An Adventure with Hermitian Operators Linear operators are linear maps from the Hilbert space to itself, i.e. the functions L from wavefunctions to wavefunctions such that L(ov+ BD) = QLV+BLd, for all wavefunctions I, D and all complex numbers o, B. A Hermitian operator A is one whose adjoint is equal to itself, At = A, i.e. for all V, D, (4, AD) = V(x)[AD](x)dx = [AV]'(x)@(x) do = ( AV, Q). (1) Physical measurables are represented by Hermitian operators. We know two such opera- tors, the position and momentum, i : [iv](x) = xV(x), p : [pv](x) = -ih- T ( ). (2) For the following maps, check whether they are linear operators, and if so, whether they are Hermitian. (a x (b) A : Av(x) = V(x)V(x) (c) p (d) B : BY = -i12021+ azz tip I, for some fixed L (e) Po : P.V(x) = $(x) f D*(y) (y) dy, for some fixed $(x) (f) Q. : QV(x) = $(x) fv*(y)$(y) dy, for some fixed $(x) (g) Ta : T. V(x) = V(x + a), for some fixed a (h) xp (i) xp + px