary conditions given in Equation (12): Jν(kn) = 0, (14) and kn = k−n is chosen as the |n|th positive root of Jν . The Equation (13) holds for n ≠ 0, the trivial case of n = 0, ε0 = 0, which occurs for κ > 0, having the form ψ01(y) = 0. ψ02(y) = A0y κ. Inserting the expansion: φ(y, τ) = +∞∑ n=−∞ an(τ)ψn(y) (15) into Equation (4) and projecting onto ψn, we obtain: iȧn(τ) = εnan(τ) −mR0(t(τ))a−n(τ), n ∈ Z, i.e. the system of ordinary differential equations for an, a−n: iȧn(τ) = knan(τ) −mR0(t(τ))a−n(τ), iȧ−n(τ) = −kna−n(τ) −mR0(t(τ))an(τ), n ∈ N, (16) which can be written in a matrix form as follows: iȧ(n)(τ) = ( kn −mR0(t(τ)) −mR0(t(τ)) −kn ) a(n)(τ), a(n)(τ) = ( an(τ) a−n(τ) ) , n ∈ N. (17) In particular, coefficients a±n remain bounded in τ for every n ∈ N. For the special case of a massless Dirac particle, i.e. for m = 0, the system (16) splits into two uncou- pled equations, whose solutions are given as: a±n(τ) = a±n(0)e∓iknτ . (18) For a linearly moving wall of the sphere, given by R0(t) = A+ Bt, by taking into account relation be- tween τ and t given in Equation (3), one can obtain the exact solution for a massless Dirac particle as: a±n(τ) = a±n(0)e∓i kn B ln(1+ B A t). (19) The solution of the original Dirac equation for time- dependent box given by Equation (2) can be found using the relation between ψ and φ in Equation (3) that yields: Ψ(r, t) = 1√ R0(t) +∞∑ n=−∞ an(0)e−i εn B ln(1+ B A t)ψn ( r R0(t) ) , where ψn are given by Equation (13) and a±n fulfill the relation following from the norm conservation: +∞∑ n=−∞ |an(0)|2 = 1. For the general case, we are able to solve the system (17) only numerically, starting from the initial values: an(0) = 1√ R0(0) ∫ R0(0) 0 ψ† n r R0(0)Ψ(r, 0)dr, (20) where ψn are given by Equation (13). For the illustrative purposes, we choose the initial condition, e.g. as a Gaussian wave packet given by: Ψ(r, 0) = f(r)√ s2 1 + s2 2 ( s1 s2 ) , with s1 and s2 being the initial spin polarisation and: f(r) = 1√ d √ 2π r(R0(0) − r) exp [ − (r − r0)2 4d2 + iv0r ] , (21) where, d, r0 and v0 are the packet’s width, initial position of the centre of mass, and initial velocity, respectively. 3. Quantum dynamics under the dynamical confinement 3.1. Fermi acceleration One of the features of the time-dependent box is the so-called Fermi acceleration, which implies an increase of the particle velocity caused by its interaction with the periodically oscillating wall of the box. An impor- tant physically observable characteristics of the Fermi acceleration in quantum regime is the average kinetic energy of the particle, which is determined as: ⟨E(t)⟩ = ∫ R0(t) 0 Ψ(r, t)†HΨ(r, t) dr. 549 K. Matchonov, D. Matrasulov, J. Dittrich Acta Polytechnica 0 20 40 60 80 100 t 20 40 60 80 100 120 140 160 < E (t )> B = 0.1 B = 0.2 B = 0.3 (a). The average kinetic energy of a Dirac particle con- fined in an expanding spherical box (R0(t) = A + Bt, A = 5) as a function of time at different values of B for m = 1. The wave packet’s width, initial position of the center of mass, initial velocity and the initial spin polarization are chosen as d = 0.1, r0 = R0(0) 2 , v0 = 0, s1 = 1 and s2 = 0, respectively. 0 20 40 60 80 100 t 100 200 300 400 500 600 700 800 < E (t )> B = -0.02 B = -0.03 B = -0.04 (b). The average kinetic energy of a Dirac particle con- fined in a contracting box (R0(t) = A + Bt, A = 5) as a function of time at different values of B. The other parameters are as in Figure 1a. Figure 1. The average kinetic energy of a Dirac particle confined in an expanding and a contracting box. 0 20 40 60 80 100 t 140 145 150 155 160 < E (t )> B = 0.1 B = 0.2 B = 0.3 (a). Time-dependence of the average kinetic energy of a Dirac particle in a spherical box (R0(t) = A + B sin ωt) at different values of amplitude for fixed A = 5, ω = 2 and m = 1. The wave packet’s width, initial position of the center of mass, initial velocity and the initial spin polarization are chosen as d = 0.1, r0 = R0(0) 2 , v0 = 0, s1 = 1 and s2 = 0, respectively. 