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## Homework Statement

So-called compatible observables correspond to operators which commute, i.e. [A, B] = 0, where [A, B] stands for AB − BA.

a) [3pt] Suppose Hermitian operators A and B represent two compatible observables, and all eigenvalues of A are different. Show that eigenstates of A are also eigenstates of B. Thus we can label the common eigenstates of A and B by the corresponding eigenvalues as |a',b'>

## Homework Equations

[A,B] = AB-BA = 0

## The Attempt at a Solution

A|a'> = a'|a'>

B|b'> = b'|b'>

Multiply A|a'> = a'|a'> by B

BA = B(a'|a'>) (1)

Since AB-BA = 0

AB = BA

I assume I can multiply by |a'> so

Multiply by |a'>

AB|a'> = BA|a'> = Ba'|a'>

I'm not sure if what I'm on the correct path but this is my attempt so far. I'm not sure how I can equate prove the question so I just tried something.

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