The Quantum Mechanical Probability Density and Probability Current

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II. PROBABILITY OPERATOR Using (5) it is straightforward to write down Bogoluybov transformation from “bare” to “dressed” operators The four component spinors of plane-wave eigen func- + ak,1 Ak,1 Bk,1 tions of free problem, described by Dirac Hamiltonian = NkI MkλC σ k + , (6) ak,2 Ak,2 − · B 2 k,2 HD = cαk + βmc are chosen for positive energy branch + + (~ = 1) in momentum representation as bk,1 Bk,1 Ak,1 + = NkI + + MkλC σ k , (7) bk,2 Bk,2 · Ak,2 1 0 εk +1 0 εk +1 1 where uk,1 =  λC kz  ,uk,2 =  λC k−  , 2εk εk +1 2εk εk+1 r C r εk +1 1 λ k+ λC kz N = ,M = . (8)  εk +1   εk +1  k k   −  2εk 2ε (ε + 1)    (1) r k k and for negative branch the eigen functions are p The probability density operator Pˆ(R) by deﬁnition is λC kz λC k− εk +1 εk+1 −λC k − + εk +1 + εk +1 λC kz Pˆ(R)= Ψ(r) δ(R r)Ψ(r)dr. (9) u˜k,1 =  εk +1  , u˜k,2 =  εk +1  , 2ε − 2ε − r k 1 r k 0 Z  0   1      Inserting Dirac ﬁeld operator (5) into the latter expres-    (2) sion we obtain 2 2 where λC = 1/mc, εk = 1+ λC k . We will need also 1 + + “bare” plane-wave representation spanned by the follow- Pˆ(R)= a ak + bk+q b p (2π)3 k+q,s ,s ,s k,s ing set of four-component spinors s=1,2 X Z Z eiqRdkdq 1 0 × 1 0 0 0 1 ˆ iqR u = ,u = , (3) = 6 P (q)e dq. (10) k,1 0 k,2 0 (2π) Z 0 0     It must be stressed that position operator is c-number     only in “bare” representation. As it was shown in [17] 0 0 the position operator in dressed representation is non- 0 0 local operator and its components are non-commuting. u˜0 = , u˜0 = , (4) k,1 1 k,2 0 In what follows we will consider only electron single particle channel. Projecting onto positive energy states and 0 1     neglecting pair contribution (stability of vacuum approx- 0 0 imation) we obtain in “dressed” representation Where uk,1, uk,2 represent “bare” electron states and u˜0 ,u ˜0 are “bare” hole states. This representation k,1 k,2 ˆ + is needed as only within this representation in position Pe (q)= dk Nk+qNkAk+q,sAk,s s=1,2 space the momentum operator is obtained by replacing X Z h k i∇ and position operators are c-numbers. The + T → − 2 Ak+q,1 Ak,1 same substitution is used for the solution of multiband + Mk+qMkλC + ((k + q) σ) (k σ) Ak+q,2 · · Ak,2 k p Hamiltonians within EFA. Thus the semiconductor i problems· become analogous to the ones in the relativis- (11) tic theory [15], allowing to extend the obtained below Using identity for Pauli matrices σiσj = Iδij +iεijkσk, we results in Dirac theory to the description of carriers in can separate the spin dependent terms in this expression semiconductors [16].

The ﬁeld operator can be decomposed into “dressed” Pˆe(q)= dk Nk+qNk + + (A , A , B , B ) creation/annihilation elec- s=1,2 k,i k,i k,i k,i X Z h tron/hole operators, diagonalizing free HD, or into 2 + + + + Mk+qMkλC ((k + q) k) Ak+q,sAk,s “bare” (ak,i, ak,i, bk,i, bk,i) ones. · + i T 2 Ak+q,1 Ak,1 0 0 + 0 0 + iλC dkMk+qMk + σ [q k] . Ψ(r)= uk,1ak,1 + uk,2ak,2 +˜uk,1ak,1 +˜uk,2ak,2 A · × Ak,2 Z k+q,2 Z eikrdk (12) × + + Such decomposition of expression for Pˆ (q) is a semi- = uk,1Ak,1 + uk,2Ak,2 +˜uk,1Bk,1 +˜uk,2Bk,2 e relativistic analog of the famous Gordon decomposition Z eikrdk (5) of Dirac current [18]. × 3

