Momentum operator in quantum mechanics

[adosar]

The momentum operator for one spation dimension is -iħd/dx (which isn't a vector operator) but for 3 spatial dimensions is -iħ∇ which is a vector operator. So is it a vector or a scalar operator ?

 

[vanhees71]

In 3D the momentum is a (polar) vector and thus must be represented by a vector operator in quantum mechanics, and that's indeed the case as you correctly wrote: In the position representation,
$$\hat{\vec{p}}=-\mathrm{i} \hbar \vec{\nabla},$$
and this transforms indeed as a vector when operating on scalar fields (and the Schrödinger wave function is a scalar field under rotations!).

 

[Vanadium 50]

Your objection, that 3-d operators in 3-d can appear as scalar operators in 1-d, isn't really specific to QM.

 

[haushofer]

Well, technically, as a covector :P

 

[vanhees71]

That's no quantum specific. Also in classical physics (canonical) momenta are co-vectors.

 

[haushofer]

I know. I'm in a pedantic mood ;)