Eigenkets of a function of a hermitian operator

[shehry1]

For a hermitian operator A, does the function f(A) have the same eigenkets as A?

This has been bothering me as I try to solve Sakurai question (1.27, part a). Some of my class fellows decided that it was so and it greatly simplifies the equations and it helps in the next part too but I don't think so because I might add anything to A in order to make it non-hermitian.

 

[Edgardo]

How is a function of an operator A defined? Have a look at page 51 of http://www.lsr.ph.ic.ac.uk/~plenio/lecture.pdf .

 

[shehry1]

How is a function of an operator A defined? Have a look at page 51 of http://www.lsr.ph.ic.ac.uk/~plenio/lecture.pdf .

The operator is completely general but I think that definition 26 would apply. But I have a few question concerning Definition 27:

1. What specifically are the Aⁱ?
2. Doesn't the power series expansion go under the continuous spectrum? I ask this because in the discrete spectrum at least, wouldn't the eigenkets of a hermitian operator be complete?

Thanks for the link btw, I printed the first two chapters :)

 

[olgranpappy]

The operator is completely general but I think that definition 26 would apply. But I have a few question concerning Definition 27:

1. What specifically are the Aⁱ?

powers of A.

E.g., 1, A, A^2, A^3, etc

2. Doesn't the power series expansion go under the continuous spectrum?

under? Don't know exactly what you mean by that...

Often the symbol
$$\sum_n$$
which "looks like" a discrete sum really means
$$\sum_n+\int dn$$
a sum over the discrete part of the spectrum and a integral over the continuous part.