We present the quantum and classical mechanics formalisms for a particle with a position-dependent mass in the context of a deformed algebraic structure (named κ-algebra), motivated by the Kappa-statistics. From this structure, we obtain deformed versions of the position and momentum operators, which allow us to define a point canonical transformation that maps a particle with a constant mass in a deformed space into a particle with a position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews–Lakshmanan oscillator, exhibiting uncertainty relations depending on the deformation.
REFERENCES
1.
S.
Hossenfelder
, Living Rev. Relativ.
16
, 2
(2013
).2.
A.
Kempf
, J. Math. Phys.
35
, 4483
(1994
).3.
R. N.
Costa Filho
, J. P. M.
Braga
, J. H. S.
Lira
, and J. S.
Andrade
, Jr., Phys. Lett. B
755
, 367
(2016
).4.
O.
von Roos
, Phys. Rev. B
27
, 7547
(1983
).5.
J. R. F.
Lima
, M.
Vieira
, C.
Furtado
, F.
Moraes
, and C.
Filgueiras
, J. Math. Phys.
53
, 072101
(2012
).6.
M. L.
Glasser
, Phys. Lett. A
384
, 126277
(2020
).7.
B. G.
da Costa
, I. S.
Gomez
, and M. A. F.
dos Santos
, Europhys. Lett.
129
, 10003
(2020
).8.
R.
Khordad
, Physica B
406
, 3911
(2011
).9.
F.
Arias de Saavedra
, J.
Boronat
, A.
Polls
, and A.
Fabrocini
, Phys. Rev. B
50
, 4248
(1994
).10.
M.
Barranco
, M.
Pi
, S. M.
Gatica
, E. S.
Hernández
, and J.
Navarro
, Phys. Rev. B
56
, 8997
(1997
).11.
N.
Aquino
, G.
Campoy
, and H.
Yee-Madeira
, Chem. Phys. Lett.
296
, 111
(1998
).12.
H. A.
Bethe
, Phys. Rev. Lett.
56
, 1305
(1986
).13.
K.
Bencheikh
, K.
Berkane
, and S.
Bouizane
, J. Phys. A: Math. Gen.
37
, 10719
(2004
).14.
H. R.
Christiansen
and M. S.
Cunha
, J. Math. Phys.
55
, 092102
(2014
).15.
O.
Cherroud
, S.-A.
Yahiaoui
, and M.
Bentaiba
, J. Math. Phys.
58
, 063503
(2017
).16.
O.
Aydoǧdu
, A.
Arda
, and R.
Sever
, J. Math. Phys.
53
, 042106
(2012
).17.
M. F.
Rañada
, Phys. Lett. A
380
, 2204
(2016
).18.
M.
Alimohammadi
, H.
Hassanabadi
, and S.
Zare
, Nucl. Phys. A
960
, 78
(2017
).19.
A. G. M.
Schmidt
and A. L.
de Jesus
, J. Math. Phys.
59
, 102101
(2018
).20.
A. L.
de Jesus
and A. G. M.
Schmidt
, J. Math. Phys.
60
, 122102
(2019
).21.
D. O.
Richstone
and M. D.
Potter
, Astrophys. J.
254
, 451
(1982
).22.
P. M.
Mathews
and M.
Lakshmanan
, Q. Appl. Math.
32
, 215
(1974
).23.
P. M.
Mathews
and M.
Lakshmanan
, II Nuovo Cimento A
26
, 299
(1975
).24.
A. K.
Tiwari
, S. N.
Pandey
, M.
Senthilvelan
, and M.
Lakshmanan
, J. Math. Phys.
54
, 053506
(2013
).25.
B.
Bagchi
, A.
Ghose Choudhury
, and P.
Guha
, J. Math. Phys.
56
, 012105
(2015
).26.
J. F.
Cariñena
, M. F.
Rañada
, and M.
Santander
, Ann. Phys.
322
, 434
(2007
).27.
N.
Amir
and S.
Iqbal
, J. Math. Phys.
56
, 062108
(2015
).28.
V.
Chithiika Ruby
, V. K.
Chandrasekar
, M.
Senthilvelan
, and M.
Lakshmanan
, J. Math. Phys.
56
, 012103
(2015
).29.
A.
Schulze-Halberg
and B.
Roy
, J. Math. Phys.
57
, 102103
(2016
).30.
S.
Karthiga
, V.
Chithiika Ruby
, M.
Senthilvelan
, and M.
Lakshmanan
, J. Math. Phys.
58
, 102110
(2017
).31.
A.
Khlevniuk
and V.
Tymchyshyn
, J. Math. Phys.
59
(8
), 082901
(2018
).32.
R.
Bravo
and M. S.
Plyushchay
, Phys. Rev. D
93
, 105023
(2016
).33.
O.
Mustafa
, Phys. Lett. A
384
, 126265
(2020
).34.
N.
Amir
and S.
Iqbal
, J. Math. Phys.
57
, 062105
(2016
).35.
B.
Midya
and B.
Roy
, J. Phys. A: Math. Theor.
42
(28
), 285301
(2009
).36.
S.
Karthiga
, V.
Chithiika Ruby
, and M.
Senthilvelan
, Phys. Lett. A
382
(25
), 1645
(2018
).37.
N.
Amir
and S.
Iqbal
, Commun. Theor. Phys.
66
, 615
(2016
).38.
M.
Tchoffo
, F. B.
Migueu
, M.
Vubangsi
, and L. C.
Fai
, Heliyon
5
(9
), e02395
(2019
).39.
V.
Chithiika Ruby
and M.
Senthilvelan
, J. Math. Phys.
51
, 052106
(2010
).40.
