Electron scattering at a potential temporal step discontinuity

We solve the problem of electron scattering at a potential temporal step discontinuity. For this purpose, instead of the Schrödinger equation, we use the Dirac equation, for access to back-scattering and relativistic solutions. We show that back-scattering, which is associated with gauge symmetry breaking, requires a vector potential, whereas a scalar potential induces only Aharonov–Bohm type energy transitions. We derive the scattering probabilities, which are found to be of later-forward and later-backward nature, with the later-backward wave being a relativistic effect, and compare the results with those for the spatial step and classical electromagnetic counterparts of the problem. Given the unrealizability of an infinitely sharp temporal discontinuity—which is of the same nature as its spatial counterpart!—we also provide solutions for a smooth potential step and demonstrate that the same physics as for the infinitely sharp case is obtained when the duration of the potential transition is sufficiently smaller than the de Broglie period of the electron (or deeply sub-period).

Electron scattering at a potential spatial step is a canonical problem that is treated in the introductory section of most textbooks on quantum mechanics [1][2][3][4][5][6] and that underpins uncountable phenomena (e.g., quantum reflection, transmission and interference, quantum tunneling, quantum wells and scattering resonances, quantum coherent transport) and applications (e.g., p-n junction diodes, transistors, semiconductor lasers and detectors, scanning tunneling microscopy, quantum computing, particle accelerators).The problem is typically addressed by resolving the Schrödinger equation [7] for nonrelativistic particles, but requires promotion to the Klein-Gordon equation [8,9] or to the Dirac equation [10] for relativistic particles, of spin 0 or 1/2, respectively.
We present here an exact and comprehensive resolution of the problem of electron scattering at a potential temporal step discontinuity.We first show that the Schrödinger equation cannot account for electron scat- * christophe.caloz@kuleuven.betering for that problem and therefore decide to resort to the (more general) Dirac equation.We then demonstrate that a scalar potential temporal step does not produce any scattering, whereas a vector potential temporal step does (see Sec.I in [58]), and explain this fact in terms of related gauge symmetry and symmetry breaking.We next derive formulas for the scattering coefficients, probabilities, and energy transitions of the electronic wave.Finally, we demonstrate that the corresponding scattering is a relativistic effect.Throughout the letter, we systematically compare the problem with its spatial counterpart, and also point out some similarities and differences with corresponding electromagnetic problems [33][34][35].
Figure 1 represents the problem of electron scattering at a potential step discontinuity, with the discontinuity being spatial in Fig. 1(a) and temporal in Fig. 1(b).The latter is the problem at hand in the letter while the former is considered as its dual reference.In both cases, the changing parameter is a component of the four-vector potential A µ = (V, A) and we shall later see why, as indicated in the figure, V and A are the most relevant components for the spatial and temporal cases, respectively.As known from textbooks, the scattered electronic waves in the spatial problem [Fig.1(a)] are reflected and transmitted waves with conserved energy (∆E = 0), as in classical electromagnetics, and we shall show that the scattered electronic waves in the temporal problem [Fig.1(b)] are generally later-backward and later-forward waves [59] with conserved momentum (∆p = 0), also as in classical electromagnetics [24,34], but with a number of differences, such as the fact the phase and group velocities are generally distinct, as represented in the figure.We shall restrict our attention to the 1+1-dimensional case, with one dimension of space (z) and the dimension of time (t).Moreover, we shall use natural units (ℏ = c = 1) and the Minkowski metric η µν = diag(1, −1, −1, −1) throughout the letter.
One may first be tempted to address the problem of the temporal step [Fig.1(b)] with the Schrödinger equation, as typically done for the spatial step [Fig.1(a)] in the nonrelativistic regime.The Schrödinger equation reads i where m is the mass of the particle, which we shall consider from now on as being the electron.This equation has unpaired spatial and temporal derivatives, with the former (∇ 2 ) being of the second arXiv:2307.08111v2[quant-ph] 13 Oct 2023 . Electron scattering at a potential (a) spatial and (b) temporal step discontinuity, in spacetime (top panels) and space/time-transverse coordinates (bottom panels).The subscripts i, r, t, b, and f stand for incident, reflected, transmitted, later backward, and later forward, while the subscripts p and g stand for phase and group (velocity), respectively.
