Comment on: “On the problem of an electron scattering in an arbitrary one-dimensional potential field”

23 October 2000

Physics Letters A 275 Ž2000. 493–494 www.elsevier.nlrlocaterpla

Comment on: AOn the problem of an electron scattering in an arbitrary one-dimensional potential fieldB wPhys. Lett. A 265 ž2000/ 294–296x q V. Gasparian Department of Physics, UniÕersity of Murcia, 30071 Murcia, Spain Received 7 April 2000; received in revised form 25 August 2000; accepted 31 August 2000 Communicated by L.J. Sham

In a recent article w1x, Sedrakian and Khachatrian ŽSK. have discussed the problem of an electron in an arbitrary one-dimensional chain when the potential energy is the sum of N individual potentials: N

VŽ x. s

Ý Vn Ž x y x n . .

Ž 1.

ns1

Using the method of the characteristic determinant, developed in Refs. w2–7x for any kind of one-dimensional potential with and without an additional homogeneous electric field, SK showed that the recurrence relation for the inverse complex amplitude of transmission DŽ x . s 1rt Ž x . can be rewritten as a second order differential equation Žsee Eq. Ž14. of Ref. w1x. Ž " s 2 m 0 s 1 and E s k 2 .: d2 dx

ž

D y 2 ik q 2

1

dV Ž x .

VŽ x.

dx

/

d dx

d dx d dx

rŽ x. s

1 2 ik

DŽ x . s y

2

V Ž x . eyi k x q r Ž x . e i k x , 1 2 ik

Ž 3.

DŽ x . V Ž x . 1 q r Ž x . e2 ik x .

Ž 4. D

y V Ž x . D s 0.

Ž 2.

This is the central result of the article w1x. It turns out that SK are not aware of the papers by Calogero and by Babikov Žsee e.g., Refs. w8–10x and

q

references therein.. Otherwise it would not make any sense in deriving Eq. Ž2., which is well known for almost 35 years. In the mentioned Refs. of Calogero and Babikov w8–10x, using the variable phase approach to potential scattering, it was shown that the complex reflection amplitude r Ž x . and the inverse transmission amplitude DŽ x . of an electron incident, e.g., from the right onto an arbitrary V Ž x . Žconfined to a finite segment yL - x - 0. obeyed the following well known first order differential equations:

PII of original article S0375-9601Ž99.00903-2 E-mail address: [email protected] ŽV. Gasparian..

Eq. Ž2. simply follows from the above equations if one takes the derivative with respect to the x coordinate of Ž4. and makes use of Ž3.. Thus from the point of view of Eqs. Ž3. and Ž4., Eq. Ž2. is trivial and does not contain any extra information. More, for some especial cases, when the potential V Ž x . has a infinite or finite discontinuity Že.g. when the potential is the set of an arbitrary N Ž .. arranged delta functions V Ž x . s Ý ns 1Vn d x y x n it is much easier to deal with the first order differen-

0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 5 9 3 - 4

494

V. Gasparianr Physics Letters A 275 (2000) 493–494

tial equations Ž3. and Ž4. Žfor details see w9x. rather than with Eq. Ž2..

References w1x D.M. Sedrakian, A.Zh. Khachatrian, Phys. Lett. A 265 Ž2000. 294. w2x V.M. Gasparian, B.L. Altshuller, A. G Aronov, Z.H. Kasamanian, Phys. Lett. A 132 Ž1988. 201. w3x A.G. Aronov, V.M. Gasparian, U. Gummich, J. Phys. Condens. Matter 3 Ž1991. 3023.

w4x V. Gasparian, M. Ortuno, ˜ J. Ruiz, E. Cuevas, M. Pollak, Phys. Rev. B 51 Ž1995. 6743. w5x P. Carpena, V. Gasparian, M. Ortuno, ˜ Phys. Rev. B 51 Ž1995. 12813. w6x P. Carpena, V. Gasparian, M. Ortuno, ˜ Z. Phys. B 102 Ž1997. 425. w7x P. Carpena, V. Gasparian, M. Ortuno, ˜ Eur. Phys. J. B 8 Ž1999. 635. w8x F. Calogero, Variable Phase Approach to Potential Scattering, Acad. Press, New York, London, 1967. w9x V.V. Babikov, Metod Fazovikh Funktsii v Kvantovoi Mexanike, Izd. Nauka, Moscow, 1976. w10x V.V. Babikov, Usp. Fizich. Nauk 92 Ž1967. 3.