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The Quantum Mechanical Probability Density and Probability Current

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The Quantum Mechanical Probability Density and Probability Current The quantum mechanical probability density and probability current density operators in the Pauli theory E. L. Rumyantsev1 and P. E. Kunavin1 1Institute of Natural Sciences, Ural Federal University, 620000 Ekaterinburg, Russia We present systematic construction of probability and probability current densities operators for one-band single particle Pauli equations starting from the operators in Dirac electron model within Second Quantized Approach. These operators are of importance as in probability interpretation of experimental data, so in establishing of boundary conditions. It is shown that derived operators differ significally from their convential Schrodinger-type counterparts. The generalization of continuity equation for probability density under external perturbations and physical meaning of additional source terms is discussed. The presented approach can be useful in analysis of carriers dynamics described within generic multicomponent k · p Hamiltonians in Envelope Function Approximation (EFA). PACS numbers: 71.70.Ej, 75.70.Tj I. INTRODUCTION nation method of Pauli [10]. As a result the electron and positron (hole) degrees of freedom are formally dis- entangled in Dirac Hamiltonian and their dynamics is The quantum mechanical concept of probability and described separatly by two spinor Pauli-like equations. probability current density is of utter importance for in- While FW approach is without doubt valid for “free” terpretation of experimental results connecting the wave- Dirac problem, it comes into conflict with Quantum like description of the quasiparticle with macroscopic ob- Field description in general, when external perturbation servables [1]. In non-relativistic quantum mechanics the is switched on. FW approximation ignores the possibil- functional form of these quantaties, as their physical in- ity (inevitable in this case) of real/virtual pair creation terpretation, accompanying the solution of corresponding processes. Simultaniously with formulation of Pauli-like Schrodinger equation, are well established. Nevertheless, equations, there arises the question of relativistic cor- in the general case, the quantum behavior of electron un- rections to Schrodinger probability and current. Usu- der external perturbation must be described by relativis- ally, the sought for expressions are obtained starting from tic Dirac equation [2]. The revival of interest in Dirac second-order in 1/c Hamiltonian, obtained through FW equation is triggered also by recent advances in spintron- transformation [1, 11]. ics and new developments in condense matter systems, especially with advent of graphene. It was shown that there exist an analogy between the Dirac description of electron kinetics in vacuum and the coupled-band for- We intend to show, that straight and physically clear way to derive the form of probability and probability cur- malism for electrons in narrow-gap semiconductors and carbon nanotubes [3]. At the same time, the relativis- rent densities operators supplementing two-component tic Dirac equation poses major open problems which are Pauli-like equations is to view this problem upon within due to the presence of negative energy spectrum inter- the second quantization method (SQM) consided as an preted as antimatter (positrons). It is of common con- alternative to FW-type approach. This is not at all sensus that the so-called single-particle solutions of Dirac new approach to the derivation of Pauli equation [12– Hamiltonian describe not so much particle states as the 14]. Nevertheless, this approach has not been applied states with the charge e which are in general multi- for derivation of the form of single particle probability particle states [4]. Relativistic± corrections to electron and probability current density operators in Pauli theory dynamics are taken into account in quantum chemistry starting from their counterparts in Dirac theory. At the arXiv:1708.04193v1 [cond-mat.mes-hall] 14 Aug 2017 calculations, especially for systems containing heavy ele- same time, these quantities are crucial for formulation of boundary conditions and probability interpretation of ments [5]. Nevertheless, the relativistic approach based on Dirac-Kohn-Sham equations and Dirac-Hartree Fock experimental data. In what follows we adopt the sim- plest “no-pair” approximation, neglecting possible pair models [6, 7] is too complex in general to handle ei- ther analytically or numerically [8]. Thus semirelativistic production while projecting Dirac operators onto single particle channel. one-band description of electron behavior described by so called two-component Pauli equation, which is much simpler to solve, is very popular. The problem of ap- proximate reduction of four component Dirac problem The paper is organized as follows. In Sec. II and Sec. to two component Pauli problem is solved usially by the III we derive the probability density and probability cur- projection method proposed by Foldy and Wouthuysen rent operators respectively. In Sec. IV we discuss the (FW) [9] and its modifications, or by so called elimi- peculiarities of obtained probability continuous equation. 2 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 par- ticle 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-established|− i − fact that backscattering is absent in 3D As seen from (17) the derived Pauli probability density Weyl or 2D graphene [19] case.
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