# Hermitian Operators and the Commutator

## Homework Statement

If A is a Hermitian operator, and [A,B]=0, must B necessarily be Hermitian as well?

olgranpappy
Homework Helper

## Homework Statement

If A is a Hermitian operator, and [A,B]=0, must B necessarily be Hermitian as well?

## The Attempt at a Solution

attempt at solution?

if Y is an eigenstate of both A and B with respective eigenvalues a and b and respective adjoints (A+) and (B+),

<Y|AB|Y> = <Y|BA|Y>
= <Y|Ab|Y> = <(B+)Y|A|Y>
= b<Y|A|Y> = (b*)<Y|A|Y>

Therefore, b=(b*), and so it follows that B=(B+), or B is Hermitian.

olgranpappy
Homework Helper
if Y is an eigenstate of both A and B with respective eigenvalues a and b and respective adjoints (A+) and (B+),

<Y|AB|Y> = <Y|BA|Y>
= <Y|Ab|Y> = <(B+)Y|A|Y>
= b<Y|A|Y> = (b*)<Y|A|Y>

Therefore, b=(b*), and so it follows that B=(B+), or B is Hermitian.

counter example:

consider a hermitian operator H. H commutes with any function of H.

For example, the function
$$U=e^{-iHt}\;.$$

Does U commute with H?

Is U hermitian?