Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided.
REFERENCES
1.
D. J.
Griffiths
, Introduction to Quantum Mechanics
(
Cambridge U. P
.,
Cambridge
, 2017
).2.
R.
Clausius
, “
XVI. On a mechanical theorem applicable to heat
,” London, Edinburgh, Dublin Philos. Mag. J. Sci.
40
, 122
–127
(1870
).3.
V.
Fock
, “
Bemerkung zum Virialsatz
,” Z. Phys.
63
, 855
–858
(1930
).4.
M. A.
Maize
,
M. A.
Antonacci
, and
F.
Marsiglio
, “
The static electric polarizability of a particle bound by a finite potential well
,” Am. J. Phys.
79
, 222
–225
(2011
).5.
B. J.
Riel
, “
An introduction to self-assembled quantum dots
,” Am. J. Phys.
76
, 750
–757
(2008
).6.
R. J.
Olsen
and
G.
Vignale
, “
The quantum mechanics of electric conduction in crystals
,” Am. J. Phys.
78
, 954
–960
(2010
).7.
D. M.
Mitnik
,
J.
Randazzo
, and
G.
Gasaneo
, “
Endohedrally confined helium: Study of mirror collapses
,” Phys. Rev. A
78
, 062501
(2008
).8.
G.
Allen
, “
Band structures of one-dimensional crystals with square-well potentials
,” Phys. Rev.
91
, 531
–533
(1953
).9.
C. I.
Mendoza
,
G. J.
Vazquez
,
M.
del Castillo-Mussot
, and
H.
Spector
, “
Stark effect dependence on hydrogenic impurity position in a cubic quantum box
,” Phys. Rev. B
71
, 075330
(2005
).10.
M.
Belloni
and
R. W.
Robinett
, “
Quantum mechanical sum rules for two model systems
,” Am. J. Phys.
76
, 798
–806
(2008
).11.
V. G.
Gueorguiev
,
A. R. P.
Rau
, and
J. P.
Draayer
, “
Confined one-dimensional harmonic oscillator as a two-mode system
,” Am. J. Phys.
74
, 394
–403
(2006
).12.
J.
Milton Pereira
,
V.
Mlinar
,
F. M.
Peeters
, and
P.
Vasilopoulos
, “
Confined states and direction-dependent transmission in graphene quantum wells
,” Phys. Rev. B
74
, 045424
(2006
).13.
S.
Bandopadhyay
,
B.
Dutta-Roy
, and
H. S.
Mani
, “
Understanding the Fano resonance through toy models
,” Am. J. Phys.
72
, 1501
–1507
(2004
).14.
P. L.
Garrido
,
S.
Goldstein
,
J.
Lukkarinen
, and
R.
Tumulka
, “
Paradoxical reflection in quantum mechanics
,” Am. J. Phys.
79
, 1218
–1231
(2011
).15.
J.
Gea-Banacloche
, “
Splitting the wave function of a particle in a box
,” Am. J. Phys.
70
, 307
–312
(2002
).16.
S. A.
Fulling
and
K. S.
Güntürk
, “
Exploring the propagator of a particle in a box
,” Am. J. Phys.
71
, 55
–63
(2003
).17.
C.
Duffin
and
A. G.
Dijkstra
, “
Controlling a quantum system via its boundary conditions
,” Eur. Phys. J. D
73
, 221
–226
(2019
).18.
R. M.
Dimeo
, “
Wave packet scattering from time-varying potential barriers in one dimension
,” Am. J. Phys.
82
, 142
–152
(2014
).19.
M.
Belloni
and
R. W.
Robinett
, “
The infinite well and Dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics
,” Phys. Rep.
540
, 25
–122
(2014
).20.
J. O.
Hirschfelder
and
J. F.
Kincaid
, “
Application of the virial theorem to approximate molecular and metallic eigenfunctions
,” Phys. Rev.
52
, 658
–661
(1937
).21.
J.
Killingbeck
, “
Methods of proof and applications of the virial theorem in classical and quantum mechanics
,” Am. J. Phys.
38
, 590–596
(1970
).22.
Z.
Dongpei
, “
Proof of the quantum virial theorem
,” Am. J. Phys.
54
, 267–270
(1986
).23.
R. M.
Schectman
and
R. H.
Good
, Jr.
, “
Generalizations of the virial theorem
,” Am. J. Phys.
25
, 219–225
(1957
).24.
A. G.
McLellan
, “
Virial theorem generalized
,” Am. J. Phys.
42
, 239
–243
(1974
).25.
E.
Weislinger
and
G.
Olivier
, “
The classical and quantum mechanical virial theorem
,” Int. J. Quantum Chem. Symp.
8
, 389
–401
(1974
).26.
E.
Weislinger
and
G.
Olivier
, “
The virial theorem with boundary conditions applications to the harmonic oscillator and to sine-shaped potentials
,” Int. J. Quantum Chem.
9
, 425
–433
(1975
).27.
E.
Weislinger
and
G.
Olivier
, “
Applications of a quantum virial theorem to Kronig and Penney's model and to a diatomic molecule in static approximation
,” Int. J. Quantum Chem.
10
, 225
–231
(1976
).28.
H.
Kalf
, “
The virial theorem in relativistic quantum mechanics
,” J. Funct. Anal.
21
, 389
–396
(1976
).29.
H.
Kalf
, “
The quantum mechanical virial theorem and the absence of positive energy bound states of Schrödinger operators
,” Israel J. Math.
20
, 57
–69
(1975
).30.
M.
Guilhem
and
W. C.
McMillan
, “
The virial theorem
,” in Advances in Chemical Physics
, edited by
I.
Prigigine
and
S.
Rice
(
John Wiley and Sons
,
New York
, 1985
), Vol.
LVIII
.31.
R. F. W.
Bader
, “
A quantum theory of molecular structure and its applications
,” Chem. Rev.
91
, 893
–928
(1991
).32.
Y.
İpekoğlu
and
S.
Turgut
, “
An elementary derivation of the quantum virial theorem from Hellmann–Feynman theorem
,” Eur. J. Phys.
37
, 045405
(2016
).33.
J. R.
Walton
,
L. A.
Rivera-Rivera
,
R. R.
Lucchese
, and
J. W.
Bevan
, “
Canonical approaches to applications of the virial theorem
,” J. Phys. Chem. A
120
, 817
–823
(2016
).34.
J.
Sivardiere
, “
Using the virial theorem
,” Am. J. Phys.
54
, 1100
–1103
(1986
).35.
P. D.
Robinson
and
J. O.
Hirschfelder
, “
Virial theorem and its generalizations in scattering theory
,” Phys. Rev.
129
, 1391
–1396
(1963
).© 2020 American Association of Physics Teachers.
2020
American Association of Physics Teachers
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
