Until recently observables have
been nothing more than self-adjoint operators. However, due to axiomatic
formulations of quantum mechanics, observables have now been placed in a more
abstract setting. With the advent of this abstract concept comes the natural
questions concerning uniqueness and existence. The uniqueness problem considered
here seeks to answer the question: if two bounded observables have the same
expectations in every state, are the observables equal? We say that an observable z is
the sum of two bounded observables x and y if the expectation of z is the sum of
the expectations of x and y for every state. The existence problem would
pose the question: does the sum of two bounded observables exist? The
author has found only partial answers to these questions. It is shown that the
uniqueness property holds for simultaneous observables and certain classes of
nonsimultaneous or complementary observables. The existence property
holds for simultaneous observables, and a counterexample is given to show
that this property does not hold in general. The last section of this paper
considers systems in which the existence and uniqueness properties are known to
hold.