2 In gapless case (λC or m 0), which can be of Ψ(r) , where Ψ(r) is an appropriate wave function. It interest while analyzing→ the ∞ behavior→ of carriers in quazi- is| stated| that for free spin one-half particle only proba- relativistic “graphene” case, this expression simplifies to bility current density undergoes modification and spin- dependent divergenceless term + T 1 Ak+q,1 Pˆ (q)= dk + 1 + (n n + ) e k k q 1 † 2 Ak+q,2 · ▽ (Ψ (r)σΨ(r)) (18) Z h 2m × Ak,1 + iσ [nk nk+q] , (13) · × Ak,2 is added to classical Schrodinger expression [11, 20] i where nk = k/ k . It is seen from (13) that probabil- 1 | | Ψ†(r)▽Ψ(r) Ψ(r)▽Ψ†(r) . (19) ity of electron scattering from the state k to the state 2m − | i k (q = 2k) is zero. This is manifestation of the |−well-establishedi − fact that backscattering is absent in 3D As seen from (17) the derived Pauli probability density Weyl or 2D graphene [19] case. It must be stressed that operator also differs essentially from its canonical coun- this result does not depend on the form of external per- terpart. In order to understand the meaning of obtained turbation. This is an inherent property of the defined result let us consider the probability density W (R), that probability operator and is a consequence of introducing is by the definition vacuum (filled valence band in k p theory) into the Dirac · ˆ theory. Note, that this result is valid in our approach We(R)= ϕ(r) Pe(R) ϕ(r) dr h | | i only if we impose the condition of vacuum stability. It Z 1 must be stressed that in the gapless case the separation = ϕ(k + q) Pˆ (q) ϕ(q) e−iqRdkdq. (2π)6 h | e | i of modified potential in Pauli equation into “bare” and Z SO parts is meaningless. As it is seen from (13), all three (20) terms, including “bare” potential term, are of the same order and do not depend on the parameters of the prob- Here the averaging is done over single particle functions lem, but only on the strength of external potential. of the type As we intend to apply proposed approach elsewhere for + + ϕ(k) = [ϕ1(k)A 1 + ϕ2(k)A 2] 0 (21) constructing density and current operators in semicon- | i k, k, | i ductors described within multi-component k p Hamil- and it is assumed that tonians, it is of interest to consider low energy· limit k p 2 2 λC k 1. In theory such approximation is known ϕ1(k) + ϕ2(k) dk =1. (22) as EFA,≪ while in· Dirac theory it is called 1/c approxi- | | | | Z mation [2] or v/c approximation [16]. As it will be seen this approximation not only simplifies the derived expres- Let us consider expansion of We(R) up to the second sions but allows the elucidation of the physical meaning order in λC k. The “classical” result is reproduced in the of various terms entering obtained expressions. Using the zeroth order in λC k. approximate expressions for N and M up to the second k k ∗ ∗ δW 0(R) [ϕ (k + q)ϕ1(k)+ ϕ (k + q)ϕ2(k)] order in λC k e, ≈ 1 2 Z−iqR 1 2 2 e dkdq Nk 1 λC k , (14) × 2 2 ≈ − 8 = ϕ1(R) + ϕ2(R) (23) | | | | 1 3 2 2 Mk 1 λ k , (15) The account of terms of the second order in λ k leads to ≈ 2 − 8 C C the following additional spin independent term δWe,1(R) 1 2 2 1 2 2 2 ε 1+ λ k (λ k ) , (16) and spin dependent term δWe,2(R) k ≈ 2 C − 8 C 1 2 † ˆ q δWe,1(R)= λC ∆RΦ (R)Φ(R), (24) the expression for Pe( ) simplifies to −8 1 1 2 † 2 2 + δW 2(R)= i λ [∇ 1 ∇ 2 ]Φ (r1)σΦ(r2) 1= 2= Pˆ (q)= dk 1 λ q A Ak e, C r r r r R e − 8 C k+q,s ,s − 4 × | s=1,2 (25) X Z h i 2 + T λC Ak+q,1 Ak,1 + i dk + σ [q k] . Here Φ(R) is two component Pauli spinor 4 A · × Ak,2 Z k+q,2 ϕ1(R) (17) Φ(R)= . (26) ϕ2(R) It is widespread opinion that in Pauli equation, considered as non-relativistic approximation to Dirac equation, The similar terms for the charge and current densities the probability density remains Schrodinger-like equal to 4 considered as the sources in the Maxwell equations were where HˆSO(R) is familiar SO operator obtained in [8]. The derivation within Lagrangian approach, which is valid up to the second order in λC k, was HˆSO(R)= σ[∇V (R) pˆ] based on a two component Pauli-like equation obtained × by Foldy-Wouthuysen transformation. and

