CHAPTER 2 - CONSERVATION LAWS IN QUANTUM MECHANICS

Abstract

This chapter discusses conservation laws in quantum mechanics. It describes the Hamiltonian operator, the differentiation of operators with respect to time, stationary states, matrices of physical quantities, and different types of momentum. The wave function completely determines the state of a physical system in quantum mechanics. This means that, if this function is given at some instant, not only are all the properties of the system at that instant described, but its behavior at all subsequent instants is determined. The mathematical expression of this fact is that the value of the derivative of the wave function with respect to time at any given instant must be deter-mined by the value of the function itself at that instant, and, by the principle of superposition, the relation between them must be linear. Further, if the system is closed or is in a constant external field, its Hamiltonian cannot contain the time explicitly. This follows from the fact that all times are equivalent so far as the given physical system is concerned. Since any operator of course commutes with itself, a conclusion can drawn that Hamilton's function is conserved for systems which are not in a varying external field. As is well known, a Hamilton's function which is conserved is called the energy. The law of conservation of energy in quantum mechanics signifies that, if in a given state the energy has a definite value, this value remains constant in time. States in which the energy has definite values are called stationary states of a system.