# Hermitian Operators and the Commutator

1. Oct 28, 2008

### njcc7d

1. The problem statement, all variables and given/known data
If A is a Hermitian operator, and [A,B]=0, must B necessarily be Hermitian as well?

2. Relevant equations

3. The attempt at a solution

2. Oct 28, 2008

### olgranpappy

attempt at solution?

3. Oct 28, 2008

### njcc7d

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.

4. Oct 28, 2008

### olgranpappy

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?

5. Oct 28, 2008

### borgwal

Much easier: how about B=iA?

6. Oct 28, 2008

### njcc7d

fair enough. thank you for answering my question, though that makes the problem a little more complicated... i hate it when that happens.

7. Nov 1, 2008

### borgwal

Or the easiest of all: B=iI (with I the identity) :-)