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Mathematical Proceedings of the Cambridge Philosophical Society

On the solution of an equation occurring in quantum mechanics

M. Anthea Grubb

Published Apr 1, 1965• 1 citation
Publisher

Abstract

The following equation arose in a study of the evolution of a one-dimensional quantum system: where α and κ are positive constants, and It will be shown that this system provides an example of exactly exponential decay. This is of interest in view of recent papers discussing the validity of the exponential decay law (see, for example, Khalfin (1)and Martin (2)).

DOI
10.1017/s0305004100004102

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The following equation arose in a study of the evolution of a one-dimensional quantum system: where α and κ are positive constants, and It will be shown that this system provides an example of exactly exponential decay. This is of interest in view of recent papers discussing the validity of the exponential decay law (see, for example, Khalfin (1)and Martin (2)).
The Pauli–Schrödinger equation for a spin 1/2, neutral particle with a nonvanishing magnetic moment μ0, interacting with an external scalar potential V and a static magnetic field B, both functions of only one of the coordinates, is solved exactly for four different choices of the potential and the field. By choosing in the examples the coordinate y, we present these solutions in the following cases: (i) V( y)=0, B( y)=(B0sin κy, 0, B0cos κy) where B0 and κ are two arbitrary constants. (ii) V( y)=λ′y2, B( y)=‖αy‖(cos 2κy, 0, sin 2κy) where 6* λ=ℏ2/2m and α,κ arbitrary constants and λ′=λ(μ0/2κλ)2. (iii) V( y)=λ′(c̄1tan αy+c̄2cot αy)2, B( y)=B( y) (sin (2κy+2δ(y)), 0, cos (2κy+2δ( y)) where B2( y)=(c̄1tan αy+c̄2cot αy)2 (+α2/4κ2)(c̄1sec2αy−c̄2csc2αy)2 and tan 2δ( y)=(2κ/λ)((c̄1tan αy+c̄2cot αy) /(c̄1sec2αy−c̄2csc2αy)); c̄1, c̄2, α, and κ are arbitrary constants. (iv) V( y)=λ′(ā tanh z+b̄)2 where z =[(y−cd)/d], B2( y)=(ā tanh z+b̄)2+(ā2/4κ2d2)sech2z, tan 2δ( y)=(2κd/ā)((ā tanh z+b̄)/sech2z); ā, b̄, c, d, and κ are arbitrary constants restricted only by (μ0/2κλ)ā2+ā/d>0. The functions B( y) and δ( y) define the vector B( y) as in (iii). Our method of solution is based on the familiar factorization technique for solving the Schrödinger eigenvalue equation. Several interesting physical results of these solutions are discussed.
An alpha field is the field that can be described by two dimensionless field parameters α and α'.A quantum alpha field is the alpha field in which quantum effects can be occurred.In classical mechanics, a nanorobot has, at every moment, an exact position and an exact momentum, following the Newton's laws of motion.In quantum mechanics, nanorobots do not have exactly determined properties and when they are measured, the result is randomly drawn from a probability distribution.The classical approaches to control of nanorobot motion in the multipotential field have been presented in the authors previous papers.In this paper we continue to investigate the related control algorithms for a nanorobot motion in a quantum alpha field.In that sense the nanorobot dynamics has been described by the Schrödinger equation in an alpha field that predicts what the probability distributions are.Meanwhile, it cannot fundamentally predict the exact result of each measurement.Thus, the problem is: how to control of a nanorobot motion in the case when quantum effects are occurred.Some approaches are discussed in this paper.
The basic requirement that, in quantum theory, the time evolution of any state is determined by the action of a unitary operator, is shown to be the underlying cause for certain ``exact'' results that have recently been reported about the time dependence of transition rates in quantum theory. Departures from exponential decay, including the ``quantum Zeno effect,'' as well as a theorem by Khalfin about the ratio of reciprocal transition rates, are shown to follow directly from such considerations. At sufficiently short times, unitarity requires that reciprocity must hold, independent of whether T invariance is valid. If T invariance does not hold, unitarity restricts the form of possible time dependence of reciprocity ratios. \textcopyright{} 1996 The American Physical Society.