[QM] Help understanding this bra-ket solution

[JBrandonS]

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.

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.

 

[JBrandonS]

Ugh I don't know how I just saw what they did. Multiply by the identity operator (a') between <a''| and A. It all falls together then. Still not 100% sure what all it means but I'll work on it.

 

[rude man]

Ugh I don't know how I just saw what they did. Multiply by the identity operator (a') between <a''| and A. It all falls together then. Still not 100% sure what all it means but I'll work on it.

I think this post belongs under Advanced Physics.

 

[JBrandonS]

I think this post belongs under Advanced Physics.

I checked the rules for the advanced physics and it said that just because it's QM doesn't mean it belongs there. So I figured this would be a good place to put it. Either way this question can be closed now as I figured it out. Just can't find out how to mark it for closure.

 

[Nickie]

Dear student,

Thank you for reaching out for help with understanding the bra-ket notation and the given problem. I can understand how it can be confusing, but I will try my best to explain it to you.

First, let's break down the notation. The bra-ket notation is a mathematical notation used in quantum mechanics to represent vectors and operators. The "bra" <a''| represents a vector in the dual space, also known as the bra space, and the "ket" |a'> represents a vector in the original space, also known as the ket space. The notation <a''|A|a'> means the inner product of the bra vector <a''| and the operator A applied to the ket vector |a'>. In simpler terms, it represents the value obtained by applying the operator A to the ket vector |a'>, and then taking the inner product with the bra vector <a''|.

Now, let's move on to the problem at hand. The first step to understanding the solution is to remember that the operator A is Hermitian, which means it is equal to its own adjoint. In other words, A=A†. This property allows us to rewrite the expression A|a'> as A|a'> = A†|a'>.

Next, we can use the fact that a' and a'' are eigenkets of A with corresponding eigenvalues a' and a''. This means that when the operator A is applied to these vectors, it simply gives back the same vector multiplied by its eigenvalue, i.e., A|a'> = a'|a'> and A†|a''> = a''|a''>.

Now, let's come back to the original expression <a''|A|a'>. We can rewrite this as <a''|A|a'> = <a''|A†|a'>, using the property A=A†. Substituting the expressions we obtained earlier, we get <a''|A†|a'> = <a''|a''>|a'> = a''<a''|a'>.

Finally, we can use the orthonormality property of eigenkets to simplify the expression further. Since a'' and a' are orthonormal, their inner product <a''|a'> is equal to 1 if a''=a' and 0 if a''≠a'. This