We discuss some basic issues that arise when one
attempts to model quantum mechanical systems on a computer, 
and we describe the mathematical structure of the resulting
discretized cannonical commutation relations. The C*-algebras  
associated with the discretized CCRs are the non-commutative
spheres of Bratteli, Elliott, Evans and Kishimoto.  

This is an expository paper with no proofs.  See 
"Discretized CCR algebras", also posted on this page, 
for details.