The “particle-in-a-box” problem is investigated for a relativistic particle obeying the Klein–Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for “shallow” wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wave function outside the well vanishes, and Klein tunneling is suppressed: Quantization is, thus, recovered, similar to the non-relativistic particle in a box. In addition, we show how wave packets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way.

1.
N.
Dombey
and
A.
Calogeracos
,
Phys. Rep.
315
,
41
58
(
1999
).
2.
A.
Wachter
,
Relativistic Quantum Mechanics
(
Springer
,
Dordrecht
,
2011
), Sec. 1.3.4.
3.
P.
Strange
,
Relativistic Quantum Mechancis
(
Cambridge U. P.
,
Cambridge
,
1998
).
4.
W.
Greiner
,
Relativistic Quantum Mechanics
, 3rd ed. (
Springer
,
Berlin
,
2000
).
5.
P.
Alberto
,
C.
Fiolhais
, and
V. M. S.
Gil
,
Eur. J. Phys.
17
,
19
24
(
1996
).
6.
V.
Alonso
,
S.
De Vincenzo
, and
L.
Mondino
,
Eur. J. Phys.
18
,
315
320
(
1997
).
7.
P.
Alberto
,
S.
Das
, and
E. C.
Vagenas
,
Eur. J. Phys.
39
,
025401
(
2018
).
8.
M.
Alkhateeb
,
X.
Gutierrez de la Cal
,
M.
Pons
,
D.
Sokolovski
, and
A.
Matzkin
,
Phys. Rev. A
103
,
042203
(
2021
).
9.
H.
Nitta
,
T.
Kudo
, and
H.
Minowa
,
Am. J. Phys.
67
,
966
971
(
1999
).
10.
A. O.
Barut
,
Electrodynamics and Classical Theory of Fields and Particles
(
Dover
,
New York
,
1980
), Sec. 3 of Chap. II.
11.
J. P.
Costella
,
B. H. J.
McKellar
, and
A. A.
Rawlinson
,
Am. J. Phys.
65
,
835
841
(
1997
).
12.
F. M. S.
Lima
,
Am. J. Phys.
88
,
1019
1022
(
2020
).
13.
J. E.
Beam
,
Am. J. Phys.
38
,
1395
1401
(
1970
).
14.
C. A.
Manogue
,
Ann. Phys.
181
,
261
283
(
1988
).
15.
T.
Cheng
,
M. R.
Ware
,
Q.
Su
, and
R.
Grobe
,
Phys. Rev. A
80
,
062105
(
2009
).
16.
M.
Barbier
,
F. M.
Peeters
,
P.
Vasilopoulos
, and
JMilton
Pereira
, Jr.
,
Phys. Rev. B
77
,
115446
(
2008
).
17.
X.
Gutierrez de la Cal
,
M.
Alkhateeb
,
M.
Pons
,
A.
Matzkin
, and
D.
Sokolovski
,
Sci. Rep.
10
,
19225
(
2020
).
18.
D.
Xu
,
T.
Wang
, and
X.
Xue
,
Found. Phys.
43
,
1257
1274
(
2013
).
19.
R. W.
Robinett
,
Am. J. Phys
68
,
410
420
(
2000
).
20.
S. P.
Gavrilov
and
D. M.
Gitman
,
Phys. Rev. D
93
,
045002
(
2016
).
21.
B. R.
Holstein
,
Am. J. Phys.
66
,
507
512
(
1998
).
23.
M.
Koehn
,
Europhys. Lett.
100
,
60008
(
2012
).
24.
O.
Hamidi
and
H.
Dehghan
,
Rep. Math. Phys.
73
,
11
16
(
2014
).
25.
S.
Colin
and
A.
Matzkin
,
Europhys. Lett.
130
,
50003
(
2020
).
26.
We thank an anonymous referee for suggesting this argument.
27.
Strictly speaking, a Gaussian in position space would have negative energy contributions not included in G(0,x) given by Eq. (30). Such contributions are negligible in the non-relativistic regime and become dominant in the ultra-relativistic regime. For more details in the context of barrier scattering, see Ref. 18.
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