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Journal of Henan Institute of Education

Discussion on the Energy Conservation for a Nonrelativistic Quantum Mechanics System

Yue Su

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The energy Conservation for a non relativistic quantum mechanics system is derived and discussed

Related papers

Comparison of Nonrelativistic Quantum and Newtonian Mechanics in terms of Information Suggesting the Latter is not a Limit of the Former

Francesco R. Ruggeri

In (1) we argued that one may derive Newton’s second law (which is equivalent to an energy conservation equation) by using information arguments. We argued that V(x), a potential, is a fixed piece of spatial information linked to the x axis being divided into equal dx units. The variable spatial information is dt(x) i.e. the time spent in each dx or v(x) =dx/dt (to put this in the same form as the time displacement generator). One may not simply compare v(x) directly with V(x) because they change differently. v(x) changes in a weighted way i.e. v(x)d/dx or d/dt while V(x) changes as d/dx V(x). Thus we suggest d/dt v(x) = C V(x) which ultimately leads to Newton’s second law. Using d/dt → v(x)d/dx and integrating yields conservation of energy: p(x)p(x)/2m + V(x) = E. A similar argument was used in (2) to deal with the special relativistic case. There one cannot use d/dt → v(x)d/dx to obtain a conservation of energy equation. The point we wish to make is that for both the nonrelativistic and relativistic classical equations there are two types of “information” equations. The first is a “force” i.e. -dV/dx equation which is based on how V(x) changes with x. The LHS of this equation is linked to d/dt p(x). The second is a corresponding energy conservation equation. The quantum case is quite different, we argue, because a free particle is described by exp(-iEt+ipx). There is still the notion of p=mo dx/dt(x) as in the nonrelativistic case because p must be defined, but exp(ipx) is additional imposed piece of spatial information. -id/dx exp(ipx) = p exp(ipx) and -d/dx d/dx exp(ipx) = pp exp(ipx). This begs the question: How does one reconcile the two? For the nonrelativistic Newtonian and relativistic cases it is possible to have two equations (one with Force the other with energy) using one x dependent variable p(x). In the quantum case one may take an average to define a kind of complex = Sum over p a(p) p exp(ipx)/ Sum over p a(p)exp(ipx, but this is complex, not real. Furthermore both +/- is included in so there seems to be a problem linking it with changes in V(x) in one direction. Thus it seems that one is forced to consider the second kind of information equation from classical mechanics and define a <pp> = Sum over p a(p) pp exp(ipx)/ {Sum over p a(p)exp(ipx)}. In the nonrelativsitic case this may be multiplied by 1/2mo and then added to V(x) to obtain a constant E. If one only looks at <pp>/2mo as a spatial function then this is identical to KE(x) i.e. kinetic energy(x) for a classical system. Thus in the nonrelativstic case, Newtonian energy conservation is always embedded in the quantum system regardless of the energy. In other words the quantum bound equation i.e. the time independent Schrodinger is not obtained independently from the classical one in terms of information.One may ask: How does information related to exp(ipx) affect the quantum system? Two immediate ways are as follows. First <pp>/2mo = -1/2mo d/dx dW/dx / W where W= Sum over p a(p)exp(ipx). This leads to a differential equation which has discrete En values verified by experiment. Secondly W is linked to spatial density W*(x)W(x) which exhibits crests and near zero regions. Using <pp> suggests that p(x) in the classical case is linked to prms(x) quantum. Thus building quantum mechanics using “informational” ideas seems to involve embedding classical conservation of energy into the quantum equation a priori. This holds both nonrelativstically or relativistically (Klein-Gordon equation). This immediately raises the question: If classical mechanics (in the form of energy conservation) is embedded in quantum mechanics for any energy, why do people seek to take some kind of limit (3) to obtain classical mechanics from quantum? We argue that quantum mechanics has more spatial information than classical, but retains classical conservation of energy so there is no limit to take. One may simply ignore ideas related to the wavelength to obtain classical mechanics, we argue.

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Speculation on Relation Between Special Relativity, Newtonian Mechanics and Simple Quantum Mechanics

Francesco R. Ruggeri

Historically, special relativity followed as a generalization of Newtonian mechanics and later quantum mechanics (both nonrelativistic and relativistic) was developed as a somewhat independent theory with some connections to Newtonian/special relativistic concepts such as energy and momentum and their conservation. For example, the Newtonian concept of a potential V(x) is used in quantum mechanics. Here we wish to consider the notion of a system seen from different frames moving with constant speed (-v) together with some basic physical ideas that are presumed to hold at any scale such as the idea of a push or pull changing the speed of a particle and the notion of speed itself. We argue that one may develop special relativistic equations in such a manner, but at the same time consider notions of dp/dt and dE/dx linked not only to Newtonian mechanical ideas, but at the same time to quantum mechanics, by generalizing these forms. We argue that one may start with the notion of energy, momentum and their conservation together with the generalizations of a linear force linked to finite dt=hbar/E and dx=hbar/p. This allows one to a priori write, based on conservation of energy for v<

