Is the energy operator (time derivative) a linear one?

[MHD93]

Typically in mathematics time derivative is linear in the sense that constants are pulled out the operator which then operates on a time dependent function. But in quantum mechanics we say linear to mean that the operator passes over the coefficients of the kets (which themselves might be time dependent, and therefore the derivative is nonlinear).

So is the energy operator linear?

 

[vanhees71]

In quantum mechanics the energy operator (or Hamilton operator) is not the time derivative but some function (or functional) of the fundamental observables (position and momentum operator in quantum mechanics or quantum fields and the conjugate momenta in quantum field theory), and is thus a self-adjoint linear operator on the Hilbert space of state vectors.

 

[MHD93]

the energy operator (or Hamilton operator) is not the time derivative

That's what I always liked and hoped to be the case, and not what wikipedia asserts:

http://en.wikipedia.org/wiki/Energy_operator

Is something wrong with this article? (which I wish to be the case).

 

[vanhees71]

Wow, that's a pretty bad Wikipedia article, indeed :-(. The following is much better:

https://en.wikipedia.org/wiki/Mathe...tum_mechanics#Postulates_of_quantum_mechanics

although also there some imprecisions are present. E.g., observables are not represented by hermitian matrices but by essentially self-adjoint (densely defined) operators on Hilbert space.

The next section on dynamics also shows the correct formulation of the meaning of the Hamiltonian!