In his first paper on wave mechanics, Schrödinger presented a heuristic argument which led from the Hamilton–Jacobi equation through the quantum variational principle to his famous wave equation. In his second paper, Schrödinger withdrew this heuristic argument as “incomprehensible.” We show by using a recently generalized form of Maupertuis’ principle that Schrödinger’s original heuristic argument can be made more logical. Aside from pedagogical interest, this path is useful as a method of quantization of general mechanical systems.

1.
E.
Schrödinger
, “
Quantisierung als eigenwert problem I
,”
Ann. Phys.
79
,
361
376
(
1926
);
E.
Schrödinger
, “
Quantisierung als eigenwert problem II
,”
Ann. Phys.
79
,
489
527
(
1926
).
Reprinted and translated in E. Schrödinger, Collected Papers on Wave Mechanics (Blackie, London, 1928)
(Chelsea reprint 1982).
2.
Wolfgang Yourgrau and Stanley Mandelstam, “Variational Principles in Dynamics and Quantum Theory,” 3rd ed. (Saunders, Philadelphia, 1968)
(Dover reprint 1979).
3.
C. G.
Gray
,
G.
Karl
, and
V. A.
Novikov
, “
The Four Variational Principles of Mechanics
,”
Ann. Phys.
251
,
1
25
(
1996
).
4.
See, e.g., Ref. 2, p. 52, and note that our notations W and S (which follow current usage) are interchanged compared to Ref. 2.
5.
N. I. Akhiezer, The Calculus of Variations (Harwood, London, 1988), pp. 4 and 146.
6.
J. W. Gibbs, The Scientific Papers of J. Willard Gibbs, Vol. I, Thermodynamics (Longmans, Green, London, 1906), p. 56
(reprinted by Dover, New York 1961).
For a modern treatment, see H. B. Callen, Thermodynamics, 2nd ed. (Wiley, New York, 1985), p. 133.
7.
C. G.
Gray
,
G.
Karl
, and
V. A.
Novikov
, “
Direct use of variational principles as an approximation technique in classical mechanics
,”
Am. J. Phys.
64
,
1172
1184
(
1996
).
8.
See, e.g., Eugen Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, New York, 1998), p. 135.
9.
S. T. Epstein, The Variation Method in Quantum Chemistry (Academic, New York, 1974).
10.
J. Frenkel, Wave Mechanics, Advanced General Theory (Oxford U.P., Oxford, 1934), p. 253.
Frenkel states that he took his argument from the appendix to the Russian edition of P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford U.P., Oxford, 1930).
11.
M.
Born
and
P.
Jordan
, “
Zur Quantenmechanik
,”
Z. Phys.
34
,
858
888
(
1925
),
reprinted and translated in B. L. van der Waerden (ed.), Sources of Quantum Mechanics (North Holland, Amsterdam, 1967)
(reprinted by Dover, New York, 1968).
12.
W. R.
Greenberg
,
A.
Klein
, and
C-t.
Li
, “
Invariant Tori and Heisenberg Matrix Mechanics: A New Window on the Quantum-Classical Correspondence
,”
Phys. Rev. Lett.
75
,
1244
1247
(
1995
).
13.
J.
Schwinger
, “
The Theory of Quantized Fields I
,”
Phys. Rev.
82
,
914
929
(
1951
).
14.
G.
Karl
and
V. A.
Novikov
, “
Variational Estimates for Excited States
,”
Phys. Rev. D
51
,
5069
5078
(
1995
);
F. M.
Fernandez
and
E.
Castro
, “
Simple Approximate Analytical Expressions for the Eigenvalues of Anharmonic Oscillators and Confining Potential Models
,”
J. Chem. Phys.
79
,
321
324
(
1983
);
J.
Dias de Deus
,
A. B.
Henriques
, and
J. M. R.
Pulido
, “
Quarkonia with a Variational Model
,”
Z. Phys. C
7
,
157
168
(
1981
);
D.
Gromes
and
I. O.
Stamatescu
, “
Baryon Spectra and the Forces Between Quarks
,”
Z. Phys. C
3
,
43
50
(
1979
);
R.
McWeeny
and
C. A.
Coulson
, “
Quantum Mechanics of the Anharmonic Oscillator
,”
Proc. Cambridge Philos. Soc.
44
,
413
422
(
1948
).
15.
B. L. van der Waerden, Ref. 11 above, p. 5.
16.
M. Born, The Mechanics of the Atom (Bell, London, 1925), pp. 59–62
(reprinted by Ungar, New York 1960).
17.
J. H.
Van Vleck
, “
Quantum Principles and Line Spectra
,”
Bull. National Research Council
10
(
4
),
18
,
23
(
1926
).
18.
The correspondences (14) are a standard result of quasi-classical (WKB) mechanics: the classical result is derived from the quantum by writing ψ(q)=A(q)exp(iW(q)/ℏ) (the equivalent of which was written down by Schrödinger in his first paper—see Introduction), and assuming that, at large energies or actions, or short wavelengths, the amplitude A(q) varies slowly compared to the phase factor, which depends on the classical action W(q)≡∫q0qpdq≫ℏ.
See, e.g., Wolfgang Pauli, General Principles of Quantum Mechanics (Springer-Verlag, Berlin, 1980), pp. 95–100.
19.
W. Moore, Schrödinger, Life and Thought (Cambridge U.P., Cambridge, 1989), p. 200;
J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory (Springer-Verlag, New York, 1987), Vol. 5, Pt. 2, p. 464;
H.
Kragh
, “
Erwin Schrödinger and the Wave Equation: The Crucial Phase
,”
Centaurus
26
,
154
197
(
1982
);
L.
Wessels
, “
Schrödinger’s Route to Wave Mechanics
,”
Stud. Hist. Philos. Sci.
10
,
311
340
(
1979
);
M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw–Hill, New York, 1966), p. 259.
This content is only available via PDF.
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.