CHEM 120A Study Guide - Midterm Guide: Linear Map
Document Summary
1. (a) an operator o is said to be linear if for any pair of function f and g and scalar c1 and c2, it satis es. O(c1f + c2g) = c1 o(f ) + c2 o(g) Use this condition to check if a, b and c are linear operator: A(cid:0)c1f (x) + c2g(x)(cid:1) = cos(x)(cid:0)c1f (x) + c2g(x)(cid:1) So we know that a is a linear operator. B(cid:0)c1f (x) + c2g(x)(cid:1) = (c1f (x) + c2g(x)(cid:1) Thus c is a linear operator. (b) in order for the observables of a and c to be known simultaneously, a and c must share the same eigenvectors, which means the commutator [ a, c] = 0. To evaluate the commutator, we should have [ a, c], which is also an operator, act on an arbitary function f (x), and check if [ a, c]f (x) = 0. C af (x) = c(cid:0) cos(x)f (x)(cid:1) = i.