order and the latter ( ∂ ∂t ) of the first order.In the case of the spatial step, the second-order derivative operator ∇ 2 provides two boundary conditions, viz., the continuity of both ψ and of ∇ψ at the spatial discontinuity [60], which leads to a fully determined problem whose resolution provides the usual reflected and transmitted scattered electronic waves [Fig.1(a)].In contrast, the firstorder derivative ∂ ∂t provides only one boundary condition in the temporal step problem [Fig.1(b)].Assuming the plane-wave ansatz ψ ∝ e ipz , the Schrödinger equation reduces to ψ = i(2m/p 2 ) ∂ ∂t ψ and ∂ ∂t ψ must therefore be finite to ensure finite ψ, which entails that ψ must be continuous at the temporal discontinuity.However, this is indeed the the only boundary condition associated with the Schrödinger equation, due to the absence of a higherorder temporal derivative.Therefore, the Schrödinger equation does not include sufficient information to account for possible scattering, which would presumably involve two unknowns, corresponding to later-forward and later-backward waves.
The Dirac equation, which reads for the free-electron case i ∂ ∂t ψ = −(iα i ∂ i − γ 0 m)ψ, where ψ is a (4 × 1) spinor and where α i and γ 0 are (4 × 4) matrices [3,10,61,62] (see Sec. II in [58]), might represent a safer avenue for searching a solution to our temporal step problem.It has also, as the Schrödinger equation, a first-order temporal derivative ( ∂ ∂t ), but it involves multiple sub-equations, which might together support sufficient information to provide a complete solution.
Let us then try to address the problem with the Dirac equation.In order to account for the potentials in Fig. 1, we extend the free-electron Dirac equation to its minimalcoupling form [3,10,61,62] where γ µ are the matrices (Dirac-Pauli representation) where q = −e (e > 0) is the charge of the electron.One may first attempt to apply the general solution (2) to the case of a pure scalar potential, i.e., A µ = (V, 0), as typically done for the spatial step [Fig.1(a)], which corresponds in the problem at hand to a temporal scalar potential step where t 0 is the switching time.However, it may be easily verified (see Sec. III B 1 in [58]) that, although providing the expected energy shift (from , of potential interest for amplification applications, such a potential does not produce any scattering!This result, which may a priori appear surprising, may be explained in terms of gauge invariance symmetry. The (external) electric and magnetic fields, E and B, associated with the potential modulation, are generally related to the potentials as E = −∇V − ∂A ∂t and B = ∇ × A, which are invariant under the gauge transformation [63,64] where Λ is an arbitrary scalar function.The potential temporal step V (t) considered in the previous paragraph is equivalent to the transformation where θ(t−t 0 ) is the Heaviside step function, and A ′ = 0, which is a particular case of the gauge transformation (3) with V = V 1 , − ∂Λ ∂t = ∆V θ(t − t 0 ), A = 0 and ∇Λ = 0, corresponding to Λ = −∆V tθ(t − t 0 ).Therefore, this potential step does not involve any change in the external fields, which explains why we found that it produces no (later-backward wave) scattering [65].The external fields are actually zero, since A = 0 and ∇V = ∇V (t) = 0; the related energy transition due to potential without field is therefore somewhat akin to the Aharonov-Bohm effect [66].This absence of scattering contrasts with the situation of the pure scalar potential spatial step V (z) [Fig.1(a)], whose (reflected wave) scattering results from the breaking of the gauge condition (3) (see Sec. IV A in [58]).
We may suspect at this point that, since V (t) fails to break the gauge symmetry (3), its pure vector potential counterpart A(t) should break it, and hence bring about scattering, as the familiar spatial step V (z).That this is indeed the case is shown as follows.Assuming A(t) = A(t)ẑ, the step function reads now A(t < t 0 ) = A 1 and A(t > t 0 ) = A 2 = A 1 + ∆A.The corresponding transformation is Consistency with the gauge (3), given the mapping A = A 1 , ∇Λ = ∆A(t) or ∂Λ ∂z = ∆Aθ(t − t 0 ), V = V 1 = 0 and − ∂Λ ∂t = 0, would now demand that Λ = ∆Aθ(t − t 0 )z = Λ(z, t) along with Λ ̸ = Λ(t).The incompatibility between the last two conditions on Λ indicates that the transformation indeed breaks the symmetry of the gauge (3), which entails transformed external fields and which may hence lead to electron scattering.
We can now solve the problem of interest, for the potential, A(t), using the later-forward and later-backward ansätze corresponding to the related temporal-step classical electromagnetic solutions [24,34][Fig.1(b)].According to Noether's theorem [67], for such a potential, momentum is conserved (∆p = 0) due to spatial translational symmetry, viz., p i = p f = p b = p, while broken temporal translational symmetry leads to energy transitions, which are given by the dispersion relation (Secs.III A and III B 4 in [58]) where the subscript labels 1 and 2 refer to the earlier and later potential regions, respectively.Equation (4) leads to the energy relations where, assuming E f > 0, the apparent negative energy E b < 0 in the last relation simply represents propagation in the negative z direction (v g,b < 0), with positive energy Figure 2 plots the dispersion relations and electronic transitions for the two problems in Fig. 1, with Figs.2(a 4) and ( 5)] exhibits perfectly dual characteristics, with horizontal dispersion shifting and vertical (energy) transitions from the earlier to the later-backward and later-forward states.Note that the orthogonal dispersion shifting in electronic scattering in Fig. 2 is a feature that does not exist in the classical electromagnetic counterparts of these problems, which rather involve (refractive index) dispersion curves that are rotated with respect to each other and that do not differ between the space and time cases [34] (see Sec. VI in [58]).Upon the basis of the energy relations (5), the scattering amplitudes and probabilities may be easily found by inserting the expression for the vector potential step function A(t) into the general solution form (2) and enforcing the continuity condition corresponding to the probabilities where (6c) Interestingly, the amplitude coefficients in Eqs.(6a) are formally identical to those for classical electromagnetic scattering at a refractive index temporal step discontinuity [24,34], with the parameter Γ t in Eq. (6c) replacing the refractive index contrast N = n 2 /n 1 (see Sec. VI in [58]).
