Study Of The Space Translation Invariance Of Quantum Systems

The behaviors of quantum system's particles obey the uncertainty theory. However, the probability that the particles are found at specifically points in space isn't changable, and the energy is invariance as the coordinates change. So, we can make use of the property to argue the conservation law of momentum at the moment of quantum system translation in space.First, the disquisition reviewed briefly Noether's theorems and the connection of symmetries and invariance in quantum mechanics, discussed the essential property of quantum system's translation, and argued the form of momentum operator, then introduced the methods of researching the invariance of translation in many textbooks and literatures. And then, according toschrddinger picture, the disquisition expanded the wave function in entire power series that is gained by infinitesimal translation in Descartes coordinates. Because the wave function is single-value, finite, continuous, the disquisition reset the second and the higher order items in expanding wave function, recurred to Lie symmetry and energy conservation, found that these items' coefficients equal zero, derived the momentum conservation of quantum system.At last, considering the general principle of covariance, the disquisition derived the method of quantization canonical momentum operator in generic curve coordinates from quantum Poission Bracket, and solved the projective momentum operators in sphere coordinate and column coordinate, and then by using the results derived space translation invariance of quantum system.