When is the norm of a state equal to 1?

[dEdt]

My textbook says that all physical vectors in the quantum mechanical vector space are unit vectors. But elsewhere, there are quantities like <a|a> which are not assumed to be equal to 1. Why the discrepancy, and under what situations does a state have/not have norm 1?

 

[dextercioby]

Normally, you should not encounter <a|a>, unless there's no reference to physics. QM is built on the normalization to unit for a state vector. It gives a simpler/nicer path to its probabilistic interpretation.

 

[Hurkyl]

My textbook says that all physical vectors in the quantum mechanical vector space are unit vectors. But elsewhere, there are quantities like <a|a> which are not assumed to be equal to 1. Why the discrepancy, and under what situations does a state have/not have norm 1?

The discrepancy is resolved by noting that |a> does not refer to a quantum state. Strictly speaking, it is not even a ket, but rather a generalized ket!

(I assume |a> is referring to a position eigen"state")

 

[ChrisLM]

Just recall why do we need the wavefunction is normalized. ===> we need the wavefunction of the system be normalized because we know that the particle must be somewhere, so <psi|psi> = 1, ( equivalently, probability over all space is 1 ) .
However, we know that any state |psi can be expresseed in linear combination of the stationary state ( the eigen vectors of the harmiltonian of the system ) ,
|psi> = |a1> + |a2> +...|an> ( for simplicity, let's consider there are finite number of eigenvector)
the |ai> is the solution of the time independent Schrödinger equation, they live in the Hilbert space.
Here comes the point, these |ai> is not normalized ( <ai|ai> = 1) but normalizable (<ai|ai> < infinite)
the explanation is the following,
if |ai> is normalized, then the |psi> cannot be normalized. however, |psi> MUST BE normalized. then |ai> can not be normalzied.
therefore, the condition for the |ai> is loose, we only restrict them to be finite.
In summary, state of the system |psi> , the one representing the particle, MUST BE normalized. while for the eigenstate of the system, the one representing stationary state, is not necessary normalized. However, for convenience, you can always find in textbook where the eigenstate is normalized, but as we see that this is impossible, therefore, if we are using normalized eigenstate, we don't write
|psi> = |a1> + |a2> +...|an>
but
|psi> = A1|a1> + A2|a2> +...+An|an>
to ensure the |psi> is normalized.