The following is an excerpt from my lecture notes, aiming to derive a more general form of the Schrodinger equation.
There has been little reason for the introduction of the commutation relations given in (3.3) regarding the operators $\hat{a}(\overrightarrow{x})$ and $\hat{a^{\dagger}}(\overrightarrow{x})$, so my question is; what is the significance of this, both mathematically and physically, and why is it necessary? I heard somewhere that these commutation relations are necessary in order for the Schrodinger equation to be quantised, but even if this is true then how so?
Also, it is not entirely clear to me as to why the ingredients of (3.2) can be written as in (3.4) and (3.5). How do I convince myself of this? Could I go about showing that (3.2) is now only satisfied, when written with the more explicit terms provided in (3.4) and (3.5), if and only if the wavefunction $\psi(t, x_1, ..., x_n)$ satisfies (3.1)? Or is there a more intuitive way to see this? Thank you :)