18.03 Lecture Notes - Lecture 15: Particle Number Operator, Identity Function, Royal Society Of Biology
Document Summary
Several different topics today, and a respite from the gain game. Just as number ------------> number operator function ------------> function. The *differentiation operator d takes x to x" : dx = x" . For example, d sin(t) = cos(t) , d x^n = n x^{n-1} , d8 = 0 . There"s also the "identity operator": ix = x. And we can take linear combinations of operators: (d^2 + 2d + 2i) x = x" + 2x" + 2x . The characteristic polynomial here is p(s) = s^2 + 2s + 2 , and it"s irresistible to write. D^2 + 2d + 2i = p(d) so x" + 2x" + 2x = p(d) x. This formalism lets us discuss linear equations of higher order with no extra work. Such an equation has the form an x^{(n)} + + a1 x" + a0 x = q(t) (*)
