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American Journal of Physics

On a Space-Time Representation of Nonrelativistic Wave Mechanics

H. L. Armstrong

Published Mar 1, 1957• 4 citations
Publisher

Abstract

A space-time representation is developed for nonrelativistic wave mechanics as was done for classical mechanics, by adding to the system a rotating body to serve as a clock, and applying the time-independent forms of the equations of motion. The usual Schrödinger equation is obtained in this way, however, only in the limit as the energy of the clock becomes very large. The reason for this restriction is found in an appendix, by comparison with the classical equations. Other appendices deal with certain aspects of the Hamiltonian representation of the problem.

DOI
10.1119/1.1934392

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Space-Time in the Time-Dependent Schrodinger Equation and Classical Mechanics

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The time-independent Schrodinger equation: En= -1/2m [d/dx dWn/dx] / Wn(x) + V(x) ((1)) matches a classical conservation of energy equation where -1/2m [d/dx dWn/dx] / Wn(x) = KEclassical(x). In classical physics, however, one has x(t) so each x point is associated with a specific t for a given energy En. In the quantum time-independent equation, one has exp(-iEnt) as the time dependent factor and does not consider any link with x although if one is comparing ((1)) to a classical conservation equation then a link between x and t could be made to complete the analogy. This is usually not considered, but we examine this point as we argue it becomes important in the time-dependent situation. For the time-dependent Schrodinger case we consider two approaches. First, a linear combination of time independent solutions is taken i.e. W(x,t) = Sum over n an(En) exp(-iEnt) Wn(x) ((2)) Again each Wn(x) has an association with KEclassical(x) and so one may think in terms of (x,t) classical pairs for a given En. The second approach is to consider the function: exp(-iEnt + m Integral (0,x) v(y) dy) ((3)). This satisfies the time-dependent Schrodinger equation and yields a classical conservation of energy equation with v(x) being the classical velocity. Noting that En= .5m v(x)v(x) + V(x), ((3)) may be rewritten as: exp(i Integral L dt) where L = T-V is the Langrangian with T=.5mv(x)v(x). At this point En has been removed from the picture, but is replaced with: x(t)= X g(t)/g(T) ((4)) where (X,T) are a space-time pair corresponding to a particular energy E value. Using exp(i Action) ((5)) in the time-dependent Schrodinger equation yields a conservation equation which holds for a particular classical energy as long as (x,t) values in the equation are classically linked to the appropriate classical (X,T) values i.e. for a classical trajectory there is a specific family of X at time T for a given energy. For another energy, there is another family of (X,T) pairs. Thus the equation obtained represents energy conservation for many different energy values depending on which simultaneous values one uses for x and t. There is, however, no overlap as a specific (X,T) family corresponds to an energy E and another nonoverlapping family of (X,T) to another energy. This, however, is the same situation as ((2)) for which there are also families of (X,T) corresponding to each En. The only difference is that the En are quantized i.e. discrete. ((3)) and ((5)) are both solutions to the time-dependent Schrodinger equation which represent a variety of energies and families of (X,T) associated to the classical solution of the energy in question. We thus suggest that ((2)) and ((4)) are equivalent for this reason without using the idea of a complete set of basis functions represented by Wn(x). We consider these ideas as the Schrodinger propagator is given by ((5)), but with ((4)) generalized to: x(t)= Xi g(tf-t)/g(tf-ti) + Xf g(t-ti)/g(tf-ti) for the case that g(0)=0 (2) so that Xi,Xf, tf and ti enter the picture. This then should be equivalent to: Sum over n exp(-iEntf) Wn(xf) Wn* (xi) ((6)) In this note, we suggest that one may link ((5)) and ((6)) based on ideas of space-time families associated with different energies as well as the two being solutions of the time-dependent Schrodinger equation.

