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The time of arrival in quantum mechanics. 1. Formal considerations
ANNALS OF
PHYSICS:52, 208-209 (1969)
Abstracts
of Papers
to Appear
in Future
Issues
The Time of Arrival in Quantum Mechanics. 1. Formal Considerations. G. R. ALLCOCK. Department of Theoretical Physics, Chadwick Laboratory, The University, Liverpool 3, England. It is argued that the complementarity of time and energy in one-dimensional wave mechanics finds its proper application in the measurement of the time at which a moving particle arrives at (or departs from) a fixed spatial point. A rigorous proof is given that the ordinary Hilbert space of a freely moving particle does not contain a set of measurement eigenstates for this measurement problem. The proof depends upon the semi-infinite extent of the energy spectrum. Both this result and the physical nature of an arrival-time situation suggest that the most suitable and general wave function for the problem might be obtained by arranging that the particle be emitted by a source situated at a finite distance from the point of detection. Such sources are introduced, and the motion over the intervening space is treated by using the one-dimensional non-relativistic Schrodinger equation. The resultant free-particle wave functions contain components with both positive and negative energies, and constitute a linear space different from the ordinary Hilbert space. The arrival-time problem then becomes one of interpreting these unfamiliar wave functions, with a view to obtaining a formula for the arrival probability per unit time. It is proved with complete generality and rigour that this modified problem also has no solution. The mathematical properties of the time-dependent Schrodinger equation make it impossible to construct any operationally meaningful and apparatus-independent probability formula. This demonstrates the existence of genuine measurement problems which lie beyond the scope of the usual measurement formalism. Such problems can only be satisfactorily discussed by making reference to the mechanical parameters of the measuring apparatus. The Rotational Wigner Function. ANTHONY G. ST. PIERRE AND WILLIAM A. STEELE. Department of Chemistry and Department of Physics, The Pennsylvania State Univ., University Park, Pennsylvania. The quantum mechanics of angular momentum is studied using the Wigner phase space distribution function. To accomplish this in a way which allows one to retain the bilinear form of the distribution function originally conceived by Wigner, a new set of generalized variables is introduced, viz. the Cayley Klein (C.K.) coordinates and momenta. The equations relating the four C.K. coordinates to the three Euler angles give rise to a constraint which is relaxed in a natural way by introducing a fourth “radial” variable p whose conjugate momentum is pD. Physical quantities in the classical mechanics of rotational systems are reexpressed in the C.K. representation; and the canonical transformation between the C.K. and Eulerian variables is established. In a similar manner, after introducing the basic quantum mechanical operators for the C.K. coordinates and momenta, the wavefimctions and operators for rotational systems are reexpressed in these variables. With the aid of the Weyl correspondence to link the operators of quantum mechanics with the dynamical functions of classical mechanics, the phase space formulation is established. Rotational Wigner distribution functions are constructed, their properties investigated, and their form is explicitly given for several quantum states. Expectation values are then calculated according to the procedure identical to the one in classical statistical mechanics for calculating phase space averages. The equation of motion for the distribution function is derived. It turns out to be the sine expansion of the Poisson bracket operator expressed in the C.K. variables and operating on the
208
PHYSICS:52, 208-209 (1969)
Abstracts
of Papers
to Appear
in Future
Issues
The Time of Arrival in Quantum Mechanics. 1. Formal Considerations. G. R. ALLCOCK. Department of Theoretical Physics, Chadwick Laboratory, The University, Liverpool 3, England. It is argued that the complementarity of time and energy in one-dimensional wave mechanics finds its proper application in the measurement of the time at which a moving particle arrives at (or departs from) a fixed spatial point. A rigorous proof is given that the ordinary Hilbert space of a freely moving particle does not contain a set of measurement eigenstates for this measurement problem. The proof depends upon the semi-infinite extent of the energy spectrum. Both this result and the physical nature of an arrival-time situation suggest that the most suitable and general wave function for the problem might be obtained by arranging that the particle be emitted by a source situated at a finite distance from the point of detection. Such sources are introduced, and the motion over the intervening space is treated by using the one-dimensional non-relativistic Schrodinger equation. The resultant free-particle wave functions contain components with both positive and negative energies, and constitute a linear space different from the ordinary Hilbert space. The arrival-time problem then becomes one of interpreting these unfamiliar wave functions, with a view to obtaining a formula for the arrival probability per unit time. It is proved with complete generality and rigour that this modified problem also has no solution. The mathematical properties of the time-dependent Schrodinger equation make it impossible to construct any operationally meaningful and apparatus-independent probability formula. This demonstrates the existence of genuine measurement problems which lie beyond the scope of the usual measurement formalism. Such problems can only be satisfactorily discussed by making reference to the mechanical parameters of the measuring apparatus. The Rotational Wigner Function. ANTHONY G. ST. PIERRE AND WILLIAM A. STEELE. Department of Chemistry and Department of Physics, The Pennsylvania State Univ., University Park, Pennsylvania. The quantum mechanics of angular momentum is studied using the Wigner phase space distribution function. To accomplish this in a way which allows one to retain the bilinear form of the distribution function originally conceived by Wigner, a new set of generalized variables is introduced, viz. the Cayley Klein (C.K.) coordinates and momenta. The equations relating the four C.K. coordinates to the three Euler angles give rise to a constraint which is relaxed in a natural way by introducing a fourth “radial” variable p whose conjugate momentum is pD. Physical quantities in the classical mechanics of rotational systems are reexpressed in the C.K. representation; and the canonical transformation between the C.K. and Eulerian variables is established. In a similar manner, after introducing the basic quantum mechanical operators for the C.K. coordinates and momenta, the wavefimctions and operators for rotational systems are reexpressed in these variables. With the aid of the Weyl correspondence to link the operators of quantum mechanics with the dynamical functions of classical mechanics, the phase space formulation is established. Rotational Wigner distribution functions are constructed, their properties investigated, and their form is explicitly given for several quantum states. Expectation values are then calculated according to the procedure identical to the one in classical statistical mechanics for calculating phase space averages. The equation of motion for the distribution function is derived. It turns out to be the sine expansion of the Poisson bracket operator expressed in the C.K. variables and operating on the
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