J.-M.
Lévy-Leblond
, Phys. Rev. A
52
, 1845
(1995
).41.
D. J.
BenDaniel
and C. B.
Duke
, Phys. Rev.
152
, 683
(1966
).42.
T.
Gora
and F.
Williams
, Phys. Rev.
177
, 1179
(1969
).43.
Q.-G.
Zhu
and H.
Kroemer
, Phys. Rev. B
27
, 3519
(1983
).44.
T. L.
Li
and K. J.
Kuhn
, Phys. Rev. B
47
, 12760
(1993
).45.
R. A.
Morrow
and K. R.
Brownstein
, Phys. Rev. B
30
, 678
(1984
).46.
O.
Mustafa
and S. H.
Mazharimousavi
, Int. J. Theor. Phys.
46
, 1786
(2007
).47.
A.
de Souza Dutra
and C. A. S.
Almeida
, Phys. Lett. A
275
, 25
(2000
).48.
R. N.
Costa Filho
, M. P.
Almeida
, G. A.
Faria
, and J. S.
Andrade
, Jr., Phys. Rev. A
84
, 050102(R)
(2011
).49.
S. H.
Mazharimousavi
, Phys. Rev. A
85
, 034102
(2012
).50.
R. N.
Costa Filho
, G.
Alencar
, B.-S.
Skagerstam
, and J. S.
Andrade
, Jr., Europhys. Lett.
101
, 10009
(2013
).51.
M. A.
Rego-Monteiro
and F. D.
Nobre
, Phys. Rev. A
88
, 032105
(2013
).52.
B. G.
da Costa
and E. P.
Borges
, J. Math. Phys.
55
, 062105
(2014
).53.
A.
Arda
and R.
Sever
, Few-Body Syst.
56
, 697
(2015
).54.
J. P. M.
Braga
and R. N.
Costa Filho
, Int. J. Mod. Phys. C
27
, 1650047
(2016
).55.
B. G.
da Costa
and E. P.
Borges
, J. Math. Phys.
59
, 042101
(2018
).56.
B. G.
da Costa
and I. S.
Gomez
, Physica A
541
, 123698
(2020
).57.
E. P.
Borges
, Physica A
340
, 95
(2004
).58.
C.
Tsallis
, Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World
(Springer
, New York
, 2009
).59.
G.
Kaniadakis
, Physica A
296
, 405
(2001
).60.
G.
Kaniadakis
, Phys. Rev. E
66
, 056125
(2002
).61.
G.
Kaniadakis
, Phys. Rev. E
72
, 036108
(2005
).62.
A. M.
Scarfone
, H.
Matsuzoe
, and T.
Wada
, Phys. Lett. A
380
(38
), 3022
(2016
).63.
G.
Kaniadakis
, Eur. Phys. J. A
40
(3
), 275
(2009
).64.
S.
Abe
, G.
Kaniadakis
, and A. M.
Scarfone
, J. Phys. A: Math. Gen.
37
, 10513
(2004
).65.
G.
Kaniadakis
, A. M.
Scarfone
, A.
Sparavigna
, and T.
Wada
, Phys. Rev. E
95
(5
), 052112
(2017
).66.
A. M.
Scarfone
, Entropy
17
, 2812
(2015
).67.
G.
Kaniadakis
, Entropy
15
(10
), 3983
(2013
).68.
A. M.
Scarfone
, Physica A
480
, 63
(2017
).69.
T.
Wada
and H.
Suyari
, Phys. Lett. A
348
, 89
(2006
).70.
T.
Wada
and H.
Suyari
, Entropy
15
(12
), 5144
(2013
).71.
N. T. C. M.
Souza
, D. H. A. L.
Anselmo
, V. D.
Mello
, and R.
Silva
, Phys. Lett. A
378
, 1691
(2014
).72.
J. L. E.
da Silva
, G. B.
da Silva
, and R. V.
Ramos
, Phys. Lett. A
384
(8
), 126175
(2020
).73.
A. M.
Scarfone
, H.
Matsuzoe
, and T.
Wada
, Entropy
20
(6
), 436
(2018
).74.
G.
Kaniadakis
, M.
Lissia
, and A. M.
Scarfone
, Phys. Rev. E
71
(4
), 046128
(2005
).75.
G.
Lapenta
, S.
Markidis
, A.
Marocchino
, and G.
Kaniadakis
, Astrophys. J.
666
, 949
(2007
).76.
J. C.
Carvalho
, R.
Silva
, J. D.
do Nascimento
, Jr., B. B.
Soares
, and J. R.
de Medeiros
, Europhys. Lett.
91
, 69002
(2010
).77.
G.
Livadiotis
, Europhys. Lett.
113
, 10003
(2016
).78.
T.
Wada
, Eur. Phys. J. B
73
, 287
(2010
).79.
F.
Clementi
, T.
Di Matteo
, M.
Gallegati
, and G.
Kaniadakis
, Physica A
387
, 3201
(2008
).80.
M.
Stella
and M.
Brede
, Physica A
407
, 360
(2014
).81.
M. O.
Costa
, R.
Silva
, D. H. A. L.
Anselmo
, and J. R. P.
Silva
, Phys. Rev. E
99
(2
), 022112
(2019
).82.
K.
Ourabah
and M.
Tribeche
, Phys. Rev. E
89
(6
), 062130
(2014
).83.
K.
Ourabah
, A. H.
Hamici-Bendimerad
, and M.
Tribeche
, Phys. Scr.
90
, 045101
(2015
).84.
I. S.
Gradshteyn
and I. M.
Ryzhik
, Table of Integrals, Series, and Products
(Academic Press
, 2014
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