•Feb 3, 2026•0 citations
In (1) we argued that one may derive Newton’s second law (which is equivalent to an energy conservation equation) by using information arguments. We argued that V(x), a potential, is a fixed piece of spatial information linked to the x axis being divided into equal dx units. The variable spatial information is dt(x) i.e. the time spent in each dx or v(x) =dx/dt (to put this in the same form as the time displacement generator). One may not simply compare v(x) directly with V(x) because they change differently. v(x) changes in a weighted way i.e. v(x)d/dx or d/dt while V(x) changes as d/dx V(x). Thus we suggest d/dt v(x) = C V(x) which ultimately leads to Newton’s second law. Using d/dt → v(x)d/dx and integrating yields conservation of energy: p(x)p(x)/2m + V(x) = E. A similar argument was used in (2) to deal with the special relativistic case. There one cannot use d/dt → v(x)d/dx to obtain a conservation of energy equation. The point we wish to make is that for both the nonrelativistic and relativistic classical equations there are two types of “information” equations. The first is a “force” i.e. -dV/dx equation which is based on how V(x) changes with x. The LHS of this equation is linked to d/dt p(x). The second is a corresponding energy conservation equation. The quantum case is quite different, we argue, because a free particle is described by exp(-iEt+ipx). There is still the notion of p=mo dx/dt(x) as in the nonrelativistic case because p must be defined, but exp(ipx) is additional imposed piece of spatial information. -id/dx exp(ipx) = p exp(ipx) and -d/dx d/dx exp(ipx) = pp exp(ipx). This begs the question: How does one reconcile the two? For the nonrelativistic Newtonian and relativistic cases it is possible to have two equations (one with Force the other with energy) using one x dependent variable p(x). In the quantum case one may take an average to define a kind of complex = Sum over p a(p) p exp(ipx)/ Sum over p a(p)exp(ipx, but this is complex, not real. Furthermore both +/- is included in so there seems to be a problem linking it with changes in V(x) in one direction. Thus it seems that one is forced to consider the second kind of information equation from classical mechanics and define a <pp> = Sum over p a(p) pp exp(ipx)/ {Sum over p a(p)exp(ipx)}. In the nonrelativsitic case this may be multiplied by 1/2mo and then added to V(x) to obtain a constant E. If one only looks at <pp>/2mo as a spatial function then this is identical to KE(x) i.e. kinetic energy(x) for a classical system. Thus in the nonrelativstic case, Newtonian energy conservation is always embedded in the quantum system regardless of the energy. In other words the quantum bound equation i.e. the time independent Schrodinger is not obtained independently from the classical one in terms of information.One may ask: How does information related to exp(ipx) affect the quantum system? Two immediate ways are as follows. First <pp>/2mo = -1/2mo d/dx dW/dx / W where W= Sum over p a(p)exp(ipx). This leads to a differential equation which has discrete En values verified by experiment. Secondly W is linked to spatial density W*(x)W(x) which exhibits crests and near zero regions. Using <pp> suggests that p(x) in the classical case is linked to prms(x) quantum. Thus building quantum mechanics using “informational” ideas seems to involve embedding classical conservation of energy into the quantum equation a priori. This holds both nonrelativstically or relativistically (Klein-Gordon equation). This immediately raises the question: If classical mechanics (in the form of energy conservation) is embedded in quantum mechanics for any energy, why do people seek to take some kind of limit (3) to obtain classical mechanics from quantum? We argue that quantum mechanics has more spatial information than classical, but retains classical conservation of energy so there is no limit to take. One may simply ignore ideas related to the wavelength to obtain classical mechanics, we argue.
The interrelationship between energy and probability conservation is explored from the point of view of statistical physics and non-relativistic quantum mechanics. The simultaneous validity of the law of conservation of energy and the continuity equation (probability conservation) breaks for an interacting dynamical system. A separate and independent description of a physical system can be obtained by requiring that the law of conservation of probability is at the heart of the derivation of the ''equations of motion''. In effect, The Schrodinger equation can be viewed as an appropriate factorization of the continuity equation instead of an energy conservation relation per se.
Historically, special relativity followed as a generalization of Newtonian mechanics and later quantum mechanics (both nonrelativistic and relativistic) was developed as a somewhat independent theory with some connections to Newtonian/special relativistic concepts such as energy and momentum and their conservation. For example, the Newtonian concept of a potential V(x) is used in quantum mechanics. Here we wish to consider the notion of a system seen from different frames moving with constant speed (-v) together with some basic physical ideas that are presumed to hold at any scale such as the idea of a push or pull changing the speed of a particle and the notion of speed itself. We argue that one may develop special relativistic equations in such a manner, but at the same time consider notions of dp/dt and dE/dx linked not only to Newtonian mechanical ideas, but at the same time to quantum mechanics, by generalizing these forms. We argue that one may start with the notion of energy, momentum and their conservation together with the generalizations of a linear force linked to finite dt=hbar/E and dx=hbar/p. This allows one to a priori write, based on conservation of energy for v<