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Volumn 38, Issue 12, 1997, Pages 6230-6248

Quantum mechanics in classical dynamics

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EID: 0031528056     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532209     Document Type: Article
Times cited : (2)

References (34)
  • 2
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    • Classical frames for a quantum theory - A bird's-eye view
    • For a survey of two approaches other than that presented here, see S. Bugajski, "Classical frames for a quantum theory - A bird's-eye view," Int. J. Theoret. Phys. 32, 969 (1993). A recent development in one of these approaches is given in E. G. Beltrametti and S. Bugajski, "A classical extension of quantum mechanics," J. Phys. A 28, 3329 (1995).
    • (1993) Int. J. Theoret. Phys. , vol.32 , pp. 969
    • Bugajski, S.1
  • 3
    • 21844494127 scopus 로고
    • A classical extension of quantum mechanics
    • For a survey of two approaches other than that presented here, see S. Bugajski, "Classical frames for a quantum theory - A bird's-eye view," Int. J. Theoret. Phys. 32, 969 (1993). A recent development in one of these approaches is given in E. G. Beltrametti and S. Bugajski, "A classical extension of quantum mechanics," J. Phys. A 28, 3329 (1995).
    • (1995) J. Phys. A , vol.28 , pp. 3329
    • Beltrametti, E.G.1    Bugajski, S.2
  • 4
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    • Complex coordinates and quantum mechanics
    • F. Strocchi, "Complex coordinates and quantum mechanics," Rev. Mod. Phys. 38, 36 (1966).
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    • Strocchi, F.1
  • 5
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    • Many-body quantum mechanics as a symplectic dynamical system
    • D. J. Rowe, A. Ryman, and G. Rosensteel, "Many-body quantum mechanics as a symplectic dynamical system," Phys. Rev. A 40, 2362 (1980).
    • (1980) Phys. Rev. A , vol.40 , pp. 2362
    • Rowe, D.J.1    Ryman, A.2    Rosensteel, G.3
  • 6
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    • Quantum mechanics as a classical theory
    • A. Heslot, "Quantum mechanics as a classical theory," Phys. Rev. D 31, 1341 (1985).
    • (1985) Phys. Rev. D , vol.31 , pp. 1341
    • Heslot, A.1
  • 7
    • 0009136679 scopus 로고
    • Classical mechanics as an example of generalised quantum mechanics
    • K. R. W. Jones, "Classical mechanics as an example of generalised quantum mechanics," Phys. Rev. D 45, 2590 (1992).
    • (1992) Phys. Rev. D , vol.45 , pp. 2590
    • Jones, K.R.W.1
  • 8
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    • MacMillan, New York
    • In the interest of brevity, and since this material is not the subject of this paper, the terms used will not be defined here, but the reader is referred to, for instance, S. MacLane and G. Birkhoff, Algebra (MacMillan, New York, 1979). Unlike some definitions of rings, here it will not be assumed at the outset that the algebra's multiplication is associative.
    • (1979) Algebra
    • MacLane, S.1    Birkhoff, G.2
  • 10
    • 85033139761 scopus 로고    scopus 로고
    • note
    • The term real is used instead of self-conjugate for the sake of brevity and because it is already familiar in this meaning. It is important to note that real is not the same as real-valued, a term which will be introduced shortly.
  • 11
    • 85033134345 scopus 로고    scopus 로고
    • note
    • With this definition, if a complex number z = x + iy, with x and y elements of ℛ, the imaginary part is iy, not just y.
  • 12
    • 85033142088 scopus 로고    scopus 로고
    • note
    • A set of states is separable if there is a dense sequence of states that arbitrarily closely approximates any state. A set of states is complete if every Cauchy sequence of states has a limit which is also a state.
  • 14
    • 85033138504 scopus 로고    scopus 로고
    • note
    • This paper will only concern itself with such ring linear operators, and so "ring linear" will not be explicitly written every time.
  • 15
    • 85033139644 scopus 로고    scopus 로고
    • note
    • For a commutative algebra, transposition is also an antiautomorphism.
  • 16
    • 85033144794 scopus 로고    scopus 로고
    • note
    • Adler (Ref. 12) discusses this in the case of quantum mechanics using quaternions.