Figure 3 plots the electron scattering probabilities versus potential strength for the two problems in Fig. 1, with Figs.3(a) and 3(b) corresponding to the (reference) spatial step and temporal step problems in Figs.1(a) and (b), respectively.The probabilities for the spatial step [Fig.3(a)], also computed here from the Dirac equation (see Sec. III B 2 in [58]), show the well-known Klein paradox [61,68], corresponding to the transmission gap in the range qV = [E − m, E + m] and increasing transmission with increasing potential beyond the gap.In contrast, the probabilities for the temporal step [Fig.3(b)] do not exhibit such a gap; they follow a monotonic trend of exchange from forward propagation at low potentials to backward propagation at high potentials [69].The asymptotic response at high potentials (qV /m, qA/m ≳ 5) is another fundamental difference: while the temporal step is mostly "reflective" (backwardwave) there, the spatial discontinuity is mostly transmissive, as a result of the double reflection-transmission crossing due to the Klein effect.Otherwise, the temporal step supports quasi-total forward transmission up to energies (qA/m ≈ 2) more than twice the cutoff of the quasi-total transmission in the spatial case (qV /m < 1) and a forward-backward crossing point (qA/m ≈ 3.4) almost identical to the transmission-reflection crossing point in the spatial case (qV /m ≈ 3.2); these two observations correspond to trends that are generally valid when the (incident) energy is sufficient to produce a transition to the backward state, as understandable from the dispersion diagram in Fig. 2(b).
The Dirac solutions in Eqs. ( 6) and Fig. 3(b) confirm the validity of the later-forward wave and later-backward wave ansatz for the scattering potential A = A(t)ẑ.In the non-relativistic regime, where Γ t = 1 (see Sec. VII in [58]), these solutions reduce to b = 0 and f = 1, actually corresponding to the purely later-forward solution of the Schrödinger equation.This fact reveals that later-backward scattering is a purely relativistic effect [70].That may a priori seem contradictory, given that the electromagnetic-counterpart problem unconditionally supports back scattering [24,34].However, considering that the particle (photon) in the latter case is inherently relativistic (v photon = c), whereas it is not necessarily in the former case (v electron < c), makes the finding a posteriori much less surprising.
In this letter, we have resolved the fundamental problem of electron scattering at a potential temporal step discontinuity, with a systematic comparison to the spatial counterpart of the problem and mention of similarities and differences with the classical electromagnetic coun-terparts of the two problems.Such a potential might be practically produced by switched magnetic coils or electric pulses in various materials.The related effects described in this letter might lead to a wide range of applications, such as electronic amplification, anharmonic conversion, electron beam splitting, quantum computing, and attosecond physics.modulation [71], so that γ vm→∞ = 1 (no boost).It is really the speed of the electron (v) (not that of the modulation (vm) in which it propagates) that may be relativistic in our problem.
Inserting Eq. (7b) into Eq.(7a) multiplied by γ 0 yields Then combining Eqs.(7f) and (7c), which gives and noting that γ 0 γ 0 = 1, simplifies Eq. ( 8) to Finally, moving the right-hand side terms of this equation to its left-hand side provides the explicit form of the free-particle Dirac equation, which assumes the Einstein summation convention, whereby the repetition of an index in a given term implies summation over that index.Note that Eq. ( 11) represents four scalar equations via the γ µ matrices.When the particle is subjected to an electromagnetic field, expressed in terms of the four-potential [63] the Dirac equation ( 11) transforms according to the minimal-coupling prescription [62] , where q is the charge of the particle.The Dirac equation ( 14) may then be solved with the general ansatz where φ and ϑ are two-component (2 × 1) spinors and with being the Minkowski metric (West Coast convention).That the ansatz ( 15) is correct will be verified in Sec.III.In Eq. ( 16), is the momentum four-vector and is the position four-vector.The expression p•x = Et−p•x is then identified as the phase of a plane wave, which reveals that the solution form in Eq. ( 15) corresponds to a plane-wave solution.We see then that two signs in that equation [Eq.( 15)] correspond to positive and negative energies, which Dirac identified as corresponding to the particle (e.g., electron) and its antiparticle (e.g., positron).Assuming that we are exclusively dealing with the former type, we shall ignore negative energies and hence drop the positive sign in the spinor waveform (15).Thus, further restricting our attention to the 1 + 1 dimensional case, with the spatial dimension being z, we find that the ansatz (15) reduces to the specific form

