We develop a systematic approach to determine the large behavior of the momentum-space wave function, , of a one-dimensional quantum system for which the position-space wave function, , has a discontinuous derivative at any order. We find that if the derivative of the potential energy function for the system has a discontinuity, there is a corresponding discontinuity in at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wave function proportional to . A number of familiar pedagogical examples are examined in this context, leading to a general derivation of the result.
REFERENCES
1.
Quote attributed to
Louis
Sullivan
, “The tall office building artistically considered
,” Lippincott’s Magazine
(March 1896
)The entire quotation is available at ⟨en.wikipedia.org/wiki/Form_follows_function⟩.
2.
For example, see
R. W.
Robinett
, “Quantum and classical probability distributions for position and momentum
,” Am. J. Phys.
63
, 823
–832
(1995
);and comments by
C. C.
Real
, J. G.
Muga
, and S.
Brouard
, “Comment on ‘Quantum and classical probability distributions for position and momentum,’ by R. W. Robinett [Am. J. Phys. 63 (9), 823–832 (1995)]
,” Am. J. Phys.
65
, 157
–158
(1997
).3.
R. W.
Robinett
, “Visualizing classical and quantum probability densities for momentum using variations on familiar one-dimensional potentials
,” Eur. J. Phys.
23
, 165
–174
(2002
).4.
G.
Yoder
, “Using classical probability functions to illuminate the relation between classical and quantum physics
,” Am. J. Phys.
74
, 404
–411
(2006
).5.
M.
Lieber
, “Quantum mechanics in momentum space
,” Am. J. Phys.
43
, 486
–491
(1975
).6.
P. W.
Langhoff
, “Schrödinger particle in a gravitational well
,” Am. J. Phys.
39
, 954
–957
(1971
);R. L.
Gibbs
, “The quantum bouncer
,” Am. J. Phys.
43
, 25
–28
(1975
);R. D.
Desko
and D. J.
Bord
, “The quantum bouncer revisited
,” Am. J. Phys.
51
, 82
–84
(1983
);D. A.
Goodings
and T.
Szeredi
, “The quantum bouncer by path integral method
,” Am. J. Phys.
59
, 924
–930
(1991
);S.
Whineray
, “An energy representation approach to the quantum bouncer
,” Am. J. Phys.
60
, 948
–950
(1992
).7.
V. V.
Nesvizhevsky
et al., “Quantum states of neutrons in the Earth’s gravitational field
,” Nature
415
, 297
–299
(2002
);
[PubMed]
V. V.
Nesvizhevsky
et al., “Measurement of quantum states of neutrons in the Earth’s gravitational field
,” Phys. Rev. D
67
, 102002
-1–9 (2002
).8.
C. G.
Aminoff
, A. M.
Steane
, P.
Bouyer
, P.
Desbiolles
, J.
Dalibard
, and C.
Cohen-Tannoudji
, “Cesium atoms bouncing in a stable gravitational cavity
,” Phys. Rev. Lett.
71
, 3083
–3086
(1993
).9.
K.
Bongs
, S.
Burger
, G.
Birkl
, K.
Sengstock
, W.
Ertmer
, K.
Rzażewski
, A.
Sanpera
, and M.
Lewenstein
, “Coherent evolution of bouncing Bose–Einstein condensates
,” Phys. Rev. Lett.
83
, 3577
–3580
(1999
).10.
G.
Della Valle
, M.
Savoini
, M.
Ornigotti
, P.
Laporta
, V.
Foglietti
, M.
Finazzi
, L.
Duò
, and S.
Longhi
, “Experimental observation of a photon bouncing ball
,” Phys. Rev. Lett.
102
, 180402
-1–4 (2009
).11.
J.
Gea-Banacloche
, “A quantum bouncing ball
,” Am. J. Phys.
67
, 776
–782
(1999
).12.
O.
Vallée
, “Comment on ‘A quantum bouncing ball’ by Julio Gea-Banacloche
,” Am. J. Phys.
68
, 672
–673
(2000
).13.
D. M.
Goodmanson
, “A recursion relation for matrix elements of the quantum bouncer. Comment on ‘A quantum bouncing ball’ by Julio Gea-Banacloche
,” Am. J. Phys.
68
, 866
–868
(2000
).14.
M.
Belloni
and R. W.
Robinett
, “Constraints on Airy function zeros from quantum-mechanical sum rules
,” J. Phys. A: Math. Theor.
42
, 075203
-1–11 (2009
).15.
R. W.
Robinett
, “The Stark effect in linear potentials
,” Eur. J. Phys.
31
, 1
–13
(2010
).16.
J. R.
Albright
, “Integrals of products of Airy functions
,” J. Phys. A
10
, 485
–490
(1977
);R. G.
Gordon
, “New method for constructing wavefunctions for bound states and scattering
,” J. Chem. Phys.
51
, 14
–25
(1969
), Appendix B.17.
D.
Branson
, “Continuity conditions on Schrödinger wave functions at discontinuities of the potential
,” Am. J. Phys.
47
, 1000
–1003
(1979
).18.
C.
Aslangul
, “ well with reflecting barrier
,” Am. J. Phys.
63
, 935
–940
(1995
).19.
K. R.
Brownstein
, “Calculation of a bound state wavefunction using free wavefunctions only
,” Am. J. Phys.
43
, 173
–176
(1975
);W. C.
Damert
, “Completeness of the energy eigenstates for a delta function potential
,” Am. J. Phys.
43
, 531
–534
(1975
);S. H.
Patil
, “Completeness of the energy eigenfunctions for the one-dimensional -function potential
,” Am. J. Phys.
68
, 712
–714
(2000
).20.
O. A.
Ayorinde
, K.
Chisholm
, M.
Belloni
, and R. W.
Robinett
, “New identities from quantum mechanical sum rules of parity-related potentials
,” J. Phys. A: Math. Theor.
43
, 235202
-1–22 (2010
).21.
See, for example, the theme issue on quantum mechanics,
Am. J. Phys.
70
, 199
–367
(2002
).22.
B.
Lohmann
and E.
Weigold
, “Direct measurement of the electron momentum probability distribution in atomic hydrogen
,” Phys. Lett. A
86
, 139
–141
(1981
);I. E.
McCarthy
and E.
Weigold
, “A real ‘thought’ experiment for the hydrogen atom
,” Am. J. Phys.
51
, 152
–155
(1983
).© 2011 American Association of Physics Teachers.
2011
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.
