Using the thin-layer quantization, we formulate the problem of a Schrödinger particle constrained to move along a coordinate surface of the bi-spherical coordinate system. In three-dimensional space, the free Schrödinger equation is not separable in this coordinate system. However, when we consider the equation for a particle constrained to a given surface, there are only two degrees of freedom. One has to introduce a geometrical potential to attach the particle to the surface. This well-known potential has two contributions: one from Gauss’ curvature and the other from the mean curvature. The Schrödinger equation leads to a general Heun equation. We solve it exactly and present the eigenfunctions and plots of the probability densities, and, as an application of this methodology, we study the problem of an electric charge propagating along these coordinate surfaces in the presence of a uniform magnetic field.

1.
S. I.
Rich
,
S.
Lee
,
K.
Fukuda
, and
T.
Someya
,
Adv. Mater.
34
,
2106683
(
2022
).
2.
H.
Terrones
and
M.
Terrones
,
New J. Phys.
5
,
126
(
2003
).
3.
Z.-M.
Huang
,
W.-Q.
Huang
,
X.-K.
Wu
,
S.-R.
Liu
, and
C.-J.
Qin
,
Sci. Rep.
7
,
17974
(
2017
).
4.
S. I.
Rich
,
Z.
Jiang
,
K.
Fukuda
, and
T.
Someya
,
Mater. Horiz.
8
,
1926
(
2021
).
5.
R.
Makiura
,
S.
Motoyama
,
Y.
Umemura
,
H.
Yamanaka
,
O.
Sakata
, and
H.
Kitagawa
,
Nat. Mater.
9
,
565
(
2010
).
7.
E.
Giglio
,
B.
Gervais
,
J.
Rangama
,
B.
Manil
,
B. A.
Huber
,
D.
Duft
,
R.
Müller
,
T.
Leisner
, and
C.
Guet
,
Phys. Rev. E
77
,
036319
(
2008
).
8.
G.
Derkachov
,
D.
Jakubczyk
,
K.
Kolwas
,
K.
Piekarski
,
Y.
Shopa
, and
M.
Woźniak
,
Measurement
158
,
107681
(
2020
).
9.
J.
Onoe
,
T.
Ito
,
H.
Shima
,
H.
Yoshioka
, and
S.-I.
Kimura
,
Eur. Phys. Lett.
98
,
27001
(
2012
).
10.
F.
Xu
,
H.
Yu
,
A.
Sadrzadeh
, and
B. I.
Yakobson
,
Nano Lett.
16
,
34
(
2016
).
11.
12.
B.-X.
Lei
,
L.-L.
Zeng
,
P.
Zhang
,
H.-K.
Qiao
, and
Z.-F.
Sun
,
J. Power Sources
253
,
269
(
2014
).
13.
H. N.
Hansen
,
R. J.
Hocken
, and
G.
Tosello
,
CIRP Ann.
60
,
695
(
2011
).
14.
R. C. T.
da Costa
,
Phys. Rev. A
23
,
1982
(
1981
).
15.
G.
Ferrari
and
G.
Cuoghi
,
Phys. Rev. Lett.
100
,
230403
(
2008
).
16.
S. H.
Mazharimousavi
,
Phys. Scr.
96
,
125245
(
2021
).
17.
H.
Zainuddin
,
C. K.
Tim
,
N. S.
Shamsuddin
, and
N. M.
Shah
,
J. Phys.: Conf. Ser.
795
,
012002
(
2017
).
18.
R. C. T.
da Costa
,
Phys. Rev. A
25
,
2893
(
1982
).
19.
J.
Tempere
,
I.
Silvera
, and
J.
Devreese
,
Surf. Sci. Rep.
62
,
159
(
2007
).
20.
B. S.
Kandemir
,
J. Math. Phys.
46
,
032110
(
2005
).
21.
A. G. M.
Schmidt
,
Braz. J. Phys.
50
,
419
(
2020
).
24.
M. D.
Oliveira
and
A. G. M.
Schmidt
,
Physica E
120
,
114029
(
2020
).
25.
Q.
Dong
,
S.-S.
Dong
,
E.
Hernández-Márquez
,
R.
Silva-Ortigoza
,
G.-H.
Sun
, and
S.-H.
Dong
,
Commun. Theor. Phys.
71
,
231
(
2019
).
27.
Heun’s Differental Equation
, edited by
A.
Ronveaux
(
Oxford University Press
,
1995
), In the first chapter Arscott develops the theory of general Heun functions as well as the augmented convergence.
28.
S. Yu.
Slavyanov
,
W.
Lay
, and
A.
Seeger
,
Special Functions: A Unified Theory Based on Singularities
(
Oxford University Press
,
2000
).
29.
M.
Hortaçsu
,
Adv. High Energy Phys.
2018
,
8621573
.
30.
S.
Dong
,
G.
Yáñez-Navarro
,
M. A. M.
Sanchez
,
C.
Mejía-García
,
G.-H.
Sun
, and
S.-H.
Dong
,
Adv. High Energy Phys.
2018
,
9824538
.
31.
32.
A. D.
Alhaidari
,
J. Math. Phys.
61
,
122102
(
2020
).
33.
A.
Ralko
and
T. T.
Truong
,
J. Phys. A: Math. Gen.
35
,
9573
(
2002
).
34.
S.
Bellucci
and
V.
Yeghikyan
,
J. Math. Phys.
54
,
082103
(
2013
).
35.
H.
Movasati
and
S.
Reiter
,
Bull. Braz. Math. Soc., New Series
43
,
423
(
2012
).
36.
B. D. B.
Figueiredo
,
Math. Methods Appl. Sci.
44
,
7165
(
2021
).
37.
A. M.
Ishkhanyan
,
Theor. Math. Phys.
202
,
1
(
2020
).
38.
W.
Lay
and
S. Y.
Slavyanov
,
Proc. R. Soc. London, Ser. A
455
,
4347
(
1999
).
39.
P.
Moon
and
D. E.
Spencer
,
Field Theory Handbook: Including Coordinate Systems, Differential Equations and Their Solutions
(
Springer
,
1971
).
41.
F. W. J.
Olver
,
D. W.
Lozier
,
R. F.
Boisvert
, and
C. W.
Clark
,
NIST Handbook of Mathematical Functions
(
Cambridge University Press
,
2010
), Chap. 31.
42.
A.
Erdélyi
,
W.
Magnus
,
F.
Oberhettinger
, and
F. G.
Tricomi
,
Higher Transcendental Functions
(
McGraw-Hill
,
1953
), Vol. III.
44.
G.
Kristensson
,
Second Order Differential Equations: Special Functions and Their Classifications
(
Springer
,
2010
), See the excellent chapter 8.
You do not currently have access to this content.