Question

3. (5 points) Chapter 3. #4 (modified). Prove the following properties related to Hermitian operators: (a) If Ô and 6 are Hermitian, so is Ê + 0. (b) If z is any complex number and if Ô is Hermitian, then zÔ is Hermitian if and only if z is real. (c) If Ê and Ộ are Hermitian and if they commute, the Ộ Ô is Hermitian. In your proof, indicate explicitly which step requires the two operators to commute. (d) Prove by explicitly examining the appropriate integral that x is a Hermitian operator. (e) Use all of the above results to argue that for any real potential energy V(x) which can be represented by a Taylor series expansion, V(x) is Hermitian. If you do not see that all parts a-d are required, you are overlooking something.

Answer 1

A3 (1) liven P = Pt ; & = at 8 Now (+8) t = pt tras = pt B :. + + 8 = He with M (b) uiven, = ģt and z is a complex number Now (26) t åt z = 2* * 3 in mly equal to zo when z = 2* a ng i-e when z is real.

1 -> P Q = BP this (e) riven, i = pt § = ſ t [ ?, d]=6 - ? 8 = ô Ĉ Now, ( 3 )- 4+ f = 3 But given p å commute so è å = 6 Ô (i åt - i å •. P6 Ha ha. (d) we are to prove I VČx) ĉ Y; (x) dx. ";(x). Cê Vila ))*dxe which isa i dentily ga Hermittian operator Now, we have a 4*() . x 4; (1) de Now we re arrange the wave function, J Yj(x) a Yi(x)* an . Noting that x=x* since it is not complex in ř 4₂(x) x 4; (x) * dx = J 4; (x) . (x-4; (x) * da and there fore: 4;*() x Y; (x) dx = I 43(x) (x. 4;(21)*dxn