Are operators always commutative with respect to operation +?

[kakarukeys]

In general AB =/= BA, for example,
orbital angular momentum operators, L_x, L_y.

but is A + B = B + A always true?

 

[Galileo]

Yes.
(A+B)f=Af+Bf=Bf+Af=(B+A)f, so A+B = B +A

You are simply working with functions (Af and Bf), so the addition is commutative.

 

[kakarukeys]

I think so, but this is puzzling to me:

put hbar = 1
L_x + L_y = L_y + L_x
so, exp(-i theta L_x) exp(-i theta L_y) = exp(-i theta L_y) exp(-i theta L_x)

that means rotation about x-axis and rotation about y-axis are commutative?


but rotate a book around x-axis 90 deg followed by y-axis 90 deg is not same as the other way round.

 

[vanesch]

Warning: for (non-commuting) operators A and B,
you cannot write:

exp(A) exp(B) = exp(A+B)

!

cheers,
patrick.

 

[kakarukeys]

I see,
things become clear when you expand the exponentials

Thank you.

 

[styler]

provided the domains of the operators allow for the sum to be defined, linear operators will respect the addition operation.

 

[turin]

You have to be careful with your order of operations (associativity). Exponentiation first and then addition vs. addition first and then exponentiation. The exponentiation makes different operators, say:

e^A = Ω and e^B = Λ

Not only do you have to worry about A + B = B + A, which is almost trivial in a physics context, you must also worry about [Ω,Λ] = 0, which is often enough untrue.