ABSTRACT

So far in our development of quantum mechanics, we have given very little consideration to problems involving the time dependence of the wave function. This is rather surprising as in classical mechanics it is time-dependent phenomenai.e., dynamics rather than statics-that command most attention. Moreover, most experimental observations necessarily involve some change in the quantity being observed, so we might expect that the successful prediction of experimental results will require a detailed understanding of the way in which a system changes with time. In fact we have already made some implicit assumptions about time dependence and the principal reason why we have got so far without discussing it in detail is that many observed quantum phenomena are associated with sudden discontinuous changes between otherwise stable states. Thus, most of our information concerning the energy levels of atoms has been obtained from measurements of the frequencies of electromagnetic radiation emitted or absorbed as the atom undergoes a transition from one energy eigenstate to another, assuming the correctness of the formula E = }ω , but not considering the mechanism of the transition in any detail. Assumptions concerning time dependence are also implicit in the quantum theory of measurement which refers to the probability of obtaining a particular result following a measurement performed on a system in a given state; thus, for example, the state vector of a spin-half particle changes from being an eigenvector Sˆz to being one of Sˆx following a measurement of the latter property using an appropriately oriented Stern-Gerlach apparatus. However, as was pointed out in chapter 4, and will be discussed further in chapters 12 and 13, the collapse of the wave function associated with a measurement is not a direct consequence of the time-dependent Schro¨dinger equation.