- #1
Demon117
- 165
- 1
So there is a theorem at the beginning of section 1.5 in Sakurai that states the following:
Given two sets of base kets, both satisfying orthonormality and completeness. there exists a unitary operator [itex]U[/itex] such that
[itex]|b^{(1)}> = U|a^{(1)}>,|b^{(2)}> = U|a^{(2)}>,...,|b^{(n)}> = U|a^{(n)}> [/itex]
By a unitary operator we mean an operator fulfilling the conditions
[itex]U^{t}U=1[/itex]
as well as
[itex]UU^{t}=1[/itex]
So this is not difficult to prove. But my real question is can we prove that [itex]U[/itex] is unique or is that just not the case and why?
Given two sets of base kets, both satisfying orthonormality and completeness. there exists a unitary operator [itex]U[/itex] such that
[itex]|b^{(1)}> = U|a^{(1)}>,|b^{(2)}> = U|a^{(2)}>,...,|b^{(n)}> = U|a^{(n)}> [/itex]
By a unitary operator we mean an operator fulfilling the conditions
[itex]U^{t}U=1[/itex]
as well as
[itex]UU^{t}=1[/itex]
So this is not difficult to prove. But my real question is can we prove that [itex]U[/itex] is unique or is that just not the case and why?