0 20 40 60 80 100 t 148 150 152 154 < E (t )> = 0.5 = 1 = 2 (b). Time-dependence of the average kinetic energy of a Dirac particle in a spherical box (R0(t) = A+B sin ωt) at different values of frequency for fixed B = 0.1. The other parameters are as in Figure 2a. Figure 2. Time-dependence of the average kinetic energy of a Dirac particle in a spherical box at different values of amplitude and frequency. Using Equations (3) and (15) for the average kinetic energy we have: ⟨E(t)⟩ = ∞∑ n=0 En(t), E0(t) = −m|a0(τ(t))|2, En(t) = εn R0(t) ( |an(τ(t))|2 − |a−n(τ(t))|2 ) − 2mℜ ( an(τ(t))a−n(τ(t)) ) , for n ∈ N. (22) For the massless particle, Equations (22) and (18) lead to: En(t) = kn R0(t) (|an(τ(t))|2 − |a−n(τ(t))|2) = kn R0(t) (|an(0)|2 − |a−n(0)|2), for n ∈ N. For the massive case (m ̸= 0), the average energy should be computed numerically. In Figure 1 the av- erage kinetic energy is plotted as a function of time for linearly expanding (Figure 1a) and contracting (Figure 1b) sphere at different values of the wall’s velocity. Figure 2 compares ⟨E(t)⟩ for the Dirac par- ticle confined in a spherical box with harmonically oscillating radius at different values of the oscillation amplitude (Figure 2a) and frequency (Figure 2b). In both cases, ⟨E(t)⟩ is periodic in time and the larger the oscillation amplitude of the wall is, the larger the amplitude of ⟨E(t)⟩ is. As the wall’s oscillation fre- quency increses, the period of ⟨E(t)⟩ decreases. Thus, no monotonic growth of average kinetic energy is pos- sible for a Dirac particle in a harmonically oscillating box, i.e. no Fermi acceleration can be observed. 3.2. Zitterbewegung One of the unusual effects in the Dirac particle’s dy- namics is the so-called “Zitterbewegung”, i.e. the man- ifestation of the trembling motion of the particle. It 550 vol. 65 no. 5/2025 Dirac fermion in a time-dependent spherical box was found, by considering the wave packet motion described by the time-dependent Dirac equation, that the average coordinate oscillates as a function of time. Here, we consider this phenomenon for a Dirac parti- cle under dynamical confinement provided by a time- dependent box. The main characteristics of Zitterbe- wegung, the average position, is determined as: ⟨r(t)⟩ = ⟨Ψ(r, t)|r|Ψ(r, t)⟩. (23) Using Equation (15) ⟨r(t)⟩ can be written as follows: ⟨r(t)⟩ = R0(t) ∑ n,m a∗ n(τ(t))am(τ(t))Vnm, (24) where the matrix elements: Vnm = ∫ 1 0 ψn(y)†yψm(y) dy were computed numerically. For a massless Dirac particle, i.e. for m = 0, the average coordinate can be defined using (18) for R0(t) = A+Bt as follows: ⟨r(t)⟩ = (A+Bt) ∑ n,m a∗ n(0)am(0) ei π(n−m) B ln(1+ B A t)Vnm. (25) In Figure 3, the average position is plotted as a func- tion of time for three regimes of the wall’s motion, for linearly expanding, contracting, and harmonically oscillating walls. One can clearly see that Zitterbewe- gung can be observed for all three regimes. 3.3. Quantum force The expectation value of the force, ⟨F (t)⟩ acting on a Dirac particle can be determined as: ⟨F (t)⟩ = − 〈 Ψ ∣∣∣∣ ∂H ∂R0(t) ∣∣∣∣Ψ〉 = − ∂ ∂R0(t) ⟨E(t)⟩, (26) where ⟨E(t)⟩ is the average kinetic energy of the par- ticle. Using the expansion (22) and Equations (16): ⟨F (t)⟩ = − 1 Ṙ0 ∂ ∂t ⟨E(t)⟩ = ∞∑ n=0 Fn(t), (27) where: Fn(t) = 1 R0(t)2 (|an(τ(t))|2 − |a−n(τ(t))|2) (28) n ∈ N. Apparently, F0(t) = 0. The signs in Equa- tion (28) suggest that an increase in the positive energy components of the particle state requires a positive force and its work, whereas an increase in the negative energy components requires a negative force. Figure 4 presents the plots of the