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College Physics

Some notes on space translation invariance

Qiong-Gui Lin

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

A generalized analysis for symmetry in quantum mechanics is presented.The associated conserved quantities are extended to time-dependent ones.It is shown that a single particle(including the relativistic case) moving in a static uniform magnetic field and a time-dependent linear potential possesses space translation invariance. The translated wave function and the time-dependent conserved quantities are obtained.

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A generalized analysis for symmetry in quantum mechanics is presented.The associated conserved quantities are extended to time-dependent ones.It is shown that a single particle(including the relativistic case) moving in a static uniform magnetic field and a time-dependent linear potential possesses space translation invariance. The translated wave function and the time-dependent conserved quantities are obtained.
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.
Abstract Two systems for a charged particle are studied using an adaptation of Lewis and Riesenfeld invariant method. The first system involves a particle under the effect of a constant electric field, and the second system adds a parallel magnetic field. In both cases, time-dependent conserved operators are identified, which can be used to derive time-dependent wave functions for the Schrödinger equation, where the time variable is not separable from the space coordinates. These conserved operators are also used to construct unitary operators, which define the symmetries of the systems. Finally, it is shown that the invariance of the wave function under the action of these unitary operators leads to the quantization of resistance equal to an integer number times the Klitzing’s constant.
A connection between quantum mechanical and classical adiabatic invariance is shown for the special case of a charged particle moving in a homogeneous, time-dependent magnetic field. The relationship consists of the fact that the transition probability between energy eigenstates corresponding to the magnnetic field at two different times is a function of a certain classical adiabatic invariant alone. From this relation follows the well-known result that during sufficiently slow changes of the magnetic field the particle will remain in one quantum state which passes continuously through the different instantaneous energies. The adiabatic invariance of this classical quantity is shown directly by solving the equations of niotion when the magnetic field changes linearly with time. (auth)
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.
The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. For example, Landau’s solution for a particle moving in a uniform magnetic field in the plane involves choosing a gauge in which neither translation nor rotation invariance are manifest. We show that locality can be made manifest by passing to a redundant description in which the particle moves on a U(1)-principal bundle over the original configuration space and that symmetry can be made manifest by passing to a corresponding central extension of the original symmetry group by U(1). With the symmetry manifest, one can attempt to solve the problem by using harmonic analysis and we provide a number of examples where this succeeds. One is a solution of the Landau problem in an arbitrary gauge (with either translation invariance or the full Euclidean group manifest). Another example is the motion of a fermionic rigid body, which can be formulated and solved in a manifestly local and symmetric way via a flat connection on the non-trivial U(1)-central extension of the configuration space SO(3) given by U(2).
The starting point of this research is a representation of a quantum particle according to the Schrodinger equation of the conventional quantum mechanics. In this representation, a quantum particle is described by a wave packet in the coordinate space and the conjugated wave packet in the momentum space. The problem is that while in the coordinate space the group velocity is in agreement with one of the Hamilton equations, the group velocity in the momentum space is in contradiction with the other Hamilton equation - a minus sign is missing. Group velocities in agreement with the Hamilton equations are obtained only when in the time dependent phase of a quantum particle wave packet, instead of the Hamiltonian coming from the conventional Schrodinger equation, the Lagrangian is considered. This suggests us to consider the relativistic Lagrangian in the time dependent phase. In this way, the conventional relativistic principle of invariance of the time-space interval gets the more physically understandable form of the invariance of the time dependent phase of a quantum particle - the time dependent phase of a quantum particle is the same in any system of coordinates. Based on the relativistic time dependent phase invariance of a quantum particle, from the group velocities of this particle the relativistic kinematics and dynamics are obtained. The interaction with an electromagnetic field is described by a modification of the time dependent phase with a scalar potential conjugated to time and a vector potential conjugated to the space coordinates. In this paper, we find that, according to the formalism of the general relativity, any matter element in a field of forces is accelerated only perpendicularly to its velocity. This means that the matter propagation of a quantum particle can be conceived in planes perpendicular to velocity, while the mater distribution can be considered in a Fourier representation - quantum waves. We study the quantum particle dynamics in gravitational waves and derive the graviton spin. We consider a quantum particle in electromagnetic field and obtain a Schrödinger-Dirac-type equation, with additional terms describing the velocity dependence of the particle dynamics.