III. SOLUTION TO THE DIRAC EQUATIONS A. General Solution in the Presence of Both Scalar and Vector Potentials
Substituting the ansatz (20) into the Dirac equation ( 14) gives Expanding this relation according to the Einstein summation convention yields which becomes after applying the derivative operators to the exponential term Inserting now the gamma matrices γ 0 and γ 3 from Eqs. (7f) and ( 9), respectively, into this relation and dropping the exponential function, leads to which splits into the two equations and Separately solving these equations for φ yields then and Equating these two relations and using the identity (σ 3 ) 2 = I that follows from Eq. (7d), entails the Dirac energymomentum or dispersion relation which reduces for A 0 = A 3 = 0 to Einstein's equation is the Lorentz factor, with v begin the velocity of the particle.
At this point, we may select either the spin-up solution or the spin-down solution, which correspond to the spinors and respectively.Choosing the spin-up solution and correspondingly inserting Eq. (25a) into Eq.(23a), yields which determines the spinor as ϑ as Finally, substituting Eq. (25a) and Eq. ( 27) into (20) yields the solution form Note that the alternative choice of Eq. (23b) would have led to the third element of this spinor being replaced by p−qA3 E−qA0+m , which can easily be shown to be equal to E−qA0−m p−qA3 .Further note that the alternative choice of the spindown solution [Eq.(25b)] would have led to the entries 1 and E−qA0−m p−qA3 being replaced by zero and shifted to the second and fourth slots of the spinor, respectively, with the sign of the latter changed.

