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Suppose I know the position or momentum basis representation $X$ of some operator $\hat{X}$ acting on the ket space. Likewise of $\hat{P}, \hat{A}, \hat{B}, ...$

Is the basis representation of $F(\hat{X}, \hat{P}, \hat{A}, \hat{B},..)$ just $F(X, P, A, B,..)$?

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    Things get complicated due to operator ordering – Slereah Jul 12 '19 at 07:20
  • Still I see it commonly used in a lot of QM exercises.. maybe it works just fine with products of operators ? –  Jul 12 '19 at 08:05
  • It works fine as long as you can split F into a function of x and a function of p. Otherwise you'll have to rely on clues such as hermitianness or symmetries – Slereah Jul 12 '19 at 08:38
  • Provide a simple example of what confuses you. – Cosmas Zachos Jul 13 '19 at 00:50
  • @Cosmas Zachos Take for example I know the momentum basis representation $X=x$ and $P=-ih\partial_x$ and I show $[X,P]=ih$. Does this hold true for the operators $\hat{P}$ and $\hat{X}$ on the ket space itself as well? I know it does, but can I generally switch into an arbitrary basis, I think not. –  Jul 13 '19 at 11:09
  • In one of my other questions https://physics.stackexchange.com/questions/490718/proof-that-rotational-symmetric-potential-operators-are-scalar-operators it worked and this got me wondering. –  Jul 13 '19 at 11:10
  • Why do you "think not"? Think of these operators as matrices, and basis changes, etc... What spooks you? – Cosmas Zachos Jul 13 '19 at 12:20
  • The dimension of the hilbert space does. In for example spin-space I do just that and everything is fine, everything makes perfect sense, the math is easy. In the infinitedimensional hilbert space of position and momentum I'm very much lost though, everything I know about linear algebra seems to break down here. Well, maybe a future course in spectral theory will help. –  Jul 13 '19 at 18:44
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    You put no emphasis on that in the question. Things work out pretty nicely, but if you want the fussing taken care of, skim the Reed & Simon book. There are, of course, surpises, but these are the exceptions (curiosities) that prove the rule. – Cosmas Zachos Jul 16 '19 at 13:15

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