- #1
JBrandonS
- 21
- 0
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: <a''|A|a'>=<a'|A|a'>\delta_{a'a''}= a'\delta_{a'a''}
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 A|a'> = a'|a'> since A is hermitian and rewrite as <a''|a'|a'>
Since a' is real a'=a'^* so we can rewrite as a'<a''|a'>
From here we can use the fact that a'' and a' are orthonormal eigenkets from the same operator so <a''|a'> = \delta_{a'a''} and we finally have a'\delta_{a'a''}
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: <a''|A|a'>=<a'|A|a'>\delta_{a'a''}= a'\delta_{a'a''}
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 A|a'> = a'|a'> since A is hermitian and rewrite as <a''|a'|a'>
Since a' is real a'=a'^* so we can rewrite as a'<a''|a'>
From here we can use the fact that a'' and a' are orthonormal eigenkets from the same operator so <a''|a'> = \delta_{a'a''} and we finally have a'\delta_{a'a''}
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.