1. show that the commutator obeys:
[A,B] = -[B,A]
[A,B+C]=[A,B] + [A,C]
Given the fundamental commutator relation between momentum and position [x,p] = ih
a. [x^n,p] = ihn*x^(n-1)
b. [x,p^n] = ihn*p^(n-1)
c. show that if f(x) can be expanded in polynomial in x and g(p) can be expanded in a polynomial in p, then [f(x),p] = ih*df/dx
and [x,g(p)] = ih * dg/dp
See attached files
Before we begin, some more commutation identities:
We shall use induction to prove that .
We show that for n=1,2,3 this statement is true:
So now we assume it is true for n, and we have to show that if so, it must be true for n+1 as well:
And this concludes the proof (we have shown that it is true for n=3, and therefore must be true for n=4. But then, if it is true for n=4 then it is true for n=5 and so forth and so on)
The same procedure is used to show that :
And if true for n, then it is true for n+1:
If the function is a polynomial expansion in ...
The 13 pages file describes in detail how to solve the three problems. The third problem deals with the Hamiltonian operatora nd expectation values of severeal operators.