Hermitian Operators and the Commutator

[njcc7d]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[olgranpappy]

Homework Statement

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

Homework Equations

The Attempt at a Solution

attempt at solution?

 

[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.

 

[olgranpappy]

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?

 

[borgwal]

Much easier: how about B=iA?

 

[njcc7d]

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

 

[borgwal]

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