Thread: Commutator algebra (matrices)

Analytic functions of operators (matrices) A are defined via their Taylor expansion about A = 0. Consider the function

g(x) = exp(x A) B exp(−x A)


where x is real and A, B are operators.

• Compute the derivatives dⁿg(x)/dxⁿ|_x=0 for integer n.

• Show that


e^ABe^-A =B+[A,B]+1/2[A,[A,B]]+1/6[A,[A,[A,B]]]+···

Hey bobby84.

Hint: Remember that the regular rules apply for differentiation in with an operator present but assuming A,B are independent of x, you will need to separate out these and treat them like a "constant" (you can do this based on the identities of linear objects and if you don't know what I mean, check the two major conditions of linearity).

Other than that, it would be helpful if you show us what you have tried.

really i am stumped actually, i havent done serious maths in three years and i feel like i am in the deep end with this.

Hint: Let e(xA) = e(x)*e(A) and e(-xA) = e(-x)*e(-A) and d/dx e(xA) = A*e(xA) = A*e(A)*e(x).

Use the chain rule where d/dx(uv) = v*u' + u*v' (the dashes are the derivatives with respect to x).

cheers that helps alot...