Generalized probability and current densities: A field theory approach
aa r X i v : . [ qu a n t - ph ] F e b Generalized probability and current densities: A field theoryapproach
M. Izadparast ∗ and S. Habib Mazharimousavi † Department of Physics, Faculty of Arts and Sciences,Eastern Mediterranean University, Famagusta,North Cyprus via Mersin 10, Turkey (Dated: February 5, 2021)
Abstract
We introduce a generalized Lagrangian density - involving a non-Hermitian kinetic term - fora quantum particle with the generalized momentum operator. Upon variation of the Lagrangian,we obtain the corresponding Schr¨odinger equation. The extended probability and particle currentdensities are found which satisfy the continuity equation. ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The idea of the generalized momentum operator has been extended in our earlier proposal[1, 2], which is subjected to the non-relativistic and non-Hermitian quantum mechanics. Thethought of the non-Hermitian Hamiltonian was discussed initially in Ref. [3, 4]. Yet, Benderand Boettcher have initiated PT -symmetric quantum physics by employing a harmonic-oscillator-like PT -symmetric potential as a toy model [5]. They have shown that even ifthe Hamiltonian is non-Hermitian the eigenvalues can be real, upon which, a PT -symmetricoperator is defined. The PT operator is composed of the parity and the time reversaloperators, namely i. e., P x P = − x and T i T = − i [6]. For a PT -symmetric Hamiltonian- a less general non-Hermitian Hamiltonian - the interaction potential is considered to becomplex. Hence, the unitarity for the unbroken PT -symmetry (under which the energyspectrum of the system is real) in a quantum mechanical system has to be conserved, whichis one of the prominent principles of quantum mechanics. Accordingly, in Ref. [7] theinfinitesimal probability density is considered and a method is introduced to find the pathintegral in the complex plane C . Earlier in Ref. [8], Bagchi et. al. discussed the generalizedcontinuity equation in accordance with the Schr¨odinger equation and its PT -symmetricconjugate. Furthermore, the modified normalization constant is obtained on the real x -axis.In the approach of the field theory upon applying the principle of stationary action, onefinds the equations of motion of a particle in the classical or quantum mechanical system.There exist various studies in the literature on the generalization of Lagrangian densitywhich well-describe different physical phenomenologies. According to Ref. [9], non-standardpower-law Lagrangians are discussed following the generalization of the fractional calculusof variation. In this framework, the dissipative classical and quantum dynamical systemsare not well-defined with the standard Lagrangian, whilst the fractional action-like schemerepresents a successful method to deal with such applications in physics. Nobre et. al. inRef. [10] expressed the non-linear Schr¨odinger equation upon imposing the non-linearityinto the kinetic energy. Furthermore, they found the corresponding Schr¨odinger equationby applying the variation of Lagrangian density, also, they identified the necessity of anew second field to generate the Schr¨odinger equation. The authors in Ref. [11–13] haveexpressed a set of non-Hermitian PT -symmetric Schr¨odinger equations corresponding to theposition-dependent mass in classical field theory. The probability density has been extracted2ased on the definition of the continuity equation for the linear Schr¨odinger equation in Ref.