  • 17
    • 85033151768 scopus 로고    scopus 로고
    • note
    • It should be noted that Greek indices will refer to the real-valued coordinates and will obey the Einstein summation convention. Indices from the second half of the Roman alphabet will refer to the orthonormal basis being used to expand a general wave function, while indices from the first half will refer to the ring generators.
  • 18
    • 85033142474 scopus 로고    scopus 로고
    • note
    • 1) square to zero - even though each generator has an inverse.
  • 19
    • 85033143799 scopus 로고    scopus 로고
    • note
    • In this treatment of quantum mechanics, it is nowhere necessary to normalize wave functions, since it is the dynamical aspects that are of primary concern and not the issue of measurement via the usual statistical interpretation. It is not, therefore, a problem that, in cases where the ring is not a division algebra, the norm of a nonzero state is not necessarily nonzero.
  • 20
    • 85033142730 scopus 로고    scopus 로고
    • note
    • Rowe, Ryman, and Rosensteel (Ref. 4) obtain the analogous result (in quantum mechanics using complex numbers) for a self-adjoint Hamiltonian, using a variational principle due to Dirac, that also leads to Eq. (4).
  • 23
    • 0000727907 scopus 로고
    • Fundamental equations of quantum mechanics
    • It is generally assumed that the quantum Poisson bracket is (up to a constant factor) the commutator as part of the correspondence principle - see, for example. P. A. M. Dirac, "Fundamental equations of quantum mechanics," Proc. R. Soc. London, Ser. A 114, 642 (1925).
    • (1925) Proc. R. Soc. London, Ser. A , vol.114 , pp. 642
    • Dirac, P.A.M.1
  • 24
    • 85033133479 scopus 로고    scopus 로고
    • note
    • p}.
  • 25
    • 0007468830 scopus 로고
    • Testing quantum mechanics
    • S. Weinberg, "Testing quantum mechanics," Ann. Phys. 194, 336 (1989).
    • (1989) Ann. Phys. , vol.194 , pp. 336
    • Weinberg, S.1
  • 27
    • 85033154738 scopus 로고    scopus 로고
    • The Schrödinger equation from three postulates
    • to appear
    • K. R. W. Jones, "The Schrödinger equation from three postulates," Mod. Phys. Lett. A (to appear).
    • Mod. Phys. Lett. A
    • Jones, K.R.W.1
  • 28
    • 85033132663 scopus 로고    scopus 로고
    • note
    • Heslot (Ref. 5) obtains the analogous result (in quantum mechanics using complex numbers) for a real function G arising from a self-adjoint operator G and uses this to define observables as real "regular functions of the state whose canonical transformations they generate are automorphisms of the whole quantum structure, i.e., are unitary transformations."
  • 29
    • 85033143838 scopus 로고    scopus 로고
    • note
    • μ elsewhere.
  • 30
    • 33947227205 scopus 로고
    • Elliptic eigenstates for the quantum harmonic oscillator
    • 4, these are - 1/2i times the constants of motion for the two-dimensional isotropic harmonic oscillator - see, for example. J. Pollett, O. Méplan, and C. Gignoux, "Elliptic eigenstates for the quantum harmonic oscillator," J. Phys. A 28, 7287 (1995).
    • (1995) J. Phys. A , vol.28 , pp. 7287
    • Pollett, J.1    Méplan, O.2    Gignoux, C.3
  • 31
    • 85033135016 scopus 로고    scopus 로고
    • note
    • 3 should be made with the complex numbers.
  • 32
    • 85033133174 scopus 로고    scopus 로고
    • note
    • Adler (Ref. 12) shows that (in quantum mechanics using quaternions) the eigenstates of an anti-self-adjoint operator divide into mutually orthogonal eigenclasses, such that all of the eigenstates in a given eigenclass are related by reraying, that is, by multiplying on the right by a quaternionic phase. Since, for the quaternions, all unit imaginaries are in the same automorphism class, it is possible to pick, for a given anti-self-adjoint operator, a particular ray representative such that the eigenvalues are all dependent on the same unit imaginary.
  • 33
    • 85033156320 scopus 로고    scopus 로고
    • note
    • (1,/2)n(n-1) under reversion.
  • 34
    • 85033158678 scopus 로고    scopus 로고
    • note
    • In Clifford algebras generated by more than two basis vectors, all blades of odd grade have a sign change in addition to that from reversing the order of products.


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.