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# States and Observables

In classical mechanics, the state of a system may be specified by giving the coordinates and velocities of all of the particles in the system. Once this is done, all measurable quantities such as the momenta , total energy and angular momentum of the system may be determined. Such measurable quantities are known as observables. Under the classical principle of determinism, once the state of the system is specified at time t = 0, the laws of motion may be solved to determine the state of the system for all later times.

The situation in quantum mechanics is not much different. We may still make measurements of the familiar physical observables mentioned above (and some new ones such as spin or polarization which we will learn about later). As we have seen in our discussions of the uncertainty principle and interference experiments, although the results of individual measurements are unpredictable on the quantum scale, measurements on a system in a given state yield a well-defined distribution of results. If we simply accept this notion and generalize our concept of ``experiment" to the procedure of determining the distribution associated with many measurements of the same observable over a set of identically prepared systems, then we may maintain the idea that the quantum state of a system determines the results of all experiments measuring physical observables.

As we shall refer to the quantum state of a system often, we shall make a special symbol for it, . The symbol is called a ``ket.'' We may place different symbols inside the ket to distinguish different states. For instance, our system may be in the state at time t = 0 and in the state at some later time.

Keep in mind that, in general, does not determine the result of any individual measurement of an observable, only probability distributions such as , the probability of finding a particle with position x in the range , or , the probability of finding the momentum of the particle in the range . In the case of photons we may also consider the observable of spin or polarization and measure , the probability of observing a right or left circularly polarized photon.

We mention the polarization of photons above to underscore the fact that we will keep the abstract portion of our discussion completely general. For the most part of the course, however, we will focus on systems of single particles which have no internal structure so that we will concern ourselves with only the observables of position , momentum and those observables which may be derived from these two, such as energy and angular momentum .

Next: Specification of Quantum Up: Notes on Quantum States Previous: Notes on Quantum States

Prof. Tomas Alberto Arias
Wed Oct 11 21:10:55 EDT 1995