Quantum Physics

Track 4: Wave function

Wave-function representations are one type of representation for a quantum theory in Hilbert space. Any observable has a spectrum associated with it, which is the variety of possible values the observable may have. Consider complex-valued functions on the spectrum of any observable for any physical system, and one can always construct a Hilbert-space representation for the quantum theory of that system. A vector space is formed by the collection of these functions. The set of complex-valued square-integrable functions on the spectrum can be formed into a Hilbert space given a measure on the observable by treating functions that vary only on a set of zero measures as equivalent (that is, the constituents of our Hilbert space are essentially equivalence classes of complex-valued square-integral functions).

A Hilbert-space representation of this kind is referred to as a wave function representation and the functions that represent quantum states, wave functions (also "wave-functions," or "wave functions"), if the spectrum of the chosen observable is a continuum (as it is, for example, for position or momentum). The most well-known illustrations of this type are momentum-space wave functions, which are functions of the momenta of the systems involved, and position-space wave functions, which are functions on the set of possible configurations of the system.