For the example of the infinite well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self-adjoint operator. We then describe the self-adjoint extensions and their spectra for the momentum and the Hamiltonian operators in different settings. Additional physical requirements such as parity, time reversal, and positivity are used to restrict the large class of self-adjoint extensions of the Hamiltonian.
REFERENCES
1.
L. E. Ballentine, Quantum Mechanics (Prentice–Hall, Englewood Cliffs, NJ, 1990).
2.
A. Z.
Capri
, “Self-adjointness and spontaneously broken symmetry
,” Am. J. Phys.
45
, 823
–825
(1977
).3.
A.
Cabo
, J. L.
Lucio
, and H.
Mercado
, “On scale invariance and anomalies in quantum mechanics
,” Am. J. Phys.
66
, 240
–246
(1998
).4.
R. Jackiw, “Delta function potentials in two and three dimensional quantum mechanics,” in M. A. Bég Memorial Volume, edited by A. Ali and P. Hoodbhoy (World Scientific, Singapore, 1991), pp. 1–16.
5.
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space (Ungar, New York, 1961).
6.
M. A. Naimark, Linear Differential Operators (Ungar, New York, 1968), Vol. 2.
7.
J. M. Lévy-Leblond and F. Balibar, Quantics (North-Holland, Amsterdam, 1990).
8.
W. Greiner, Quantum Mechanics (Springer-Verlag, Berlin, 1989).
9.
Notice that the positive function (4) is nearly equal to the eigenfunction as
10.
11.
12.
C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, New York, 1977).
13.
L. Schiff, Quantum Mechanics (McGraw–Hill, New York, 1965), 3rd ed.
14.
An extended version, with mathematical details, is available from the preprint PAR/LPTHE/99-43.
15.
H. B.
Rosentock
, “Specific heat of a particle in a box
,” Am. J. Phys.
30
, 38
–40
(1962
).16.
V.
Granados
and N.
Aquino
, “Comment on specific heat revisited
,” Am. J. Phys.
67
, 450
–451
(1999
).17.
D. H.
Berman
, “Boundary effects in quantum mechanics
,” Am. J. Phys.
59
, 937
–941
(1991
).18.
To make things simple we say that a function is absolutely continuous for if it can be written in the form where is absolutely integrable for any Absolute continuity in a finite interval implies uniform continuity, whereas the converse is not true. The interested reader is referred to Ref. 19, p. 337.
19.
A. Kolmogorov and S. Fomine, Eléments de la Théorie des Fonctions et de L’analyse Fonctionnelle (Mir-Ellipses, Paris, 1994).
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2001
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