JBrandonS
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Hello,
I am working my way though Sakurai's book on Quantum MEchanics and am having some problems understanding the bra-ket notation. I keep believing I understand everything there is to it but then he will do something in a single line that I cannot understand. This is one of them. If someone could help me out it would be great.
Show why the following in correct: [itex]<a''|A|a'>=<a'|A|a'>\delta_{a'a''}= a'\delta_{a'a''}[/itex]
A is an hermitian operator. a' and a'' are the eigenkets and eigenvalues of A.
The only method I can think of to coming up with the final solution is the following, which may not even be correct.
Use [itex]A|a'> = a'|a'>[/itex] since A is hermitian and rewrite as [itex]<a''|a'|a'>[/itex]
Since a' is real [itex]a'=a'^*[/itex] so we can rewrite as [itex]a'<a''|a'>[/itex]
From here we can use the fact that a'' and a' are orthonormal eigenkets from the same operator so [itex]<a''|a'> = \delta_{a'a''}[/itex] and we finally have [itex]a'\delta_{a'a''}[/itex]
However this method does no provide the middle expression which has me really thrown off. I am not sure if I am doing everything correct and I do not know how Sakurai came to that. I am also not 100% on what all this means either.
I am working my way though Sakurai's book on Quantum MEchanics and am having some problems understanding the bra-ket notation. I keep believing I understand everything there is to it but then he will do something in a single line that I cannot understand. This is one of them. If someone could help me out it would be great.
Homework Statement
Show why the following in correct: [itex]<a''|A|a'>=<a'|A|a'>\delta_{a'a''}= a'\delta_{a'a''}[/itex]
A is an hermitian operator. a' and a'' are the eigenkets and eigenvalues of A.
Homework Equations
The Attempt at a Solution
The only method I can think of to coming up with the final solution is the following, which may not even be correct.
Use [itex]A|a'> = a'|a'>[/itex] since A is hermitian and rewrite as [itex]<a''|a'|a'>[/itex]
Since a' is real [itex]a'=a'^*[/itex] so we can rewrite as [itex]a'<a''|a'>[/itex]
From here we can use the fact that a'' and a' are orthonormal eigenkets from the same operator so [itex]<a''|a'> = \delta_{a'a''}[/itex] and we finally have [itex]a'\delta_{a'a''}[/itex]
However this method does no provide the middle expression which has me really thrown off. I am not sure if I am doing everything correct and I do not know how Sakurai came to that. I am also not 100% on what all this means either.