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I've recently started studying quantum field theory and I'm trying to understand the field operators.

One thing that bugs me is the difference between field operators and wave mechanics operators. For instance, let's take the kinetic energy operator in wave mechanics for a single particle

[tex] T_1(\mathbf{r}) = -\frac{\hbar^2}{2m} \nabla^2 + V_1(\mathbf{r}) [/tex]

And then we have the standard field operator, usually written as

[tex] \psi(\mathbf{r}) = \sum_k c_k u_k(\mathbf{r}) [/tex]

where [itex]c_k[/itex] are lowering and raising operators from single particle QM and [itex]u_1(\mathbf{r})[/itex] are single particle wavefunctions in state [itex]k[/itex].

I've been told that they are two totally different things. But I'm not sure in what manner. For example, what would their commutator give?

Cheers,

S