- #1
Ravi Mohan
- 195
- 21
I am reading an intriguing article on rigged Hilbert space
http://arxiv.org/abs/quant-ph/0502053
On page 8, the author describes the need for rigged Hilbert space. For that, he considers an unbounded operator A, corresponding to some observable in space of square integrable functions \mathcal{H}, with the domain \mathcal{D}(A). The author states that in general, \mathcal{D}(A) does not remain invariant under the action of A.
Now the author claims that such non-invariance makes expectation values ill-defined on the whole Hilbert space \mathcal{H}.
I am not able to understand the claim.
Let us consider \phi\in\mathcal{D}(A). Due to invariance, \psi=A\phi may not belong to \mathcal{D}(A), but it remains in \mathcal{H}. Thus the expectation value (\phi,A\phi)=(\phi,\psi) should be well defined (or am I doing something wrong?).
http://arxiv.org/abs/quant-ph/0502053
On page 8, the author describes the need for rigged Hilbert space. For that, he considers an unbounded operator A, corresponding to some observable in space of square integrable functions \mathcal{H}, with the domain \mathcal{D}(A). The author states that in general, \mathcal{D}(A) does not remain invariant under the action of A.
Now the author claims that such non-invariance makes expectation values ill-defined on the whole Hilbert space \mathcal{H}.
I am not able to understand the claim.
Let us consider \phi\in\mathcal{D}(A). Due to invariance, \psi=A\phi may not belong to \mathcal{D}(A), but it remains in \mathcal{H}. Thus the expectation value (\phi,A\phi)=(\phi,\psi) should be well defined (or am I doing something wrong?).
