Dirac's Poisson-bracket-to-commutator analogy for the transition from classical to quantum mechanics assures that for many systems, the classical and quantum systems share the same algebraic structure. The quantum side of the analogy (involving operators on Hilbert space with commutators scaled by Planck's constant ) not only gives the algebraic structure but also dictates the average values of physical quantities in the quantum ground state. On the other hand, the Poisson brackets of nonrelativistic mechanics, which give only the classical canonical transformations, do not give any values for physical quantities. Rather, one must go outside nonrelativistic classical mechanics in order to obtain a fundamental phase space distribution for classical physics. We assume that the values of physical quantities in classical theory at any temperature depend on the phase space probability distribution that arises from thermal radiation equilibrium including classical zero-point radiation with the scale set by Planck's constant . All mechanical systems in thermal radiation will inherit the constant from thermal radiation. Here, we note the connections between classical and quantum theories (agreement and contrasts) at all temperatures for the harmonic oscillator in one and three spatial dimensions.

1.
P. A. M.
Dirac
, “
The fundamental equations of quantum mechanics
,”
Proc. R. Soc. A
109
,
642
653
(
1925
).
2.
P. A. M.
Dirac
,
The Principles of Quantum Mechanics
, 4th ed. (
Clarendon Press
,
Oxford
,
1958
), Chap. IV, p.
87
;
L. I.
Schiff
,
Quantum Mechanics
, 3rd ed. (
McGraw-Hill
,
New York
,
1968
), pp.
175
177
;
L. E.
Ballentine
,
Quantum Mechanics
(
Prentice Hall
,
Englewood Cliffs
,
NJ
,
1990
), pp.
67
68
;
J. J.
Sakurai
,
Modern Quantum Mechanics Revised Edition
(
Addison-Wesley
,
New York
,
1994
), pp.
50
51
.
3.
J. P.
Dahl
, “
Physical origin of the Runge-Lenz vector
,”
J. Phys. A
30
,
6831
6840
(
1997
), notes that the “relation … is generally valid when [f,g] may be written as a linear combination of Poisson brackets between functions of which at least one is no more than quadratic in the components of r or p.”
4.
T. H.
Boyer
, “
The contrasting roles of Planck's constant in classical and quantum theories
,”
Am. J. Phys.
86
,
280
283
(
2018
).
5.
T. H.
Boyer
, “
Blackbody radiation in classical physics: A historical perspective
,”
Am. J. Phys.
86
,
495
509
(
2018
).
6.
See, for example,
D. J.
Griffiths
,
Introduction to Quantum Mechanics
, 2nd ed. (
Pearson Prentice Hall
,
Upper Saddle River
,
NJ
,
2005
), p.
43
.
7.
See, for example,
H.
Goldstein
,
Classical Mechanics
, 2nd ed. (
Addison-Wesley
,
Reading, MA
,
1981
), beginning on p.
397
.
8.
See, for example, Sakurai in Ref. 2, pp.
50
51
.
9.
See, for example,
A.
Zee
,
Group Theory in a Nutshell for Physicists
(
Princeton U. P
.,
Princeton
,
2016
), p.
210
.
10.
See, for example, Griffiths in Ref. 6, p.
42
.
11.
See, for example, Griffiths in Ref. 6, p.
46
.
12.
See, for example, H. Goldstein in Ref. 7, p.
421
.
13.
W.
Pauli
, “
Über das Wasserstoffspecktrum vom Standpunkt der neuen Quantenmechanik
,”
Z. Phys.
36
,
336
363
(
1926
).
14.
The classical point of view begins with the careful work of
T. W.
Marshall
, “
Random electrodynamics
,”
Proc. R. Soc. London, Ser. A
276
,
475
491
(
1963
);
T. W.
Marshall
Statistical electrodynamics
,”
Proc. Cambridge Philos. Soc.
61
,
537
546
(
1965
).
For recent work, see
T. H.
Boyer
, “
Stochastic electrodynamics: The closest classical approximation to quantum theory
,”
Atoms
7
(
1
),
29
39
(
2019
);
W. C-W.
Huang
and
H.
Batelaan
, “
Discrete excitation spectrum of a classical harmonic oscillator in zero-point radiation
,”
Found. Phys.
45
,
333
353
(
2015
);
D. C.
Cole
and
Y.
Zou
, “
Quantum mechanical ground state of hydrogen obtained from classical electrodynamics
,”
Phys. Lett. A
317
,
14
20
(
2003
).