•Nov 9, 2021•0 citations
A space-time representation is developed for nonrelativistic wave mechanics as was done for classical mechanics, by adding to the system a rotating body to serve as a clock, and applying the time-independent forms of the equations of motion. The usual Schrödinger equation is obtained in this way, however, only in the limit as the energy of the clock becomes very large. The reason for this restriction is found in an appendix, by comparison with the classical equations. Other appendices deal with certain aspects of the Hamiltonian representation of the problem.
Ordinarily, the Jacobi formulation of classical mechanics describes a system by giving the path of a “system point” in “configuration space,” the path being a geodesic if a metric is suitably defined. Nothing is said, however, about the motion in time. It is suggested here that, by adding a degree of freedom in uniform motion to serve as a clock, and identifying the time with the coordinate of that degree of freedom, the configuration space can be extended to a space-time, in which the motion is completely described, both in space and in time, as being along a geodesic. Also, by this representation, the idea of “absolute time” is made unnecessary. A representation such as this is, of course, normal in relativistic theory, but its use here in nonrelativistic mechanics is believed to have some novelty.
In this second paper in a series, we show that the the general statistical approach to nonrelativistic quantum mechanics developed in the first paper yields a representation of quantum spin and magnetic moments based on classical nonrelativistic spinning top models, using Euler angle coordinates. The models allow half-odd-integer spin and predict supraluminal speeds only for electrons and other leptons, which must be treated relativistically. The spin operators in the space-fixed frame satisfy the usual commutation rules, while those in the rotating body-fixed frame satisfy "left-handed" rules. The commutation rules are independent of the structure of the top, so all nonrelativistic rigidly rotating objects must have integer or odd-half-integer spin. Physical boundary conditions restrict all mixed spin states to involve only half-odd-integer or only integer spin eigenstates. For spin 1/2, the theory automatically yields a modified Pauli-Schrödinger equation. The Hamiltonian operator in this equation contains a rigid rotator term and a term involving the square of the magmetic field, as well as an interaction term having the usual form in spherically symmetric and some cylindrically symmetric models, valid for any magnetogyric ratio.
The time-independent Schrodinger equation: En= -1/2m [d/dx dWn/dx] / Wn(x) + V(x) ((1)) matches a classical conservation of energy equation where -1/2m [d/dx dWn/dx] / Wn(x) = KEclassical(x). In classical physics, however, one has x(t) so each x point is associated with a specific t for a given energy En. In the quantum time-independent equation, one has exp(-iEnt) as the time dependent factor and does not consider any link with x although if one is comparing ((1)) to a classical conservation equation then a link between x and t could be made to complete the analogy. This is usually not considered, but we examine this point as we argue it becomes important in the time-dependent situation. For the time-dependent Schrodinger case we consider two approaches. First, a linear combination of time independent solutions is taken i.e. W(x,t) = Sum over n an(En) exp(-iEnt) Wn(x) ((2)) Again each Wn(x) has an association with KEclassical(x) and so one may think in terms of (x,t) classical pairs for a given En. The second approach is to consider the function: exp(-iEnt + m Integral (0,x) v(y) dy) ((3)). This satisfies the time-dependent Schrodinger equation and yields a classical conservation of energy equation with v(x) being the classical velocity. Noting that En= .5m v(x)v(x) + V(x), ((3)) may be rewritten as: exp(i Integral L dt) where L = T-V is the Langrangian with T=.5mv(x)v(x). At this point En has been removed from the picture, but is replaced with: x(t)= X g(t)/g(T) ((4)) where (X,T) are a space-time pair corresponding to a particular energy E value. Using exp(i Action) ((5)) in the time-dependent Schrodinger equation yields a conservation equation which holds for a particular classical energy as long as (x,t) values in the equation are classically linked to the appropriate classical (X,T) values i.e. for a classical trajectory there is a specific family of X at time T for a given energy. For another energy, there is another family of (X,T) pairs. Thus the equation obtained represents energy conservation for many different energy values depending on which simultaneous values one uses for x and t. There is, however, no overlap as a specific (X,T) family corresponds to an energy E and another nonoverlapping family of (X,T) to another energy. This, however, is the same situation as ((2)) for which there are also families of (X,T) corresponding to each En. The only difference is that the En are quantized i.e. discrete. ((3)) and ((5)) are both solutions to the time-dependent Schrodinger equation which represent a variety of energies and families of (X,T) associated to the classical solution of the energy in question. We thus suggest that ((2)) and ((4)) are equivalent for this reason without using the idea of a complete set of basis functions represented by Wn(x). We consider these ideas as the Schrodinger propagator is given by ((5)), but with ((4)) generalized to: x(t)= Xi g(tf-t)/g(tf-ti) + Xf g(t-ti)/g(tf-ti) for the case that g(0)=0 (2) so that Xi,Xf, tf and ti enter the picture. This then should be equivalent to: Sum over n exp(-iEntf) Wn(xf) Wn* (xi) ((6)) In this note, we suggest that one may link ((5)) and ((6)) based on ideas of space-time families associated with different energies as well as the two being solutions of the time-dependent Schrodinger equation.