Why do ##t## and ##-i\hbar\partial_t## not satisfy the definition of a linear map/operator in Hilbert space?

  • #1
Dr_Nate
Science Advisor
264
148
TL;DR Summary
It is common to say that ##t## and ##-i\hbar\partial_t## are not operators in quantum mechanics. But an unambiguous mathematical justification seems lacking.
It is common to say that ##t## and ##-i\hbar\partial_t## are not operators in quantum mechanics. But I haven't seen a satisfying justification.

As an example of the precision of our discourse, someone has said that ##-i\hbar\partial_t## satisfies the definition of Hermicity, but it is not an operator in quantum mechanics. That seems wrong to me because Hermicity requires the above expression to be a linear map/operator.

Alternatively, some say that time is a parameter and not a variable in Hilbert space, so it can't be an operator. However, when I look at the definition of a linear map, I don't see the words \emph{parameter} or \emph{variable} used, so there seems to be a gap in the justification.

Interestingly, no one directly answered this post in that same thread linked above.

Applying ##t## or ##-i\hbar\partial_t## to kets, I don't see a case where additivity and scalar multiplication are not preserved. I don't see how they violate the requirements of a linear mapping back to Hilbert space. Note, though, that I'm an experimentalist, so I can't tell you the difference between a vector space and a field in these definitions (wikipedia link).

Question: How does the application of ##t## and ##-i\hbar\partial_t## to a ket not satisfy the mathematical definition of a linear map/operator in Hilbert space as used in the mathematical formalism of quantum mechanics?
 
Last edited:
  • Like
Likes dextercioby
Physics news on Phys.org
  • #3
The Hilbert space in QM is the space of functions of ##x##, not a space of functions of ##x## and ##t##. The time derivative operator is therefore not an operator in this Hilbert space. The functions in the Hilbert space must have a finite norm, if they were functions of both ##x## and ##t## then the computation of norm would involve integration over both ##x## and ##t##, but the integral over ##t## would not converge because the wave functions satisfying Schrodinger equation do not vanish at ##t\to\pm\infty##. Of course, the wave functions do depend on both ##x## and ##t##, but only the dependence on ##x## is associated with a Hilbert space structure. The dependence on ##t## is also associated with some space of functions, but this space is not a Hilbert space, because the integration over ##t## does not lead to a finite norm.
 
  • Like
  • Informative
Likes Dr_Nate, weirdoguy, haushofer and 2 others
  • #4
PeterDonis said:
Yes, there was an answer given, in post #20 of that thread--which was before the one you linked to here (post #30).
The question I am asking is pretty much about straight math. That answer invokes observables, so isn't what I am looking.
 
  • #5
Demystifier said:
The Hilbert space in QM is the space of functions of ##x##, not a space of functions of ##x## and ##t##. The time derivative operator is therefore not an operator in this Hilbert space. The functions in the Hilbert space must have a finite norm, if they were functions of both ##x## and ##t## then the computation of norm would involve integration over both ##x## and ##t##, but the integral over ##t## would not converge because the wave functions satisfying Schrodinger equation do not vanish at ##t\to\pm\infty##. Of course, the wave functions do depend on both ##x## and ##t##, but only the dependence on ##x## is associated with a Hilbert space structure. The dependence on ##t## is also associated with some space of functions, but this space is not a Hilbert space, because the integration over ##t## does not lead to a finite norm.
Thank you. This is the answer I am looking for. If I recall correctly something like this was said in the linked thread, but now it is clear to me.

I have read many lists of postulates and have never noticed before in those lists that references to Hilbert space were at fixed time. Now, that I know, I can see it's clear on Wikipedia. I'm curious to see how often it's clearly stated in other more formal places.
 
  • Like
Likes dextercioby and Demystifier

Similar threads

Back
Top