Commutator of Operators: A Case Study with d/dx and x2 - Prof. William Reinhardt, Assignments of Physical Chemistry

Download Commutator of Operators: A Case Study with d/dx and x2 - Prof. William Reinhardt and more Assignments Physical Chemistry in PDF only on Docsity! Chem 455A Spring 2008 Reinhardt & Stanich Set #5: C) Additional problem: do the operators (a) and (b) of problem 3-5, p 121, commute? If yes, demonstrate this, if not, what is the value of the “commutator”? Remember to “apply the operators” to an arbitrary function of x, say f(x). Solution: to part C) i) many students got this right; ii) many got it almost right!; iii) some students were not sure what to do, even though they correctly figured out which pairs of operators commuted and which didn’t, but didn’t know how to compute the value of the “commuator,” So a worked sample problem may be useful. See also McQ Example 3-5, page 103, where the operators d/dx and x2 are considered. We will compute the “value of the operator” which is the commutator [d/dx, x2]. Some comments two operators, A, B, are “equal” if and only if (or, “iff” as math folks would write) Af(x) = Bf(x) (*) for any suitable function f(x)….note that we are here only thinking one dimension, x, and like the operators in the McQ example, are only considering operators which “act” on f(x), x being the dimension at hand. Note Carefully: it could be that Ag(x) = Bg(x) for some “specially designed function “g(x)”, for example g(x) = 0,…but that would not imply that the operators were equal, they must satisfy the above relationship (*) for any and all suitable f(x)…that’s why you should NEVER choose some particular function like “x”, or “x2”, or sin(x) to apply your operators on: you might get an equality “by accident” and come to an incorrect conclusion. Iff (*) holds for any suitable function f(x), the we write A=B, and this indicates that A, B are the SAME operator, so we can set them equal, NOTE that the “f(x)” has “gone” away in this statement of operator equality…..this is a place where many students were confused!