What is the inner product in state (ket/Hilbert) space.

[jdstokes]

In Quantum mechanics books they usually first introduce a vector space called the ket-space and then associate using (Riesz representation theorem I believe) to each ket a corresponding element of the linear dual space.

Then they write the inner product of $|x\rangle$ and $|y\rangle$ (say) by calling on the dual to $|x\rangle$:

$\langle x | y\rangle$.

There appears to be a flaw in the logic here. To employ the Riesz rep theorem we must already have knowledge of the inner product on the state space.

How is this inner product explicitly defined in QM?? I can't see it written anywhere in Sakurai.

 

[George Jones]

In Quantum mechanics books they usually first introduce a vector space called the ket-space and then associate using (Riesz representation theorem I believe) to each ket a corresponding element of the linear dual space.

Then they write the inner product of $|x\rangle$ and $|y\rangle$ (say) by calling on the dual to $|x\rangle$:

$\langle x | y\rangle$.

There appears to be a flaw in the logic here. To employ the Riesz rep theorem we must already have knowledge of the inner product on the state space.

How is this inner product explicitly defined in QM?? I can't see it written anywhere in Sakurai.

Often it's enough to say state space is a Hilbert space, and to use formally the properties of Hilbert spaces, inner products, operators, etc. Sometime concrete realizations are useful, like, for example, the completion of the inner product space of square-integrable functions.

 

[Marco_84]

In Quantum mechanics books they usually first introduce a vector space called the ket-space and then associate using (Riesz representation theorem I believe) to each ket a corresponding element of the linear dual space.

Then they write the inner product of $|x\rangle$ and $|y\rangle$ (say) by calling on the dual to $|x\rangle$:

$\langle x | y\rangle$.

There appears to be a flaw in the logic here. To employ the Riesz rep theorem we must already have knowledge of the inner product on the state space.

How is this inner product explicitly defined in QM?? I can't see it written anywhere in Sakurai.

more formally a system is described by the state space which is: H/C*.

H= Hilbert space
C*= C-(0)

IN other word is a projective space... sometimes called tha rays space.
a state vector is always normalized and what is important in QM is the relative phase between state vectors.

regards
marco