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Homework Statement
Suppose A^ and B^ are linear quantum operators representing two observables A and B of a physical system. What must be true of the commutator [A^,B^] so that the system can have definite values of A and B simultaneously?
Homework Equations
I will use the bra-ket notation for the inner product (sorry for lack of latex)
The Attempt at a Solution
So I assumed that for observables A and B to have a definite value, <psi*|A^|psi> and <psi*|B^|psi> have to be normalizable. Call a and b normalization constants of <psi*|A^|psi> and <psi*|B^|psi> respectively, then: a<psi*|A^|psi> = b<psi*|B^|psi> = 1.
Here I made my sloppy assumption that the equation above implies that A^ and B^ are proportional (A^=(b/a)*A^), which leads to that the commutator must be zero.
I made this assumption as both A^ and B^ are being "operated" by the same inner product, so the bolded equation can be reduced to aA^=bB^. Is this an assumption I can make?
Thanks in advance.