Time as a Hermitian operator in QM?

Question

In non-relativistic QM, on one hand we have the following relations:

$\langle x | P | \psi \rangle = -i \hbar \frac{\partial}{\partial x} \psi(x)$

$\langle p | X | \psi \rangle = i \hbar \frac{\partial}{\partial p} \psi(p)$

On the other hand, despite the similarities, the relations cannot be directly applied to energy and time:

$\langle t | H | \psi \rangle = i \hbar \frac{\partial}{\partial t} \psi(t)$

$\langle E | T | \psi \rangle = -i \hbar \frac{\partial}{\partial E} \psi(E)$

Just wondering, how can one mathematically prove that the "classical time" (which means no QFT or relativity involved), unlike its close relative "position", is not a Hermitian operator?

I ask your pardon if you feel the question clumsy or scattered. But to be honest, if I can clearly point out where the core issue of the problem is, I may have already answered it by myself :/

Thank you in advance.

Answer 1

Time is not a variable in Quantum Mechanics (QM), it's a parameter — much in the same way as it is in Classical (Newtonian) Mechanics.

So, if you have a Hamiltonian, e.g., for the harmonic oscillator, you have $\omega$ as a parameter, as well as the masses of the particle(s) involved, say $m$, and you also have time — even though it's not something that shows up explicitly in the Hamiltonian (remember explicit time dependency from Classical Mechanics: Poisson Brackets, Canonical Transformations, etc — in fact, you could get your answer straight from these kinds of arguments).

In this sense, just like you don't have a 'transformation pair' between $m$ and $\omega$, you also don't have one between time and Energy.

What do you say to convince yourself that $\omega \neq -i\, \partial_m$? Why can't you use this same argument to justify $E \neq -i\, \partial_t$? ;-)

I think Roger Penrose makes a nice illustration of how this whole framework works in his book The Road to Reality: A Complete Guide to the Laws of the Universe: check chapter 17.

Answer 2

The energy spectrum is bounded from below. A time operator would contradict the Stone–von Neumann theorem. This isn't really a problem. All it means is we have limits as to how accurate clocks can be in quantum mechanics.

Answer 3

Just to add to the answers. I believe that the first time it was shown that time cannot be described by an operator was a theorem of Pauli. You can google it.