Klein Paradox in Bosons

Transcription

1 Klein Paradox in Bosons arxiv:quant-ph/ v1 18 Feb 2003 Partha Ghose and Manoj K. Samal S. N. Bose National Centre for Basic Sciences,Block JD, Sector III, Salt Lake, Kolkata Animesh Datta Department of Electrical Engineering, Indian Institute of Technology, Kanpur, March 22, 2008 Abstract We give yet another confirmation of the fact that negative energy states cannot exist in the realm of bosons. We show analytically that the Zitterbewegung and Klein Paradox, such well known aspects of the Dirac Equation are not found in the case of Bosons. We use the Kemmer- Duffin- Harish Chandra formalism with β matrices to arrive at our results. 1 Introduction In discussing problems and interactions in which the electron (or any fermion) is spread out over distances large compared to its Compton wavelength, we may simply ignore the existence of the uninterpreted negative-energy solutions of the one-particle relativistic Dirac Equation [1] and hope of obtaining physically sensible and accurate results. This will not wirk, however, in situations which find electrons (or fermions) localized to distances comparable with h mc. The negative frequency amplitudes will then be appreciable, then the Zitterbewegung [2] terms in the current are important and indeed we shall find ourselves beset by paradoxes and dilemmas which defy interpretation within the framework of the one particle Dirac equation. A celebrated example of these difficulties is the Klein Paradox [3] for electrons, and in general for fermions. It is discussed in detail in numerous texts [4],[5] and resolved using the hole-theory which provides the treatment of the negative energy states which had deep-rooted implications. What the actual paradox is that in the situations under question, the coefficient of reflection for an electron beam from a barrier is more than one. In this paper we address the same problem for bosons. We explicitly show that the reflection coefficient is less than one and hence these scope of any paradox is nonexistant. Corresponding author: animesh.datta@iitk.ac.in 1

2 2 Formalism In order to address the problem of Klein Paradox, a single-particle relativistic quantum mechanics is necessary. Such a theory is provided for massive spin-0 and spin-1 bosons by one of us in [6] and also massless spin-1 bosons i.e., photons by the Kemmer- Duffin- HarishChandra formalism. It has been shown [6] that a conserved four-current with a positive definite time component does exist for relativistic bosons, and is associated not with charge current but the flow of energy. It is based on the first order Kemmer equation [7] where the β matrices satisfy the algebra (i hβ µ µ + m 0 c)ψ = 0 (1) β µ β ν β λ + β λ β ν β µ = β µ g νλ + β λ g νµ. (2) These are matrices for spin 1 and 5 5 for spin 0 partcles. Eqn.1 leads to where and i h Ψ t = [ i hc β i i m 0 c 2 β 0 ]Ψ (3) β i = β 0 β i β i β 0 (4) i hβ i β 2 0 i Ψ = m 0 c(1 β 2 0)Ψ. (5) S i = Ψ βi Ψ is the current density in the i direction while S 0 = Ψ Ψ > 0 [8]. 3 Bosonic Systems 3.1 Spin 1 Massive Bosons Let Then S inc S ref Ψ inc = A m (ψ inc 1, ψ inc 2,, ψ inc 10 ) (6) Ψ ref = A m (ψ ref 1, ψ ref 2,, ψ ref 10 ) = Ψ inc β i Ψ inc Ψ ref β i Ψ ref for i = 1, 2, 3. In particular case that we treat we may take i = 1, without loss of generality. Then some algebra gives S inc S ref = B 2. If we consider the wave A 2 propagating along the positive x-axis with a potential step at x=0, Ψ inc = A m (0, 0, E 0, 0, E 0, 0, 0, 0, 0, 0) T e i(kx ωt) (7) Ψ ref = B m (0, 0, E 0, 0, E 0, 0, 0, 0, 0, 0) T e i(kx+ωt) (8) Ψ trans = C m (0, 0, E 0, 0, ǫe 0, 0, 0, 0, 0, 0) T e i(k x ωt). (9) 2

