Expectation Value: My Understanding vs. Prof.

[dEdt]

My understanding was that the expectation value of an observable H for a state |a> is just <a|H|a>. But in a homework problem, my prof. used <H> = <a|H|a>/<a|a>. I'm a little confused by the discrepancy, why the discrepancy?

 

[comote]

The first formula assumes a ray(a vector of norm 1). The second one does not.

 

[dEdt]

How do you derive the second equation then?

 

[dEdt]

And what's the point of discussing the expectation value of a state if its norm isn't 1 (ie isn't physical)?

 

[ChrisLM]

Are you dealing with variation principle (method) of QM ?
Variation Principle : the expectation value of any hermitain operator w.r.t. any vector in hilbert space is larger than the smallest eigenvalue of that hermitain operator.
how given any hermitain operator H, chossing any state |a> will give the expectation value of H if (1) if |a> normalized, use the first equation
(2) if |a> is not normalized or don't know if it is normalized ( that's the general vector in hilbert space. then use the second equation.

 

[arkajad]

How do you derive the second equation then?

You derive the first equation from the second. Why the second? Because the expectation value of the identity operator H=I (eigenvalue 1 and every nonzero vector is an eigenvector) should be 1.