−ikR We can use our density operator to incorporate ex- K(R)= Mke . ternal field into Pauli equation. The expectation value Z of interaction with the classical potential field V (R) in Alternatively, the arrived result can be viewed upon as electron sector can be written as the “smearing” of the wave function. Within both in- terpretations the physical origin of the effect can be at- V (R)We(R)dR. (27) tributed to peculiarity of position operator in relativistic Z Dirac theory. As it has been shown in [23] in Dirac theory the position operator must be considered as a non- The spin independent terms δW 1(R) and δW 2(R) gen- e, e, local integral operator in configuration space. We can erate classical Schrodinger potential term arrived to the same conclusion within Foldy-Wouthuysen approach [24]. It was shown that within our alternative † V (R)Φ (R)Φ(R)dR (28) approach it can be inferred that position operator compo- Z nents do not commute in general [17]. Thus any function and Darwin term of coordinates becomes operator and consequently will be “smeared” as regards its classical counterpart due to 1 λ2 Φ†(R)∆V (R)Φ(R)dR. (29) emerging uncertainty relation. The spatial behavior of 8 C Z the “smearing” kernel K(R) can be estimated following the line of arguments of [25]. The transformation kernel The spin dependent term generates spin-orbit interaction in 3D case can be written as term 1 eikR 1 R k 2 † ∇ K( )= 3 d λC Φ (R)σ[ V (R) pˆ]Φ(R)dR. (30) (2π) 2ε (ε + 1) 4 × Z k k Z 1 eikR = p G(k)dk, (34) Thus within proposed approach, it follows that the exter- (2π)3 ε nal potential modification is contained implicitly in the Z k

“proper” defined density probability expression, owning where G(k) = √εk/ 2(εk + 1). This function varies its complicated form to the account for the existence of very slowly from G(0) = 0.5 to G( )=1/ (2) 0.707. p vacuum (filled valence band). It must be stressed that Thus, following [25] we approximate∞ G(k) by some≈ effec- we can arrive at the same result within projection opera- p tive constant G0. Using the following expressions [26] tor approach (Foldy-Wouthuysen method) in the considered λC approximation. Within our approach the general ∞ eitz expression for spin-dependent contribution to probabil- dz =2K0(az), (35) √t2 + a2 ity density operator, responsible for Rashba-like inter- Z−∞ ∞ 2 2 action, predicts non linear dependence on wave vector 2 K1( α + β ) tJ0(βt)K0(α t + 1)dt = , (36) [21, 22]. More generally, for arbitrary potential its in- 2 2 0 pα + β teraction with spin is non-local. Really, using (12) we Z p obtain the following term in Pauli Hamiltonian where K1(z) is the modified Bessel (Macp Donald) function, we obtain 2 ˜ † ˜ ∆HSO = iλC Φ (k + q)V (q)σ [q k]Φ(k)dkdq, · × G0 λC R Z K(R) K1 . (37) (31) ≈ 2(2π)2λ3 R λ C C ˜ where Φ(k) = MkΦ(k). In configuration space ∆HSO Using asymptotic behavior of K1(z) [27] can be written as π e−z 1 2 † ˆ K1(z) 1+ O . (38) ∆HSO = λC Φ (R1)HSO(R1, R2)Φ(R2)dR1dR2, ∞ 2 √z z r Z (32) We see that the smearing wave function (potential) func- where non-local operator HˆSO(R1, R2) is tion have a finite width of the order of λC .

