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

Dr_Nate
Science Advisor
Messages
264
Reaction score
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
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 Albertus Magnus, Dr_Nate, weirdoguy and 3 others
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.
 
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
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In her YouTube video Bell’s Theorem Experiments on Entangled Photons, Dr. Fugate shows how polarization-entangled photons violate Bell’s inequality. In this Insight, I will use quantum information theory to explain why such entangled photon-polarization qubits violate the version of Bell’s inequality due to John Clauser, Michael Horne, Abner Shimony, and Richard Holt known as the...
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...
I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
Back
Top