average quantum force, ⟨F (t)⟩ as a function of time for three regimes of time-dependence of the radius, linearly expanding, contracting, and harmonically oscillating ones. The force grows as a function of time for the contracting box, while it is observed to decay for the expanding box. It oscillates for the oscillating wall. 0 10 20 30 40 50 t 0.04 0.06 0.08 0.1 0.12 0.14 0.16 < r( t) > R 0 (t)=5+0.1t R 0 (t)=5-0.05t R 0 (t)=5+0.1sin(2t) Figure 3. Time-dependence of the average position of a Dirac particle in a box with three regimes of wall’s motion (R0(t) = 5 + 0.1t, R0(t) = 5 − 0.05t and R0(t) = 5 + 0.1 sin 2t) for m = 1. The wave packet’s width, initial position of the centre of mass, initial velocity and the initial spin polarisation are chosen as d = 0.1, r0 = R0(0) 2 , v0 = 0, s1 = 1 and s2 = 0, respectively. 0 10 20 30 40 50 t 0 10 20 30 40 50 60 < F (t )> R 0 (t)=5+0.05t R 0 (t)=5-0.05t R 0 (t)=5+0.1sin(0.5t) Figure 4. Time-dependence of the average force of a Dirac particle in a box with three regimes of the wall’s motion (R0(t) = 5 + 0.05t, R0(t) = 5 − 0.05t and R0(t) = 5 + 0.1 sin 0.5t) for m = 1. The wave packet’s width, initial position of the centre of mass, initial velocity, and the initial spin polarisation are chosen as d = 0.1, r0 = R0(0) 2 , v0 = 0, s1 = 1 and s2 = 0, respectively. 3.4. Berry phase During an adiabatic change of the radius R0, and consequently the effective mass M , along a closed curve C, the eigenfunctions can aquire a geometric (Berry) phase [21]. Self-adjoint Hamiltonians corre- sponding to T form a trivial analytic family of the type A (e.g. [22, par. XII.2]) with respect to the parameter M . Therefore, the eigenfunctions ϕn are analytic in M by the Kato-Rellich theorem, which justifies the interchange of differentiation and integra- tion in the formula for the Berry phase [21]. Taking 551 K. Matchonov, D. Matrasulov, J. Dittrich Acta Polytechnica into account that the components of eigenfunctions ϕn have constant phases and their normalisation, we obtain: γn(C) = i ∮ C (∫ 1 0 ϕn(M,y)†∂Mϕn(M, z) dy ) dM = i 2 ∮ C ( ∂M ∫ 1 0 ϕn(M,y)tϕn(M, z) dy ) dM = 0, i.e. there is no Berry phase for a Dirac particle with time-dependent mass in a spherical box. 4. Conclusion We studied the dynamics of a Dirac particle under dynamical confinement. The latter is assumed to be caused by a hard-wall sphere with a time-varying radius that ensures the spherical symmetry of the system. The system is modelled in terms of the ra- dial Dirac equation with time-dependent boundary conditions. Analytical solutions of the problem are ob- tained for some special cases of the time-dependence of the sphere radius, while the for general case, the problem is solved numerically. Physically observable characteristics, such as average kinetic energy, aver- age coordinate, and the average quantum force, are computed as functions of time for different regimes of the wall’s motion. It is shown that for a harmonically breathing sphere, the average kinetic energy oscillates in time without a monotonic growth, which implies the absence of the Fermi acceleration. The absence of the Berry phase is shown by the direct calculation of the geometric phase. 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Academic Press, San Diego, 1978. 553 https://doi.org/10.1063/1.528469 https://doi.org/10.1016/0375-9601(91)90117-Q https://doi.org/10.1007/s11785-021-01090-x https://doi.org/10.1098/rspa.1984.0023 Acta Polytechnica 65(5):546–553, 2025 1 Introduction 2 Dirac particle in a spherical box 2.1 Radial operator domain 2.2 Solution of the Dirac equation on time-dependent spherical box 3 Quantum dynamics under the dynamical confinement 3.1 Fermi acceleration 3.2 Zitterbewegung 3.3 Quantum force 3.4 Berry phase 4 Conclusion Acknowledgements References