Hˆ (R1, R2)= K(R1 R)Hˆ (R)K(R R2)dR, SO − SO − In 2D case the kernel describing non-local interaction Z (33) with external potential within the same approximation 5

[26] is where

ikR 1 e Jˆ1,e(q)= cλC dk Nk+qMkk K(R)= 2 G(k)dk (2π) εk s=1,2 Z Z X + G0 J0(kR) G0 1 − C + N M + (k + q) A A , (41) dk = e R/λ . k k q k+q,s k,s ≈ 4π ε 4π λ R Z k C

It must be pointed out that as follows from above, the Jˆ2,e(q)= icλC dk Nk+qMk “relativism” (Lorentz invariance) per se does not enter Z + T presented consideration. All consideration is carried out Ak+q,1 Ak,1 NkMk+q + [k σ] ,(42) in the fixed frame of reference. The critical point is the − A × Ak,2 k+q,2 exsistance of two different types of states with positive and negative energies. So, the proposed approach must be valid for the analysis of k p problems, which are by · + T the definition considered in the fixed rest frame of crystal. Ak+q,1 Ak,1 Jê,Z (q)= icλC dkNkMk+q + [q σ] . − A × Ak,2 Two main aspects of proposed form of probability den- Z k+q,2 sity operator must be underlined As it was shown, all (43) ˆ information about “relativism” (SO-like and Darwin-like The first term J1,e(q) in the zero order approximation in terms) is contained in defined probability density oper- λC k is ator, while external potential is considered as classical. 1 + Secondly, the obtained functional form of probability op- Jˆ1 (q) λ dk[2k + q]A Ak . (44) ,e ≈ 2m C k+q,s ,s erator implies that wave function, obtained as the solu- s=1,2 Z tion of derived two-component Pauli equation, plays sub- X sidiary role. The probabilistic interpretation of outcome It is easy to recognize “classical” result in this term after of experiment is given by averaged value of found proba- averaging over corresponding wave functions bility density operator over the corresponding single particle wave functions, which are the solutions of Pauli-like ϕ(r) Jˆ1 (q) ϕ(r) dr h | ,e | i equation. One more reson for implementation of second Z i quantization scheme will be jusified below while consid- = Ψ†(R)▽Ψ(R) Ψ(R)▽Ψ†(R) . (45) ering the time dependence of density operator, which is −2m − crucial for formulation of continuity equation. The last divergenceless (q Jê,Z (q) 0) term Jê,Z (q) represents so called spin magnetization· ≡ current. The necessity of its presence was established long ago by G.Breit[28]. The addition of this current in considered III. CURRENT DENSITY OPERATOR approximation to the classical Schrodinger expression has been justified in [8, 20]. The necessity for accounting of this term in current is supported by the fact that the Zee- Following the outlined procedure the charge current man term arises due to it in Pauli Hamiltonian. Really, density operator in Dirac theory is by the definition in by choosing bare representation is 1 A(R)= [B R] Jˆ(q)= c Ψ+(r)αδ(R rˆ)Ψ(r)e−iqRdrdR 2 × − Z + T + T for constant magnetic field B, the corresponding term ak+q,1 bk,1 bk+q,1 ak,1 = c + σ + + σ dk. in expression for expectation value of interaction with a b bk+q,2 ak,2 Z " k+q,2 k,2 # external electromagnetic