Principles of Quantum Mechanics - mathematical introduction

[AngrySaki]

I'm going through the book: Principles of Quantum Mechanics 2nd edition by R. Shankar

1.
In the mathematical introduction to projection operators (page 22), it writes:
"Consider the expansion of an arbitrary ket $|V\rangle$ in a basis:
$|V\rangle=\sum_{i}^{} |i\rangle\langle i|V\rangle$"


My understanding is that that's only true if the basis is normalized. Is this correct?

If my understanding is correct:
I can't find where the other spots were, but there's been other times where it's felt like what was written would only true for orthanormal bases, but it didn't seem to be explicitly stated. For people who have gone through this book, is it normally assumed for that textbook that you're usually working in an orthanormal basis?


2.
The book often writes kets as $|n\rangle$ where n is an integer (eg. $|1\rangle$, $|2\rangle$, etc...). Does using an integer number have a special meaning, or is it just an identifier?


Thanks

 

[dextercioby]

I'm going through the book: Principles of Quantum Mechanics 2nd edition by R. Shankar

1.
In the mathematical introduction to projection operators (page 22), it writes:
"Consider the expansion of an arbitrary ket $|V\rangle$ in a basis:
$|V\rangle=\sum_{i}^{} |i\rangle\langle i|V\rangle$"


My understanding is that that's only true if the basis is normalized. Is this correct? [...]

Yes, the vectors have to unit norm.

[...]If my understanding is correct:
I can't find where the other spots were, but there's been other times where it's felt like what was written would only true for orthanormal bases, but it didn't seem to be explicitly stated. For people who have gone through this book, is it normally assumed for that textbook that you're usually working in an orthanormal basis? [...]

In QM there's always the assumption that any 2 distinct basis vectors are orthogonal one on another.

[...] 2.
The book often writes kets as $|n\rangle$ where n is an integer (eg. $|1\rangle$, $|2\rangle$, etc...). Does using an integer number have a special meaning, or is it just an identifier? [...]

It's an indentifier, a shortand notation for some countable set of basis vectors.

 

[AngrySaki]

Thanks, that helps a ton.