For two particles’ relative position and total momentum we have introduced the entangled state representation |η?, and its conjugate state |ξ?. In this work, for the first time, we study them via the integration over ket–bra operators in Q-ordering or P-ordering, where Q-ordering means all Qs are to the left of all Ps and P-ordering means all Ps are to the left of all Qs. In this way we newly derive P-ordered (or Q-ordered) expansion formulas of the two-mode squeezing operator which can show the squeezing effect on both the two-mode coordinate and momentum eigenstates. This tells that not only the integration over ket–bra operators within normally ordered, but also within Pordered (or Q-ordered) are feasible and useful in developing quantum mechanical representation and transformation theory.
. [J]. Frontiers of Physics, 2014, 9(4): 460-464.
Hong-Yi Fan,Sen-Yue Lou. Studying bi-partite entangled state representations via the integration over ket–bra operators in Q-ordering or P-ordering. Front. Phys. , 2014, 9(4): 460-464.
A. Einstein, B. Podolsky, and N. Rosen, Can quantummechanical description of physical reality be considered complete? Phys. Rev., 1935, 47(10): 777 doi: 10.1103/PhysRev.47.777
2
H. Y. Fanand J. R. Klauder, Eigenvectors of two particles’ relative position and total momentum., Phys. Rev. A, 1994, 49(2): 704 doi: 10.1103/PhysRevA.49.704
3
H. Y. Fanand Y. Fan, Dual eigenkets of the Susskind–Glogower phase operator, Phys. Rev. A, 1996, 54(6): 5295 doi: 10.1103/PhysRevA.54.5295
4
P. A. M.Dirac, The Principles of Quantum Mechanics, Oxford: Clarendon Press, 1930
5
H. Y. Fanand J. Zhou, Coherent state and normal ordering method for transiting Hermite polynomials to Laguerre polynomials, Science China: Physics, Mechanics & Astronomy, 2012, 55(4): 605 doi: 10.1007/s11433-012-4677-x
6
H. Y. Fan, New fundamental quantum mechanical operatorordering identities for the coordinate and momentum operators, Science China: Physics, Mechanics & Astronomy, 2012, 55(5): 762 doi: 10.1007/s11433-012-4699-4
7
H. Y. Fanand S. Y. Lou, Science China: Physics, Mechanics and Astronomy, 2013 (to appear)
8
H. Weyl, Quantenmechanik und gruppentheorie, Z. Phys., 1927, 46(1-2): 1 doi: 10.1007/BF02055756
9
E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 1932, 40(5): 749 doi: 10.1103/PhysRev.40.749
10
H. Y. Fan, S. Wang, and L. Y. Hu, Evolution of the singlemode squeezed vacuum state in amplitude dissipative channel, Front. Phys doi: 10.1007/s11467-013-0367-z
11
H. Y. Fan, Squeezed states: Operators for two types of one- and two-mode squeezing transformations, Phys. Rev. A, 1990, 41(3): 1526 doi: 10.1103/PhysRevA.41.1526
12
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., 2006, 321(2): 480 doi: 10.1016/j.aop.2005.09.011
13
H. Y. Fan, Operator ordering in quantum optics theory and the development of Dirac’s symbolic method, J. Opt. B, 2003, 5(4): R147 doi: 10.1088/1464-4266/5/4/201
14
H. Y. Fan, H. C. Yuan, and N. Q. Jiang, Deriving new operator identities by alternately using normally, antinormally, and Weyl ordered integration technique, Science China: Phys Mechanics & Astronomy, 2010, 53(9): 1626 doi: 10.1007/s11433-010-4071-5