field is (39) e −iqR H 1 = A(R) Jˆ (q) e dqdR Z, c h e,Z i The truncation to electron channel leads to the following Z expression e = i [B ∇δ(q)] Jˆ (q) dq. (46) 2c × h e,Z i + T Z ˆ Ak+q,1 Ak,1 Je(q)= c Nk+qMkλC + σ(σ k) dk After some manipulation using well-known equality A · Ak,2 Z k+q,2 + T [A B][C D] = (AC)(BD) (AD)(BC) (47) Ak+q,1 Ak,1 + c NkMk+qλC + (σ (k + q)) σ dk × × − A · Ak,2 Z k+q,2 and expression (43), the expectation value of interaction = Jˆ1,e(q)+ Jˆ2,e(q)+ Jê,Z (q), (40) with external electromagnetic field can be presented as 6 the Zeeman term It is easy to recognize in the term ∆SˆF W the one ap- pearing in spin operator of Foldy-Wouthuysen theory. In e Bσ HZ,1 = dk. (48) accord with our no-pair approximation we retain only 2c mεk Sêven Sˆ Z even part F W of defined by them spin operator F W [31, 32] which is in our notation Note that in relativistic mechanics ε = 1 v2/c2. k − This result means that in Zeeman term the rest mass in êven 1 1 2 k [σˆ k] p SF W = σˆ λC × × . (51) the expression for spin magnetic moment of Pauli elec- 2 − 2 εk(1 + εk) tron must be replaced by expectation value of energy- dependent Lorentz mass in the corresponding quantum For massless situation describing charged “neutrinos” 2 state. Compare this result with [29]. The λC expan- (graphene case) the Zeeman term is sion of this expression up to the second order produces the relativistic correction term to the Zeeman interaction 1 [B σ] H 1 = e × dk, (52) gµB Z, 2 k 2 Z | | 1 [BD k][k Eσ] e 2 2 HZ,2 = e × 3 × dk. (53) HZ,1 λ k Bσ dk. (49) 2 k ≈−4mc C h i Z | | Z IV. CONTINUITYD EQUATIONE This addition to intrinsic magnetic moment due to the k4/8m3c2 term in expansion of relativistic kinetic energy was considered in [30]. This dependence of mass on ve- In the first quantization scheme the probability den- locity must be accounted for if spin-orbit interaction is sity current in Dirac theory Jˆ(k,t)= cα exp( ikrˆ(t)) − taken into account, as its contribution to single particle is connected with the probability density Hamiltonian is of the same order in λ2 . Pˆ(k,t) = exp( ikrˆ(t)) via continuity equation C − Nevetheless this result cannot be considered as final. ∂Pˆ(k,t) The point is that there is additional spin-dependent term = i[H,ˆ Pˆ(k,t)] = ikPˆ(k,t). (54) Jˆ2,e(q), depending also on B(r), but not explicitly on ∂t − A(r). Thus it must be added to written above Zeeman term. The time dependence of Pˆ(k,t) in the Heisenberg representation is determined in SQM by the time dependence e H 2 = [k σ](Nk+qMk NkMk+q) of creation/annihilation operators. In the free of external Z, 2mc h × − i Z perturbations Dirac problem [B δ′(q)]dkdq 2 × × 2 −imc εk t 3 2 (B k)(k σ) (B σ)k Ak(t)= e Ak, (55) = λ | | − | 2 C 2 + imc εk t + −4 εk(1 + εk) A (t)= e A . (56) D E k k e ∆Sˆ B = F W . (50) Thus, the time derivative of (12) is −2c mεk D E

∂Pˆe(q,t) 2 2 + = imc (ε + ε )[Nk+qNk + Mk+qMkλ ((k + q) k)]A (t)Ak (t)dk ∂t k q − k C · k+q,s ,s s=1,2 X Z + T 1 Ak+q,1(t) Ak,1(t) Mk+qMk(εk+q εk) + σ [q k] dk. − m − A (t) · × Ak,2(t) s=1,2 k+q,2 X Z