[10, 11, 13]. Rego-Monteiro et. al., in Ref. [14], have analyzed a general family of non-self-adjoint Hamiltonians corresponding to the position-dependent mass which describe somephenomenological aspects of the condensed matter. Also in Ref. [15], a Lagrangian, whichis constructed by definition of two complex fields, has been introduced. With the variationalapproach, the generalized continuity equations have been derived in the Hermitian and non-Hermitian quantum mechanical systems.Here in this paper, we introduce a generalized Lagrangian density appointed to the gen-eralized momentum operator [1, 2]. We also study the physical aspects of the Lagrangiandensity including the corresponding Schr¨odinger equation stress-energy tensor and the prob-ability density.The present paper is organized as follows. In Sec. II we introduce the generalized La-grangian density which corresponds to the extended definition of the generalized momen-tum operator. Besides that, by applying the principle of stationary action through theEuler-Lagrange equation, we obtain the generalized Hamiltonian and the correspondingSchr¨odinger equation. In Sec. III the continuity equation is obtained under the concept ofthe generalized probability and current densities. Finally, we summarize our paper in theConclusion. II. THE LAGRANGIAN DENSITY
Upon recalling the generalized momentum operator presented in Ref. [1, 2], i.eˆ p = − i ~ (cid:18) (1 + µ ) ∂ x + µ ′ (cid:19) , (1)where the auxiliary function µ ( x ) is a PT -symmetric function of x , we introduce the gen-eralized Lagrangian density to be L = i ~ φ ˙ ψ − ~ m (cid:20) (1 + µ ) φ ′ ψ ′ + (1 + µ ) µ ′ φψ ) ′ + + µ ′ φψ (cid:21) − V ( x, t ) φψ − i ~ φ ∗ ˙ ψ ∗ − ~ m (cid:20) (1 + µ ∗ ) φ ∗′ ψ ∗′ + (1 + µ ∗ ) µ ∗′ φ ∗ ψ ∗ ) ′ + µ ∗′ φ ∗ ψ ∗ (cid:21) − V ∗ ( x, t ) φ ∗ ψ ∗ . (2)Herein, a dot and a prime stand for the derivative with respect to t and x , respectively, and* implies the complex conjugate. We note that, following [10–15], the Lagrangian density32) is constructed using two field functions φ and ψ and their complex conjugate, such thatby choosing φ = ψ ∗ and µ = 0 with a real interaction potential V ( x, t ), it reduces to thestandard Schr¨odinger Lagrangian density which is given by L = i ~ (cid:16) ψ ∗ ˙ ψ − ψ ˙ ψ ∗ (cid:17) − ~ m ψ ∗′ ψ ′ − V ( x, t ) ψψ ∗ . (3)Referring to (2) the potential is assumed to be PT -symmetric i. e. V ( x, t ) = V ∗ ( − x, t ).Variation of the action with respect to φ and ψ yields the field equations which are expressedas i ~ ˙ ψ = − ~ m (cid:18) (1 + µ ) ∂ x + 2 (1 + µ ) µ ′ ∂ x + µ ′′ µ ) + µ ′ (cid:19) ψ + V ( x, t ) ψ (4)and − i ~ ˙ φ = − ~ m (cid:18) (1 + µ ) ∂ x + 2 (1 + µ ) µ ′ ∂ x + µ ′′ µ ) + µ ′ (cid:19) φ + V ( x, t ) φ, (5)respectively. Eq. (4) is the generalized Schr¨odinger equation studied in Ref. [1, 2] while (5)is the PT conjugate equation of (4) provided φ = ψ PT . Therefore, the field equations (4)and (5)can be summarized as i ~ ˙ ψ = ˆ Hψ − i ~ ˙ φ = ˆ Hφ , (6)where the Hamiltonian operator ˆ H is defined to beˆ H = ˆ H PT = − ~ m (cid:18) (1 + µ ) ∂ x + 2 (1 + µ ) µ ′ ∂ x + µ ′′ µ ) + µ ′ (cid:19) . (7)Moreover, the field equations corresponding to ψ ∗ and φ ∗ are obtained to be − i ~ ˙ ψ ∗ = − ~ m (cid:18) (1 + µ ∗ ) ∂ x + 2 (1 + µ ∗ ) µ ′∗ ∂ x + µ ∗′′ µ ∗ ) + µ ∗′ (cid:19) ψ ∗ + V ∗ ( x, t ) ψ ∗ (8)and i ~ ˙ φ ∗ = − ~ m (cid:18) (1 + µ ∗ ) ∂ x + 2 (1 + µ ∗ ) µ ′∗ ∂ x + µ ∗′′ µ ∗ ) + µ ∗′ (cid:19) φ ∗ + V ∗ ( x, t ) φ ∗ . (9)Next, using the Lagrangian density (2) and the definition of Hamiltonian density H = Σ σ π σ ˙ f σ − L , (10)where f σ ∈ { ψ, ψ ∗ , φ, φ ∗ } and π σ is the momentum-density conjugate to f σ we calculate theexplicit