B. Particular Solutions in the Presence of Specific Potentials
We shall now apply the general solution in Eq. 28 to the different step potential introduced in Sec.I.

Scalar Potential Temporal Step, V (t)
The scalar potential temporal step, shown in Fig. 4(b), may be written Substituting this potential into the solution form (28) and using ansätze corresponding to the related temporal-step electromagnetic solutions [24,34], we assume the incident, or earlier (t < t 0 ), spinor wavefunction and the later (t > t 0 ) spinor wavefunction whose first and second terms correspond to later-forward and later-backward waves, respectively, with corresponding amplitude coefficients f and b.
According to Noether's theorem [67], momentum is conserved (∆p = 0) due to spatial translational symmetry, viz., whereas the breaking of temporal symmetry entails energy transformations, which are found from Eq. ( 24) with A 0 = V (t) and A 3 = 0 to be and where the apparent negative energy in the last relation simply represents propagation in the negative z direction, with positive energy.According to Eq. ( 14), the spinor wavefunction must be continuous at the temporal discontinuity, viz., Inserting Eqs.(30) into this relation yields and Substituting then Eqs.(31) into Eq.(33b), and solving the system formed by the resulting equation and Eq.(33a), finally leads, after some algebraic manipulations, to which reveals that the scalar potential temporal step, V (t), does not produce any scattering.

Scalar Potential Spatial
Step, V (z) The scalar potential spatial step, shown in Fig. 4(a), may be written Substituting this potential into the solution form (28) and using ansätze corresponding to the related spatial-step electromagnetic solutions [34,63], we assume the left (z < z 0 ) spinor wavefunction whose first and second terms correspond to incident and reflected waves, respectively, with reflection amplitude coefficient r, and the transmitted, or right (z > z 0 ), spinor wavefunction with transmission amplitude coefficient t.
According to Noether's theorem [67], energy is conserved (∆E = 0) due to temporal translational symmetry, viz., whereas the breaking of spatial symmetry implies momentum transformations, which are found from Eq. ( 24) with A 0 = V (z) and A 3 = 0 to be and where the apparent negative momentum in Eq. (37c) simply represents propagation in the negative z direction, with positive momentum.According to Eq. ( 14), the spinor wavefunction must be continuous at the spatial discontinuity, viz., Inserting Eqs.(36) in this relation yields and Substituting then Eqs.(37) into Eq.(39b), and solving the system formed by the resulting equation and Eq.(39a), finally leads, after some algebraic manipulations, to where The probabilities associated with the reflected and transmitted waves may then be found from the ratios of the corresponding components of the conserved Dirac current.The Dirac conserved current is where ψ is the Dirac adjoint, with ψ † being the Hermitian conjugate of ψ.In our problem [Fig.4(a)], j µ has only a z spatial component, and takes then the forms and in the left and right regions, respectively.Substituting Eqs. ( 36), (7f) and ( 9) into these relations yields with and with The reflection and the transmission probabilities are then obtained as and where Γ s was defined in Eqs.(40b).Note that these probabilities verify the probability conservation formula An alternative, pragmatic way to determine the probabilities ( 45) is to write where the parameters C R and C T are "momentum-transition" coefficients, associated with change of region (from V 1 to V 2 ).Therefore, we must have C R = 1 and C T ̸ = 1.The coefficient C T may then be determined from the probability conservation, specifically by inserting R = |r| 2 and T = |t| 2 C T with Eq. ( 40) into Eq.( 46), which leads to C T = Γ s and hence retrieves the results in Eqs.(45).One may distinguish three potential regions in plotting R and T , assuming (E − qV 1 ) > m so that p i [Eq.(37b)] is real, depending on p t [Eq.(37d)] [61]: The vector potential spatial step, shown in Fig. 4(c), may be written Substituting this potential into the solution form (28) and using ansätze corresponding to the related spatial-step electromagnetic solutions [34,63], we assume the left (z < z 0 ) spinor wavefunction whose first and second terms correspond to incident and reflected waves, respectively, with reflection amplitude coefficient r, and the transmitted, or right (z > z 0 ), spinor wavefunction with transmission amplitude coefficient t.
According to Noether's theorem [67], energy is conserved (∆E = 0) due to temporal translational symmetry, viz., whereas the breaking of spatial symmetry implies momentum transformations, which are found from Eq. ( 24) with A 3 = A(z) and A 0 = V = 0 to be and where the apparent negative momentum in Eq. (50c) simply represents propagation in the negative z direction, with positive momentum.According to Eq. ( 14), the spinor wavefunction must be continuous at the spatial discontinuity, viz., Inserting Eqs.(49) in this relation yields and Substituting then Eqs.(50) into Eq.(52b), and solving the system formed by the resulting equation and Eq.(52a), finally leads, after some algebraic manipulations, to r = 0 and t = 1, (53) which reveals that the vector potential spatial step, A(z), does not produce any scattering.