The “Zeeman” current Jê,Z (q,t) being divergenceless point is that we have not taken into account changing of does not contribute to the continuity equation. Never- equations of motion for annihilation/creation operators e theless, it must be retained because it is responsible for due to the Peierls substitution p c A(r), when electro- Zeeman interaction. magnetic field is switched on. It− can be shown in our approach, that the following term appears accordingly in As the Dirac probability current density αδ(R rˆ) is single particle Pauli equation (in the first order in pa- not affected by switching on of electromagnetic field,− the famous term A(r)Ψ∗(r)Ψ(r) can not appear in our non- relativistic expression for Pauli current. Nevertherless, this term is essential for the concerving of the gauge invariance. So, the question arises where do we lost it? The 7 rameter λC k) Rearranging this term to the left side we have

+ ∂Pˆ (q,t) ∆HA = eλC A qA(k q)Aq,sdkdq e ˆ ˆ k,s − iq Je(q,t)+ Ie,A(q,t) =0. (61) s Z ∂t − X 1 + ′ ′ eλC Ak,s σB(k q) s,s Aq,s dkdq. Which is now in the form of the standard sourceless con- − 2 ′ { − } tinuity equation if we redefine current as Xs,s Z (57) Jê,total(q,t)= Jê(q,t)+ Iê,A(q,t). (62) ˆ The time dependence of annihilation operators with ac- The source term Ie,A(q,t) is a material property and count to this addition is must vanish outside the sample. It is analogous to the contribution to the charge density from divergence of ∂Aq,s(t) 2 = imc εkAq,s(t) the polarization in electrodynamics of continuous matter ∂t − [33]. Here the SQM vacuum plays the role of “polarized” perturbated medium. Such essential physical difference ieλ Aq (t)qA(k q)dq − C ,s − s in the origin of these two contributions provides the pos- X Z 1 sibility to impose different constrains on their spatial and + eλ σB(k q) ′ Aq ′ (t)dq. temporal behavior. For example, it is of importance in 2 C { − }s,s ,s s′ the theory of superconductivity, where these two contri- X Z (58) butions to the current are treated on different grounds. The first paramagnetic contribution to the current is set The last term describing interaction of spin with mag- to zero assuming rigidity of macroscopic superconduc- netic field (Zeeman term) was discussed in detail for con- tor wave function or in another words proposing the sta- stant magnetic field above. Thus we will be interested in bility of Bose vacuum of Couper pairs under external the second term. It is streightfarward to show that this perturbation. The second diamagnetic term, dependent correction can be rewritten in the form on vector potential, is considered to be different from ∂Pˆ(q,t) zero and describes the penetration of external magnetic ∆ = iqIˆ A(q,t) field into superconducting media. It must be noted once ∂t e, more, that obtained within proposed approach simple + e ∗ = iecλ A (t)qA(k˜ k q)Ak (t)dk˜dk. form divA(r)Ψ (r)Ψ(r) of material current is valid C k˜,s − − ,s mc s only in the zeroth approximation in λ k. In general, the X Z C (59) expression for this current depends non-locally on the vector potential in position space. There is a different It is easy to verify that in position space this term aver- approach to the derivation of probability current density aged over single electron wave function is just the sought outlined in classic textbook of Landau and Lifshitz [34]. e ∗ m divA(r)Ψ (r)Ψ(r). It must be stressed that the ap- It is based on vector potential variation of phenomeno- pearance of such addition to Pê(q,t) time derivative can logically written down approximate Pauli equation for not be compensated in continuity equation by the pro- electron. While their approach gives the same expres- posed truncated Jˆ(q,t). In order to preserve the con- sion for Jê,total, it does not reveal the essential physical stancy of the total probability P = Pˆ(R) dR 1 we difference in the origin of these two contributions. must complement continuity equation byh thei source≡ term If the external scalar potential perturbation V (r) is R applied to the system, the equations of motion for cre- ∂Pê(q,t) ation/annihilation operators are governed in electron iqJˆ (q,t)= iqIˆ A(q,t). (60) ∂t − e e, channel by the Hamiltonian