form of H . To do so, let’s calculate π σ as4 π ψ = ∂ L ∂ ˙ ψ = i ~ φ, π φ = ∂ L ∂ ˙ φ = 0 ,π ψ ∗ = ∂ L ∂ ˙ ψ ∗ = i ~ φ ∗ , π φ ∗ = ∂ L ∂ ˙ φ ∗ = 0 . (11)After the substitution into Eq. (10), the one dimensional Hamiltonian density is obtainedto be H = ~ m (cid:20) (1 + µ ) φ ′ ψ ′ + µ ′ (1 + µ )2 ( φψ ) ′ + µ ′ φψ (cid:21) + 12 V ( x, t ) φψ + (12) ~ m (cid:20) (1 + µ ∗ ) φ ∗′ ψ ∗′ + µ ∗′ (1 + µ ∗ )2 ( φ ∗ ψ ∗ ) ′ + µ ∗′ φ ∗ ψ ∗ (cid:21) + 12 V ∗ ( x, t ) φ ∗ ψ ∗ . Having hamiltonian density found, we apply E = R H dx to find the energy of the system.Explicitly, one finds E = Z (cid:20) − ~ m ψ (cid:18) (1 + µ ) ∂ x + 2 µ ′ (1 + µ ) ∂ x + µ ′′ µ ) + µ ′ (cid:19) φ + ψ V ( x, t ) φ (cid:21) dx + Z (cid:20) − ~ m φ (cid:18) (1 + µ ) ∂ x + 2 µ ′ (1 + µ ) ∂ x + µ ′′ µ ) + µ ′ (cid:19) ψ + φ V ( x, t ) ψ (cid:21) dx + Z (cid:20) − ~ m ψ ∗ (cid:18) (1 + µ ∗ ) ∂ x + 2 µ ∗′ (1 + µ ∗ ) ∂ x + µ ∗′′ µ ∗ ) + µ ∗′ (cid:19) φ ∗ + 14 ψ ∗ V ∗ ( x, t ) φ ∗ (cid:21) dx + Z (cid:20) − ~ m φ ∗ (cid:18) (1 + µ ∗ ) ∂ x + 2 µ ∗′ (1 + µ ∗ ) ∂ x + µ ∗′′ µ ∗ ) + µ ∗′ (cid:19) ψ ∗ + 14 φ ∗ V ∗ ( x, t ) ψ ∗ (cid:21) dx. Now, we supposes that the field ψ is PT -symmetric i. e., ψ ( x ) = ψ ∗ ( − x ) and φ = ψ .Hence, the energy reduces to E = 12 (cid:18)Z ψ ˆ Hψdx + Z ψ ∗ ˆ H † ψ ∗ dx (cid:19) , (13)and upon considering [ H, PT ] = 0 the two terms become identical such that E = Z ψ ˆ Hψdx = Z ψ PT ˆ Hψdx = D ˆ H E , (14)in which D ˆ H E is the expectation value of H in PT -symmetric quantum mechanics.The next quantity which can be discussed is the stress-energy tensor. In this line, first,we obtain the energy flux density corresponding to the Cartesian coordinate which is definedby S = ˙ ψ ∂ L ∂ψ ′ + ˙ φ ∂ L ∂φ ′ + ˙ ψ ∗ ∂ L ∂ψ ∗′ + ˙ φ ∗ ∂ L ∂φ ∗′ . (15)The explicit calculation reveals that 5 = − ~ m (cid:20) (1 + µ ) (cid:16) ˙ ψφ ′ + ˙ φψ ′ (cid:17) + (1 + µ ) µ ′ ψφ ) (cid:21) + − ~ m (cid:20) (1 + µ ∗ ) (cid:16) ˙ ψ ∗ φ ∗′ + ˙ φ ∗ ψ ∗′ (cid:17) + (1 + µ ∗ ) µ ∗′ ψ ∗ φ ∗ ) (cid:21) . (16)Second, we obtain the momentum density which is defined by P = ψ ′ ∂ L ∂ ˙ ψ + φ ′ ∂ L ∂ ˙ φ + ψ ∗′ ∂ L ∂ ˙ ψ ∗ + φ ∗′ ∂ L ∂ ˙ φ ∗ . (17)Our detailed calculations lead to P = i ~ ψ ′ φ − ψ ∗′ φ ∗ ) . (18)Let’s remind that the momentum density implies the amount of energy in a unit of volumepassing through a surface in the unit of time. It represents the physical spatial momentacorresponding to a field mutually with the canonical momentum of a quantum particle. III. CONTINUITY EQUATION
Next, we utilize (4) and (5) and their complex conjugates (8) and (9) to find the continuityequation. Let’s multiply by φ , φ ∗ , ψ and ψ ∗ from the left the equations (4), (8), (5) and(9), respectively. Then by simple addition and subtraction of the results, we obtain i ~ ∂ t ( φψ + φ ∗ ψ ∗ ) = − ~ m ∂ x (cid:0) (1 + µ ) ( ψφ ′ − φψ ′ ) + (1 + µ ∗ ) ( φ ∗ ψ ∗′ − ψ ∗ φ ∗′ ) (cid:1) . (19)This is the continuity equation provided we define ρ = φψ + φ ∗ ψ ∗ , (20)to be the probability density and j = ~ im (cid:0) (1 + µ ) ( φψ ′ − ψφ ′ ) − (1 + µ ∗ ) (cid:0) φ ∗ ψ ∗′ − ψ ∗ φ ∗ (cid:1)(cid:1) , (21)to be the particle current density. Hence the continuity equation ∂ρ ( x, t ) ∂t + ∂j ( x, t ) ∂x = 0 (22)6olds. We would like to comment that the present outcomes in (20) and (21) are significantsince the conservation of probability density is confirmed. We note that with ψ ( x, t ) = ψ PT ( x, t ) = φ ( x, t ), the particle current density vanishes and consequently, ddt Z dxρ = 0 , (23)which is the conservation of the total probability on x ∈ R [8]. IV. CONCLUSION
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