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- 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?
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?
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