Vector Potential Temporal
Step, A(t) The vector potential temporal step, shown in Fig. 4(d), may be written Substituting this potential into the solution form (28) and using ansätze corresponding to the related temporal-step electromagnetic solutions [24,34], we assume the incident, or earlier (t < t 0 ), spinor wavefunction and the later (t > t 0 ) spinor wavefunction whose first and second terms correspond to later-forward and later-backward waves, respectively, with corresponding amplitude coefficients f and b.
According to Noether's theorem [67], momentum is conserved (∆p = 0) due to spatial translational symmetry, viz., whereas the breaking of spatial symmetry implies energy transformations, which are found from Eq. ( 24) with A 3 = A(t) and A 0 = V = 0 to be and where the apparent negative energy in the last relation simply represents propagation in the negative z direction, with positive energy.According to Eq. ( 14), the spinor wavefunction must be continuous at the temporal discontinuity, viz., Inserting Eqs.(30) into this relation yields and Substituting then Eqs.(56) into Eq.(58b), and solving the system formed by the resulting equation and Eq.(58a), finally leads, after some algebraic manipulations, to where This expression may be alternatively written in terms of E i and A 1,2 only upon first using Eq.(56c) to eliminate E f and then substituting in the resulting expression which was obtained from Eq. (56c), which yields The probabilities associated with the later-forward and later-backward waves cannot be found from the ratios of the corresponding Dirac currents, contrary to the case of the V (z) problem in Sec.III B 1, because the Dirac current is not conserved here, due to the non-conservation of energy.However, we may resort to an alternative approach similar to that also used in Sec.III B 1, writing where the parameters C F and C B are now "energy-transition" coefficients, associated with change of region (from A 1 to A 2 ).Therefore, we must have C F = C B = C, since the two probabilities correspond to the same change of region, from the earlier region (A 1 ) to the later region (A 2 ), so that At the same, probability must be conserved, since the particle can only either keep moving forward or move backward, viz., Substituting then Eqs.(61) with Eqs.(59a) into Eq.( 62) yields then so that the later forward and backward probabilities are finally obtained from Eq. ( 61) as and where Eq. (59a) was used in the second equalities and where Γ t was defined in Eq. (59d).

V. PHASE AND GROUP VELOCITIES
The phase and group velocities may be computed from the dispersion relation (24), i.e., The phase velocity is defined as In general, it may be found by solving Eq. ( 73) for p and substituting the result into Eq.( 74), which yields For the cases of the scalar potential spatial step [V (z)] (Sec.III B 2) and the vector potential temporal step [A(t)] (Sec.III B 4), the scattered phase velocities may be directly obtained from Eq. (74) as with p t and p r given by Eq. (37d) and Eq.(37b), and with E f and E b given by Eq. (56c) and Eq.(56d), respectively.The group velocity is defined as Its general expression may be found by taking the derivative of Eq. ( 73) versus p and isolating ∂E/∂p, which results into For the cases of the scalar potential spatial step [V (z)] (Sec.III B 2) and the vector potential temporal step [A(t)] (Sec.III B 4), the scattered group velocities may be obtained from Eq. (79) as with p t and p r given by Eq. (37d) and Eq.(37b), and with E f and E b given by Eq. (56c) and Eq.(56d), respectively.

VI. SPATIAL AND TEMPORAL STEP ELECTROMAGNETIC PROBLEMS
We provide here the main results pertaining to the 1+1D spatial and temporal step electromagnetic problems for the sake of comparison.Related details are available in [33,34].

