Question on notation in Shankar's Quantum Mechanics - math intro on vector spaces I'm just beginning Shankar's 2nd edition Quantum Mechanics and having some trouble with notation. He defines his vectors as "$\left|V\right>$" . And with a scalar multiplier as "$a\left|V\right>$" . When he gets into inner products a couple of pages later he does this:
$$\left<V\right|\Bigl(a\left|W\right>+b\left|Z\right>\Bigr) \equiv \left<V|aW+bZ\right>$$ 
My question is whether $aW$ and $bZ$ has meaning? Or, is this just a definition of a new notation to simplify $\left<V\right|\Bigl(a\left|W\right>+b\left|Z\right>\Bigr)$? If so, does this mean that aW shouldn't be used outside of this specific case?
 A: It is sloppy formulation. Either he introduces silently the abbreviation 
$|aW+bZ\rangle$ for $a|W\rangle+b|Z\rangle$, or he meant to write 
$a⟨V|W⟩+b⟨V|Z⟩$ in place of $⟨V|aW+bZ⟩$. As you didn't provide more context, it is difficult to say more.
A: Bra/Ket notation can be used in different ways. (1) It takes the $W$ in $\left|W\right>$ to be a vector itself, so that the brackets just indicate whether we are dealing with the vector or covector form, $\left|W\right>$ or $\left<W\right|$. In this form, $aW$ makes sense, it is a scalar multiple of the vector $W$, and so does $aW+bZ$.
(2) Bra/Ket notation sometimes takes $\left|W\right>$ to be an eigenvector of some operator, say $\hat A$, with eigenvalue $W$, so that $\hat A\left|W\right>=W\left|W\right>$. In this case, $W$ is a number, and $\left|W\right>$ is the eigenvector of $\hat A$ with the eigenvalue $W$. One has to keep track of which operator one is working with, which is sometimes done by using arrows, distinguishing between $\left|\leftarrow\right>,\left|\rightarrow\right>$ and $\left|\uparrow\right>,\left|\downarrow\right>$, say, instead of writing $\left|+1\right>,\left|-1\right>$ for both, but not infrequently it's obvious to experts which operator is intended but not necessarily so obvious to others. [The vacuum vector in quantum field theory is often denoted $\left|0\right>$, for example, because it is the zero-eigenvalue eigenstate of every annihilation operator, but it is not an eigenstate of any creation operator. EDIT: note that the vacuum vector $\left|0\right>$ is not the zero vector.] If this is how the notation is being used here, then it's problematic because in general $\hat A(a\left|W\right>+b\left|Z\right>)\not=(aW+bZ)(a\left|W\right>+b\left|Z\right>)$. This is so egregious that I cannot imagine this is the way that Shankar is using the notation.
