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Chongqing Shifan Daxue xuebao. Ziran kexue ban

An Exact Derivation of Conservation of Momentum for Quantum System Under Invariance of Translation in Space

Hu Xian

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

Symmetry in space and time lead to the conservation laws of energy and momentum,which are a general principle across all fields of physics So.it is of great importance to study symmetrization and conservation laws of mechanical system,While in most of general literatures and textbooks the derivations of conservation of momentum for quantum systems are made by infinitesimal translations in space,by approximate expansion of wave functions as the first order,or by means of the Lie symmetries.In this paper,the conservation of momentum is derived from the picture Schrdinger and the wave functions belonging to the translations in space are made to expand in power series and with the help of Lie symetries,the conservation of momentum is obtained.It is more precise than that in the literatures.

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Symmetry in space and time lead to the conservation laws of energy and momentum,which are a general principle across all fields of physics So.it is of great importance to study symmetrization and conservation laws of mechanical system,While in most of general literatures and textbooks the derivations of conservation of momentum for quantum systems are made by infinitesimal translations in space,by approximate expansion of wave functions as the first order,or by means of the Lie symmetries.In this paper,the conservation of momentum is derived from the picture Schrdinger and the wave functions belonging to the translations in space are made to expand in power series and with the help of Lie symetries,the conservation of momentum is obtained.It is more precise than that in the literatures.
The free Schrodinger equation is shown to be a consequence of space-time homogeneity in the non-relativistic domain. First, we show that space-time translation invariance of natural laws, which is a conseqiuence of space-time homogeneity, entails that the state of a free particle with definite momentum and energy assumes the plane wave form when the time evolution of the state is linear. Next, we show that the conservation of energy and momentum, which is a consequence of space-time translation invariance, may further determine the energy-momentum relation In the non-relativistic domain. These results then lead to the free Schrodinger equation naturally. The new integrated analysis may help understand the origin of the wave equations in quantum theory.
It is often claimed that Einstein is wrong about quantum mechanics. However, when comparisons are made with respect to theoretical foundations rather than experimental results Einstein’s theories are found to be superior. Although quantum mechanics correctly predicts what it is possible to observe (the emissions) with remarkable accuracy it ignores the other half of natural phenomena, that which cannot be observed (the absorptions), thereby violating the conservation of energy. Similarly, the conservation of momentum is violated by disregarding unobservable molecular impulses. Despite irrefutable proof that molecular impulses transfer momentum asymmetrically as shown by Einstein’s derivation of the A and B coefficients, the wave function is assumed to apply symmetrically in time. The deficiencies are corrected by introducing Hamilton’s principle and deriving relativistic equations of motion. This allows non-relativistic quantum mechanics to be described with a physical model and wave function behavior to be interpreted as the combined action of a particle and its associated quantized field. The calculus of variations is applied to the wave function to show that it is an incomplete equation of motion because it yields twice the allowable action minimum. The concept of reality is redefined by showing that reductionism applies to classical phenomena, but not quantum phenomena.
Introduction The application of group theory to study physical problems and their solutions provides a formal method for exploiting the simplifications made possible by the presence of symmetry. Often the symmetry that is readily apparent is the symmetry of the system/object of interest, such as the three-fold axial symmetry of an NH 3 molecule. The symmetry exploited in actual analysis is the symmetry of the Hamiltonian. When alluding to symmetry we usually include geometrical, time-reversal symmetry, and symmetry associated with the exchange of identical particles. Conservation laws of physics are rooted in the symmetries of the underlying space and time. The most common physical laws we are familiar with are actually manifestations of some universal symmetries. For example, the homogeneity and isotropy of space lead to the conservation of linear and angular momentum, respectively, while the homogeneity of time leads to the conservation of energy. Such laws have come to be known as universal conservation laws. As we will delineate in a later chapter, the relation between these classical symmetries and corresponding conserved quantities is beautifully cast in a theorem due to Emmy Noether. At the day-to-day working level of the physicist dealing with quantum mechanics, the application of symmetry restrictions leads to familiar results, such as selection rules and characteristic transformations of eigenfunctions when acted upon by symmetry operations that leave the Hamiltonian of the system invariant.
General remarks As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics – the Schrödinger equation – lies beyond the potentialities of modern mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. Symmetry concepts come about in physics in two ways. First, since any physical process occurs in a real space, we have to make use of one or another coordinate system. The isotropy and homogeneity of space make physically meaningful only those mathematical relationships that remain unchanged under rotations of the axes of the coordinate system, and that impose fairly rigorous constraints on the possible physical laws. Second, every physical object and process features a symmetry which should be taken into account by the physical theory. A common idea underlying particular forms of symmetry is the invariance of a system under a certain set (group) of transformations. The normally considered forms of symmetry are rotational symmetry, which is based on the equivalence of all directions in space, and permutation symmetry, which is caused by identical particles. The operations of the geometrical symmetry group are responsible for appropriate conservation laws. So, the rotational symmetry of a closed system gives rise to the law of conservation of angular momentum. The group of rotations of a three-dimensional space stands apart in atomic spectroscopy. This is mostly due to the high accuracy of the central field approximation, on which the entire modern theory of complex atoms and ions is based.
We propose six principles as the fundamental principles of quantum mechanics: principle of space and time, Galilean principle of relativity, Hamilton's principle, wave principle, probability principle, and principle of indestructibility and increatiblity of particles. We deductively develop the formalism of quantum mechanics on the basis of them: we determine the form of the Lagrangian that satisfies requirements of these principles, and obtain the Schroedinger equation from the Lagrangian. We also derive the canonical commutation relations. Then we adopt the following four guide lines. First, we do not premise the relations between dynamical variables in classical mechanics. Second, we define energy, momentum, and angular momentum as the constants of motion that are derived from homogeneity and isotropy in space and time on the basis of principle of space and time. Since energy and momentum are quantitatively defined in classical mechanics, we define them in quantum mechanics so that the corresponding conservation laws are satisfied in a coupling system of a quantum particle and a classical particle. Third, we define Planck's constant and the mass of a particle as proportionality constants between energy and frequency due to one of Einstein-de Broglie formulas and between momentum and velocity, respectively. We shall obtain the canonical commutation relations and the Schroedinger equation for a particle in an external field in the definitive form. We shall also prove that relations between dynamical variables in quantum mechanics have the same forms for those in classical mechanics.