A. Spatial Step Problem
The wave equation may be written as The E and H fields before (z < z 0 , region 1) and after (z > z 0 , region 2) the step are where is the intrinsic medium impedance, and Inserting Eqs.(83) to (85) into the boundary condition relation yields then the scattering amplitude coefficients while the reflectance and transmittance are found, using the Poynting vector definitions and substituting Eqs.(83) in these relations, as where , assuming nonmagnetic materials.
The space-time and dispersion diagrams corresponding to these results are provided in Fig. 5(a).

B. Temporal Step Problem
The wave equation may be written as The D and B fields before (t < t 0 , region 1) and after (t > t 0 , region 2) the step are and Inserting Eqs. ( 91) to (92) into the boundary condition relation yields then the scattering amplitude coefficients while the reflectance and transmittance are found, using the Poynting vector definitions and substituting Eqs.(91) in these relations, as where , assuming nonmagnetic materials.
The space-time and dispersion diagrams corresponding to these results are provided in Fig. 5(b).In order to determine the non-relativistic limit, of the Dirac equation, we shall first recast that equation in its (non-covariant) Schrödinger form because the covariant form, introduced in Sec.II and applied in the letter to address the core of the problem, does not provide explicit access to the dynamic energy, which needs to be compared with the rest energy to determine that limit.
Let us start by recalling the covariant form of the Dirac equation, given by Eq. ( 14): Separating in this equation the temporal and spatial operators according to Einstein summation convention yields or, factoring out γ 0 , Dropping γ 0 and subsequently using Eq. ( 9) with the fact that, according to Eq. (7f), γ 0 −1 = γ 0 , simplifies that equation to or, isolating the temporal derivative term, We may now write the last equation in vector form using the definitions and This yields which, using p = −i∇, becomes with the Hamiltonian where with σ = (σ 1 , σ 2 , σ 3 ) (105d) being a vector whose components are the Pauli matrices given in Eq. (7d); specifically, it is a 1 × 3 vector of 2 × 2 matrices, so that α is a 4 × 4 matrix.Note that, in the above relations, the '•' symbol represents the scalar product defined as u • v ≜ uv T , where T represents the transpose operation.The Hamiltonian in Eq. (105b) has eigenvalues corresponding to the total relativistic energy of the particle, where E k , m and are the relativistic kinetic energy, rest mass energy and potential energy, respectively.The wavefunction ψ solution to Eq. ( 105) for positive energies may then be expressed as with separate the time evolution due to the rest mass, e −imt , and kinetic and spatial energy dependencies, embedded in the modified wavefunction, ψ.That separation will next allow us to get rid of the mass-related temporal dependence e −imt .Indeed, inserting Eqs. ( 108), (105c) and (7f) into Eq.( 105) yields which, upon applying the product rule to the left-hand side derivative and multiplying the resulting equation by e imt (and hence breaking covariance), simplifies to The suppression of the exponential time evolution associated with the rest mass energy in this relation corresponds to a redefinition of the zero of energy that will not change the observable physics in the considered non-relativistic limit [72].The observable physics will thus depend only on the kinetic and potential energies, associated with the modified wavefunction ψ, and hence with φ and ϑ.Equation ( 110) is in fact a system of two coupled equations, which splits into and Let us now consider the non-relativistic limit.For this purpose, let us write the relativistic energy, in standard units, viz., where In the non-relativistic limit [Eq.( 97)], v ≪ c, the latter relation becomes and inserting this new relation into Eq.( 112a) yields Since v ≪ c, the square root in the last expression may be approximated by its second-order Taylor expansion, which leads to or, in natural units (c = 1), where (in that non-relativistic limit approximation) the rest energy term (m) is much larger than the (now simply p 2 /(2m)) kinetic energy (E k ) and potential energy (E p ) contributions (see numerical example in Sec.VII B).
With the above redefinition of the zero-energy level, the eigen-equation corresponding to the function ϑ in Eq. ( 111) may written where, according to Eq. ( 114), is the total mass-shifted energy, corresponding to the sum of the kinetic and potential energies.In the non-relativistic limit, since the kinetic and potential energies are negligible compared to the mass term, i.e., so that, according to Eq. (115a), we find, with E p = qV ≪ m, that Eq. (111b) reduces to whose insertion into Eq.(111a) yields The first term on the right-hand side of this equation may be written as (σ • a)(σ • b)φ/(2m), where σ is the 1 × 3 vector of 2 × 2 matrices given by Eq. (105d) and a = b = (p − qA) are 1 × 3 vectors.This expression may be decomposed into components with which is a 2 × 2 matrix, since a i and b j are scalar vector components and σ i and σ j are 2 × 2 matrix vector components [the Pauli matrices, given by Eq. (7d)].The term σ i σ j in the last equality can be written in an alternative, more convenient fashion, upon using the commutation and anticommuntation relations between the Pauli matrices [73], viz., where ε ijk is the Levi-Civita symbol, and where δ ij is the Kronecker delta, whose summation leads to Inserting Eq. (122) into Eq.( 120) yields which can alternatively be written, using again Einstein summation convention, in the vectorial form Applying this result to the first term of the right-hand side of Eq. ( 119) yields where p = −i∇ has been used in the last equality.In the final expression, the term [(∇×A)+(A×∇)] is an operator and must therefore be evaluated conjointly with the wavefunction upon which it acts.Calling that wavefunction f , we find then Setting in this relation ∇ × A = B, where B is the magnetic field, according to Eq. (65b), and inserting the result into Eq.(125), we get Finally, substituting this identity into Eq (119) yields which is the Schrödinger-Pauli equation, where q = −|q| = −e in the case of the electron.In the absence of magnetic field (B = 0), the term σ • B disappears, and Eq. ( 129) reduces to the ordinary Schrödinger equation, The non-relativistic limit v ≪ c [Eq. ( 97)] is naturally valid in a large range of velocities (∼ v < c/10), which we hereafter refer to as the non-relativistic regime.
• potential energy for V = 7 V (10 times the built-in potential of a silicon p-n junction): For the sake of completeness, let us first consider the relativistic scattering coefficients for the scalar potential spatial step, which were given by Eq. ( 40) as where Squaring Eq. (131b) and algebraically manipulating the resulting expression yields and the kinetic energy p 2 i /2m is simply where E (nr) is non-relativistic total energy.Inserting Eqs. ( 133) and (134) into Eq.(132) yields then and Since the non-relativistic total energy is E (nr) = p 2 /2m + qV = ℏ 2 k 2 /2m + qV , where ℏ = 1 in natural units, where k 1,2 are wave numbers, so that Eq. ( 136) reduces to Inserting Eq. (138) into Eq.(131a) leads to the non-relativistic reduction of the Dirac reflection and transmission coefficients to the Schrödinger scattering coefficients Note that the corresponding solution to the Schrödinger equation [Eq.( 130)] is here [for the potential given in (35)] readily found as ψ 1 = e −iEit e ipiz + re −iErt e iprz (140a) and where which implies the same scattering coefficients r and t as in Eq. (139).

