Consider the position and momentum vector sets

$$X= \{|x\rangle |\ x \in \Bbb R^3\}$$

$$P= \{|p\rangle |\ p \in \Bbb R^3\}$$

By the assumption of quantum mechanics, both $X$ and $P$ are total orthonormal sets. In addition to that, $\langle x|p\rangle = e^{ip \cdot x} \neq 0$, but this conflicts with Kreyszig's functional analysis Lemma 3.5-3 which states: a vector in an inner product space can only has countably many non-zero Fourier coefficients with respect to an orthonormal family.

How's this possible?

Is the position/momentum vector representation in quantum mechanics incorrect?


An arbitrary basis $\{v_\alpha\}_{\alpha\in A}\subset \mathscr H$ may be uncountable, but $S=\{\alpha\in A:h\in \mathscr H;\ \langle h,v_\alpha\rangle \neq 0 \}$ is countable, and in this case, $h=\sum_{\alpha\in S}\langle h,v_\alpha\rangle v_\alpha. $

I am not a physicist, but isn't this fact exactly why $p$ and $x$ are quantized?

  • $\begingroup$ But S in my case is uncountable since both $X$ and $P$ are uncountable and their mutual product are all non zero $\endgroup$ – JokingBear Oct 29 '18 at 2:44
  • $\begingroup$ If you are working in a Hilbert Space, $S$ can not be uncountable. There is a good introduction to these ideas in Rudin's Real and Complex Analysis. $\endgroup$ – Matematleta Oct 29 '18 at 2:48
  • $\begingroup$ Then how can you explain the inner product of $|p>$ and $|x>$ $\endgroup$ – JokingBear Oct 29 '18 at 4:05
  • 1
    $\begingroup$ Please define 1). the Hilbert Space you are working in. 2). The inner product you are using $\endgroup$ – Matematleta Oct 29 '18 at 4:16
  • $\begingroup$ Consider the $L^2(R)$ space and vector $|x>$ is eigen vector of $x$ operator, and |p> is eigen vector of $-i\partial_x$ operators with their inner product $<x|p> = e^{ip \cdot x}$ $\endgroup$ – JokingBear Oct 29 '18 at 4:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.