Hermitian Operators and the Commutator

  • Thread starter njcc7d
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  • #1
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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

 

Answers and Replies

  • #2
olgranpappy
Homework Helper
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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?
 
  • #3
3
0
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
olgranpappy
Homework Helper
1,271
3
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
[tex]
U=e^{-iHt}\;.
[/tex]

Does U commute with H?

Is U hermitian?
 
  • #5
367
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Much easier: how about B=iA?
 
  • #6
3
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fair enough. thank you for answering my question, though that makes the problem a little more complicated... i hate it when that happens.
 
  • #7
367
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Or the easiest of all: B=iI (with I the identity) :-)
 

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