D. Temporal Scattering Coefficients in the Non-relativistic Regime
The relativistic scattering coefficients for the vector potential temporal step were given by Eq. ( 59) as with I and σ i being the (2×2) unit and Pauli matrices, respectively.Inserting the positive-energy traveling planewave ansatz ψ = φ ϑ e −i(Et−pz) into Eqs.(1) yields the general solution form (see Sec. III A of [58])

) and 2
(b) corresponding to the (reference) spatial step and temporal step problems in Figs.1(a) and (b), respectively, and with indications of the phase and group velocities (see Sec. V in [58]), corresponding to those in Fig. 1.The spatial step [Fig.2(a)] features the wellknown vertical dispersion shifting with horizontal (momentum) transitions from the incident to the reflected and transmitted states (see Sec. III B 2 in [58]).The temporal step [Figs.2(b)] [Eqs.(
. The resulting later-backward and later-forward amplitude coefficients are (see Sec. III B 4 in [58]) b

Figure 4
Figure 4 depicts the different scalar and vector potential spatial and temporal steps considered in the paper.

FIG. 5 .
FIG. 5. Electromagnetic wave scattering at (a) a spatial and (b) a temporal refractive index step discontinuity, represented in space-time diagrams (left panels) and dispersion diagrams with transitions (right panels).