Very elementary notation question

[romistrub]

Shankar p68-69 gives a mathematical "derivation" of the action of the X (position) operator, the summary of which is as follows:

\left\langle x \left| \textbf{X} \right| f \right\rangle = \dots = xf(x)

I followed the logic without a problem, since it only involves using the matrix elements of X in the basis of eigenfunctions of X. However, the next paragraph reads:

We can summarize the action of X in Hilbert space as
\textbf{X} \left| f(x) \right\rangle = \left|xf(x)\right\rangle.

Similarly, he writes, of the action of X in the K basis

\textbf{X} \left| g(k) \right\rangle = \left|i\frac{dg(k)}{dk}\right\rangle

Now, to me

f(x) = \left\langle x | f \right\rangle

and

g(k) = \left\langle k | g \right\rangle

are scalars. Hence I cannot comprehend what is intended by Shankar's notation. Any insight?

 

[romistrub]

I am wondering if the next computation gives some insight: Shankar then computes the matrix elements of X in the K basis as:

\left\langle k \left| \textbf{X} \right| k' \right\rangle = \frac{1}{2\pi}\int^{\infty}_{-\infty}e^{-ikx}xe^{ikx}dx

where, again, to me

\left\langle x|k\right\rangle \propto e^{ikx}

and not

\left|k\right\rangle \propto e^{ikx}

 

[kanato]

Inside the integral you have

e^{-ikx} x e^{ik'x}

= e^{-ikx} \frac{1}i \frac{d}{dk'} e^{ik'x}

which is what leads him to write that \textbf{X} \left| g(k) \right\rangle = \left|i\frac{dg(k)}{dk}\right\rangle.The notation |f(x)\rangle as the ket corresponding to f(x) (which he says near the top of pg. 69) is sloppy, but I don't think there is anything wrong with it. He's not saying that |k\rangle \propto e^{ikx}, he's just using it as a notation to express the ket that comes out of the operation X|f\rangle.

 

[humanino]

\left\langle x|k\right\rangle \propto e^{ikx}

and not

\left|k\right\rangle \propto e^{ikx}

This is correct. To go from left to right, simply insert
1=\int_{-\infty}^{\infty}\text{d}x\left|x\right\rangle\left\langle x\right|