A review of the work on classical electromagnetic zero-point radiation up to 1996 is provided by
L.
de la Pena
and
A. M.
Cetto
,
The Quantum Dice: An Introduction to Stochastic Electrodynamics
(
Kluwer Academic
,
Dordrecht
,
1996
).
Further suggestions linking quantum mechanics and zero-point radiation appear in the work of
L.
de la Pena
,
A. M.
Cetto
, and
A. V.
Hernandez
,
The Emerging Quantum: The Physics Behind Quantum Mechanics
(
Springer
,
New York
,
2015
).
15.
See, for example,
M.
Planck
,
The Theory of Heat Radiation
(
Dover
,
New York
,
1959
).
16.
H. B. G.
Casimir
, “
On the attraction between two perfectly conducting plates
,”
Proc. K. Ned. Akad. Wet.
51
,
793
795
(
1948
).
17.
M. J.
Sparnaay
, “
Measurement of the attractive forces between flat plates
,”
Physica
24
,
751
764
(
1958
);
S. K.
Lamoreaux
, “
Demonstration of the Casimir force in the 0.6 to 6μm range
,”
Phys. Rev. Lett.
78
,
5
8
(
1997
);
S. K.
Lamoreaux
,
Phys. Rev. Lett.
81
,
5475–5476
(
1998
);
U.
Mohideen
, “
Precision measurement of the Casimir force from 0.1 to 0.9μm
,”
Phys. Rev. Lett.
81
,
4549
4552
(
1998
);
H. B.
Chan
,
V. A.
Aksyuk
,
R. N.
Kleinman
,
D. J.
Bishop
, and
F.
Capasso
, “
Quantum mechanical actuation of microelectromechanical systems by the Casimir force
,”
Science
291
,
1941
1944
(
2001
);
[PubMed]
G.
Bressi
,
G.
Carugno
,
R.
Onofrio
, and
G.
Ruoso
, “
Measurement of the Casimir force between parallel metallic surfaces
,”
Phys. Rev. Lett.
88
,
041804(4)
(
2002
).
18.
T. H.
Boyer
, “
Retarded van der Waals forces at all distances derived from classical electrodynamics with classical electromagnetic zero-point radiation
,”
Phys. Rev. A
7
,
1832
1840
(
1973
).
19.
See, for example,
T. H.
Boyer
, “
Random electrodynamics: The theory of classical electrodynamics with classical electromagnetic zero-point radiation
,”
Phys. Rev. D
11
,
790
808
(
1975
).
20.
See, for example, the work of T. W. Marshall (
1963
) in Ref. 14, or T. H. Boyer in Ref. 19, or
B. H.
Lavenda
,
Statistical Physics: A Probabilistic Approach
(
Wiley
,
New York
, 1991), pp.
73
74
.
21.
T. H.
Boyer
, “
General connection between random electrodynamics and quantum electrodynamics for free electromagnetic fields and for dipole oscillator systems
,”
Phys. Rev. D
11
,
809
830
(
1975
).
22.
For a graph showing the gradual change in the classical oscillator energy when interacting with zero-point radiation in a simulation, see
W.
Huang
and
H.
Batelaan
, “
Dynamics underlying the Gaussian distribution of the classical harmonic oscillator in zero-point radiation
,”
J. Comput. Methods Phys.
2013
,
308538
.
23.
See, for example,
R.
Eisberg
and
R.
Resnick
,
Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
, 2nd ed. (
Wiley
,
New York
,
1985
);
K. S.
Krane
,
Modern Physics
, 2nd ed. (
Wiley
,
New York
,
1996
);
R.
Taylor
,
C. D.
Zafiratos
, and
M. A.
Dubson
,
Modern Physics for Scientists and Engineers
, 2nd ed. (
Pearson
,
New York
,
2003
);
S. T.
Thornton
and
A.
Rex
,
Modern Physics for Scientists and Engineers
, 4th ed. (
Brooks/Cole, Cengage Learning
,
Boston, MA
,
2013
).
24.
T. H.
Boyer
, “
Understanding zero-point energy in the context of classical electromagnetism
,”
Eur. J. Phys.
37
,
055206(14)
(
2016
);
T. H.
Boyer
, “
Thermodynamics of the harmonic oscillator: Derivation of the Planck blackbody spectrum from pure thermodynamics
,”
Eur. J. Phys.
40
,
025101
(
2019
);
T. H.
Boyer
, “
Diamagnetic behavior in random classical radiation
,”
Am. J. Phys.
87
,
915
923
(
2019
);
T. H.
Boyer
, “
Particle Brownian motion due to random classical radiation: Superfluid-like behavior in classical zero-point radiation
,”
Eur. J. Phys.
(published online, 2020).
25.

We can calculate the results for L2T by using the average values for powers of displacements and momenta at positive temperature quoted in Table II of Ref. 21.

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