3 Matching Ψ inc + Ψ ref = Ψ trans at x = 0, t = 0, readily gives A + B = C, A B = ǫc whence B 2 = (1 ǫ) 2 A 2 (1+. Thus the ratio S ref ǫ) 2 S inc < 1 which means that no effect analogous to Klein paradox occurs in the case of Spin 1 massive bosons. 3.2 Spin 0 Massive Bosons Since S i = Ψ β i Ψ for all i and β 1 = i 0 i (10) For a wave propagating along i = 1, with x 0 = t, we have Ψ inc Ψ ref = A(k, 0, 0, ik 0, 1)e i(k 0x 0 kx) = B( k, 0, 0, ik 0, 1)e i(k 0x 0 +kx) Ψ trans = C(k, 0, 0, ik 0, 1)ei(k 0 x 0 k x) (11) which are special cases of Ψ = (φ 1, φ 2, φ 3, φ 0, φ) where φ i = i φ and φ = e ikµxµ = e i(k 0x 0 k x). Thus S ref S inc = B 2 = B A 2 A 2. The set of Eqns.[11] evaluated at x = 0, x 0 = 0 give A B = C(k /k) (12) A + B = C(k 0 /k) A + B = C Now since E 2 = h 2 c 2 (k0 2 k 2 ) + m 2 0c 4 for x < 0 and E 2 = h 2 c 2 (k 0 2 k 2 ) + m 2 0c 4 for x < 0. If the frequency of the wave does not change on changing the media which is indeed the case, then k 0 = k 0 and consequently k 2 > k 2. Finally, Eqn.[12] gives B = /k A (1 k 1+k /k )2. Hence there is no evidence of Klein Paradox in the system under question as the reflection coefficient is less than Spin 1 Massless Bosons - Photons Using the Harish Chandra formalism [9], we have for a beam of photons propagating along the positive x axis, γψ = (0, 0, E z, 0, H y, 0, 0, 0, 0, 0). (13) Let E i z = E 0 e i(kx ωt φ) (14) E r z = E 0 Re i(kx ωt+φ) E t z = E 0 Te i(k x ωt+χ). 3

4 Maxwell s Equations then at once give (using x E z = t H y ) H i y = k ω E 0e i(kx ωt φ) (15) Hy r = k ω E 0 Re i(kx+ωt+φ) H t y = k ω E 0 Te i(k x ωt χ). Now, S i = [E H] i is the (Poynting vector) current density as given by [8]. Thus from 15 and 16 ref = R. (16) inc Matching E i z Er z = Et z at x = 0, t = 0 gives at once R = Sref x inc = ( k /k 1 k /k + 1 )2 < 1. (17) Thus there is no signature of Klein Paradox in the case of photons. 4 Conclusion This paper conclusively proves that the Klein paradox does not exixt in the framework of the Kemmer-Duffin formalism for bosons. Ensuing from this one can safely conclude that although a single particle relativistic theory is impossible for fermions, that is not the case for bosons. Although there are recent results [10] using Bohmian trajectories that show that the Klein Paradox is absent in the case of outgoing scattering asymptotics. This basically stems from the fact that bosons have nothing called negative energy states or vaccum energy levels. This explains the stability of bosonic matter and the resulting consequences. References [1] P. A. M. Dirac, Proc. Roy. Soc. (London), A 117, 610 (1928), ibid A 118, 351 (1928). Principles of Quantum Mechanics, op. cit. [2] E. Schrodinger, Sitzber. Prevss. Akad. Wiss. Physik-Math, 24, 418 (1930). [3] O. Klein, Z. Physik, 53, 157 (1929). [4] J. D. Bjorken, S. D. Drell, Relativistic Quantum Mechanics, McGraw Hill, New York, [5] W. Greiner, Relativistic Quantum Mechanics, Springer- Verlag, Berlin, [6] P. Ghose, Found. of Physics, 26 (11), 1441, [7] N. Kemmer, Proc. Roy. Soc A, 173, 91,

5 [8] P. Ghose, M. K. Samal, Phys. Rev. E, , [9] Harish Chandra, Proc. Roy. Soc(London), 186, 502, [10] G. Grubl et al, J. Phys. A: Math, Gen, 34, (2001). 5