How to Evaluate \exp (i f(A)) in Ket-Bra Form for a Hermitian Operator?

[jdstokes]

Evaluate $\exp (i f(A))$ in ket-bra form, where A is a Hermitian operator whose eigenvalues are known.

$\exp (i f(A)) = \exp(i f(\sum_i a_i \langle a_i |))$. I'm a little bit stuck on where to go from here. Is f supposed to be a matrix values function of a matrix variable or what?

 

[George Jones]

Evaluate $\exp (i f(A))$ in ket-bra form, where A is a Hermitian operator whose eigenvalues are known.

$\exp (i f(A)) = \exp(i f(\sum_i a_i \langle a_i |))$. I'm a little bit stuck on where to go from here. Is f supposed to be a matrix values function of a matrix variable or what?

Assume $f:\mathbb{R} \rightarrow \mathbb{R}$ can be expressed using a power series expansion:

$$f \left( x \right) = \sum_j c_j x^j.$$

In a standard abuse (should use a different symbol, maybe $\hat{f}$) of notation, define

$$f \left( A \right) = \sum_j c_j A^j.$$

Now use

$$A = \sum_i a_i \left| a_i \right> \left< a_i \right|$$

with $\left\{ \left| a_i \right> \right\}$ chosen to be orthonormal.

 

[jdstokes]

Interesting. I've seen this done for exp, but never thought about generalizing to arbitrary $f: \mathbb{R} \to\mathbb{R}$. I wish physicists would be more careful with their notation sometimes. Thanks.

 

[George Jones]

Interesting. I've seen this done for exp, but never thought about generalizing to arbitrary $f: \mathbb{R} \to\mathbb{R}$. I wish physicists would be more careful with their notation sometimes. Thanks.

Sometimes it might be $f: \mathbb{C} \rightarrow \mathbb{C}$. I think $f: \mathbb{R} \rightarrow \mathbb{R}$ is okay here, since the eigenvalues of $A$ are real.