The appearance of the source term induced by Hê = Hˆ0,e + Hînt,eV , (63) qIê,A(q,t), signifies that probability current operator derived directly from Dirac current is not conserved when where Hˆ0 - the Hamiltonian of free Dirac problem and external electromagnetic field is applied to the system. Hînt,eV is of the form

+ Hˆ = V (q)[Nk+qNk + Mk+qMk((k + q) k)] A Ak dkdq int,eV · k+q,s ,s s=1,2 X Z + T 2 Ak+q,1(t) Ak,1(t) + iλC V (q)Mk+qMk + σ [q k] dk. (64) A (t) · × Ak,2(t) Z k+q,2 8

Following the procedure outlined above, it is easily veri- ductivity) requires e.g. the stability of bulk state under fied that the terms of the zeroth order in λC k do not con- application of electric field and existence of dissipation- ∂ tribute to ∂t P (q,t). Only the second order term, which less surface currents. Exactly this situation is realized in is spin dependent, remains topological insulators. ∂ δ Pˆ(q,t)= iqIe,V (q,t) ∂t ≈ V. SUMMARY 2 λC + V (q˜)A (t)q[q˜ σ] ′ AQ−q−q˜ ′ (t)dQdq˜ ≈ 2 Q,s × s,s ,s Z Z We proposed SQM approach for construction of proba- (65) bility and probability current density operators for single particle Pauli-like equation as an alternative to Foldy- In configuration representation, averaged over two- Wouthuysen consideration. We partitioned the Hamilto- component electron Pauli spinors Φ(r), this contribution nian (external perturbation accounted for) and operators leads to the appearance in continuity equation additional of interest into the part acting within electron (hole) sta- ˆ material current Ie,V (r,t) of the form tionary states and resudial part responsible for pair creation/annihilation processes. The effect of resudial part ˆ 1 2 ∇ † Ie,V (r,t) λC [ V (r) Φ(r,t) σΦ(r,t)]. (66) is fully neglected in the present paper, assuming that ≈ 2 × considered external perturbations are weak enough and Compare expression for Iˆ (r,t) with [8, 11]. It must be does not depend on time. The semirelativistic probabil- e,V ity and probability current operators, defined within such underlined that coincidence with their results occurs only for this expansion, obtained from general one (64) up to approximation, demonstrate Gordon-like structure [18], splitting into spin dependent and spin-independent parts. the second order in λC k. The expressions in the cited papers are valid only within this approximation, while Proposed approach allows to go beyond the commonly used 1/c2 approximation. Thus, for example, the de- within proposed approach the only constrain is imposed upon is the potential strength. It must be weak enough fined probability operator predict non-linear dependence to create real electron/positron (hole) pairs. One more on particle momentum for Rashba-like interaction. It striking difference lies in the approach to derivation of was also inferred that in Zeeman term the rest mass in this term. “Classical” consideration outlined in Landau the expression for spin magnetic moment of Pauli elec- and Lifshitz textbook [34] is based on variation of applied tron must be replaced in general by expectation value external vector potential. It requires the single particle of energy-dependent Lorentz mass in the corresponding quantum state. The application of external perturbation Hamiltonian, written up to the second order in λC k, accounting for classical spin-orbit interaction and Peierls leads to violation of simple form of continuation equation. substitution p p + e/cA(r). As it follows from the It is shown that within proposed approach the continuity considerations→ presented above, the account for vector equation is valid in general form with sources, dependent on external fields. The source terms can be represented potential, while deriving this term, is superfluous. This current term is solely due to external potential. More- as a divergence of some “material” currents, in full analogy with the description of electro-magnetic response in over, as in the case of redefinition of probability current under electromagnetic action, we are to expect that dif- the theory of continuous matter [8, 11, 33]. Based on semi-relativistic similarity of Dirac problem ferent spatial and temporal conditions can be imposed on k p Iˆ (q,t) and Jˆ (q,t). Such situation (as in supercon- and multicomponent Hamiltonians, we intend to ex- e,V e tend the considered approach· for the derivation of single quasipaticle probability operators in semiconductors.

