By finding that the projection operators of the basic quantum mechanical representations can be written as operator delta functions, we derive the explicit forms of quantum-mechanical fundamental representations in Fock space by directly carrying out the integrals within the normally ordered product of operators.

1.
R.
Shankar
,
Principles of Quantum Mechanics
, 2nd ed. (
Plenum
,
New York
,
1994
).
2.
J. J.
Sakurai
and
J.
Napolitano
,
Modern Quantum Mechanics
, 2nd ed. (
Addison-Wesley
,
Boston
,
2011
).
3.
D. J.
Griffiths
,
Introduction to Quantum Mechanics
, 2nd ed. (
Pearson
,
London
,
2004
).
4.
P. A. M.
Dirac
,
The Principle of Quantum Mechanics
, 4th ed. (
Oxford U. P
.,
Oxford
,
1958
).
5.
H. Y.
Fan
,
H. R.
Zaidi
, and
J. R.
Klauder
, “
New approach for calculating the normally ordered form of squeeze operators
,”
Phys. Rev. D
35
(
6
),
1831
1834
(
1987
).
6.
H. Y.
Fan
and
J. R.
Klauder
, “
Eigenvectors of two particles' relative position and total momentum
,”
Phys. Rev. A
49
(
2
),
704
707
(
1994
).
7.
H. Y.
Fan
, “
Operator ordering in quantum optics theory and the development of Dirac's symbolic method
,”
J. Opt. B: Quantum Semiclass. Opt.
5
(
4
),
R147
R163
(
2003
).
8.
H. Y.
Fan
,
H. L.
Lu
, and
Y.
Fan
, “
Newton–Leibniz integration for ket–bra operators in quantum mechanics and derivation of entangled state representations
,”
Ann. Phys.
321
(
2
),
480
494
(
2006
).
9.
H. Y.
Fan
and
C. H.
, “
Wick calculus using the technique of integration within an ordered product of operators
,”
Am. J. Phys.
77
(
3
),
284
286
(
2009
).
10.
L. F.
Barragán-Gil
and
R.
Walser
, “
Harmonic oscillator thermal density matrix: First-order differential equations for the position representation
,”
Am. J. Phys.
86
(
1
),
22
24
(
2018
).
11.
M. O.
Scully
and
M. S.
Zubairy
,
Quantum Optics
(
Cambridge U. P.
,
Cambridge
,
1997
).
12.
R. J.
Glauber
, “
Coherent and incoherent states of radiation field
,”
Phys. Rev.
131
(
6
),
2766
2788
(
1963
).
13.
J. P.
Gazeau
,
Coherent States in Quantum Physics
(
Wiley-VCH
,
Weinheim
,
2009
).
14.
M. A.
Man'ko
,
V. I.
Man'ko
, and
R. V.
Mendes
, “
Tomograms and other transforms: A unified view
,”
J. Phys. A: Math. Gen.
34
(
40
),
8321
8332
(
2001
).
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