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## Homework Statement

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

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- Thread starter njcc7d
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If A is a Hermitian operator, and [A,B]=0, must B necessarily be Hermitian as well?

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olgranpappy

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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

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<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

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<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

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Much easier: how about B=iA?

- #6

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- #7

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

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