[1] F. Bottegoni, H.-J. Drouhin, G. Fishman, and J.-E. We- tivistic Eﬀects in Heavy-Element Chemistry and Physics growe, Phys. Rev. B 85, 235313 (2012). (Wiley, New York, 2003). [2] V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, [8] A. Dixit, Y. Hinschberger, J. Zamanian, G. Manfredi, Relativistic Quantum Theory (Pergamon Press, 1971). and P.-A. Hervieux, Phys. Rev. A 88, 032117 (2013). [3] T. M. Rusin and W. Zawadzki, [9] L. L. Foldy and S. A. Wouthuysen, Journal of Physics: Condensed Matter 19, 136219 (2007). Phys. Rev. 78, 29 (1950). [4] A. I. Nikishov, JETP 64, 922 (1986). [10] W. Pauli, Die allgemeinen Prinzipien der Wellen- [5] S. Keller, C. T. Whelan, H. Ast, H. R. J. Walters, and mechanik (Springer, Berlin, 1958). R. M. Dreizler, Phys. Rev. A 50, 3865 (1994). [11] W. B. Hodge, S. V. Migirditch, and W. C. Kerr, [6] G. D. Kenneth and F. J. Knut, Introduction to Relativis- American Journal of Physics 82, 681 (2014). tic Quantum Chemistry (Oxford University Press, Ox- [12] T. Helgaker, A. C. Hennum, and W. Klopper, ford, 2007). The Journal of Chemical Physics 125, 024102 (2006). [7] E. Engel, R. M. Dreizler, S. Varga, and B. Fricke, Rela- [13] T. U. Helgaker and J. Almlof, 9

International Journal of Quantum Chemistry 26, 275 (1984). Phys. Rev. A 84, 062124 (2011). [14] T. Helgaker and P. Jo/rgensen, [26] I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, The Journal of Chemical Physics 95, 2595 (1991). Series and products (Academic Press, New York, 2007). [15] W. Zawadzki, Q. H. F. Vrehen, and B. Lax, [27] M. Abramovitz and S. I., Handbook of Mathematical Phys. Rev. 148, 849 (1966). Functions with fomulas, graphs and mathematical tables [16] W. Zawadzki, ArXiv e-prints (2017), (National Bureau of Standards, Applied Mathematical arXiv:1701.07067 [cond-mat.mtrl-sci]. Series 55, 1964). [17] E. L. Rumyantsev and P. E. Kunavin, ArXiv e-prints [28] G. Breit, Phys. Rev. 53, 153 (1938). (2016), arXiv:1611.00900 [cond-mat.mes-hall]. [29] W. Zawadzki, Phys. Rev. D 3, 1728 (1971). [18] W. Gordon, Zeitschrift f¨ur Physik 50, 630 (1928). [30] T. F. Meister and B. U. Felderhof, [19] N. Shon and T. Ando, Journal of Physics A: Mathematical and General 13, 129 (1980). J. Phys. Soc. Jpn. 67, 2421 (1998). [31] H. Bauke, S. Ahrens, C. H. Keitel, and R. Grobe, [20] M. Nowakowski, American Journal of Physics 67, 916 (1999). Phys. Rev. A 89, 052101 (2014). [21] W. Yang and K. Chang, [32] P. Caban, J. Rembieli´nski, and M. W lodarczyk, Phys. Rev. B 73, 113303 (2006). Phys. Rev. A 88, 022119 (2013). [22] W. Yang and K. Chang, [33] L. D. Landau and E. M. Lifshitz, Electrodynamics of Phys. Rev. B 74, 193314 (2006). continuous matter (Pergamon Press, Oxford, England, [23] T. D. Newton and E. P. Wigner, 1963). Rev. Mod. Phys. 21, 400 (1949). [34] L. D. Landau and E. M. Lifshitz, Quantum mechanics: [24] T. M. Rusin and W. Zawadzki, Non-relativistic Theory (Pergamon Press, Oxford, Eng- Phys. Rev. A 84, 062124 (2011). land, 1977). [25] T. M. Rusin and W. Zawadzki,