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On the operators for time of motion and delay time induced by scattering. (English) Zbl 0179.30001

Summary: Recently, Lippmann showed that a time operator might exist in quantum theory and that its explicit form is \(i \,\partial/\partial E\) in the energy representation. In this paper, we wish to stress that \(i \,\partial/\partial E\) actually represents only the time which the particle used to move from the scattering center to a certain point at the outside. (Later on, we shall call this time the “time of motion”.) We also show that, although it is possible to define an operator for the “time of motion” of the particle in the energy representation, it is necessary to introduce a new conjugate representation of the energy representation, the “time representation”, to overcome difficulties which may arise in the evaluation of the mean time of motion \(\langle T\rangle\). If the position representation is used for the calculation instead, we shall meet these difficulties. The reason is that time of motion \(T\), position \(x\), and energy \(E\) are not two-by-two compatible variables. Bearing in mind that \(i \,\partial/\partial E\) only represents the time of motion of the particle, we can then deduce the explicit form of the delay time operator \(Q\) in terms of the scattering matrix \(S\) directly from the time of motion operator by simple reasoning. This form is \(Q = -i(\partial S/\partial E)\cdot S^\dagger\) as was found previously. Some interesting features of the delay time operator will also be sketched at the end of the paper.

MSC:

81-XX Quantum theory

Keywords:

quantum theory
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