Quantization on momentum space

[lokofer]

I have this doubt..quantization in momentum space using G(p) as the Fourier transform of the wave function was not common (at least when i studied Q. Physics) my doubt is, if we have that:

$$x |G(p)>=i \hbar \frac{ \partial G(p)}{\partial p}$$

But..what would happen if we apply:

$$\dot x |G(p)> = ?$$ here the "dot" means time derivative...

$$G(p)= \int_{-\infty}^{\infty}dx \psi (x) e^{i\omega t -ipx}$$

 

[Hurkyl]

The position operator X isn't time varying; it doesn't make sense to ask for its time derivative! Also, there's no reason here to restrict yourself to stationary states.


Anyways, you know that:

$$| p \rangle = \int_{-\infty}^{+\infty} e^{-ipx} |x \rangle \, dx$$

(but I might have a sign wrong, or a constant multiple missing)

So, you can always transform your state:

$$\int_{-\infty}^{+\infty} G(p) |p \rangle \, dp$$

into

$$\int_{-\infty}^{+\infty} \left( \int_{-\infty}^{+\infty} G(p) e^{-ipx} \, dp \right) | x \rangle \, dx$$

if you really need to. (e.g. if you only know how to apply an operator to position states)

 

[reilly]

It's been well known, at least since the 1930s, that dx/dt is a perfectly respectable operator in QM -- provided one is working in the Heisenberg or interaction representations (or pictures as they are sometimes called). This is pretty standard stuff, and thus is explained in countless books and papers. Dirac, in his Quantum Mechanics, discusses the dx/dt operator in section 30. He also gives an extensive discussion of the momentum represention in section 23.(For a non-relativistic free particle, dx/dt = p/m in the Heisenberg rep.)

Regards,
Reilly Atkinson

 

[Hurkyl]

He's working in the Schrödinger picture, though -- G(p) is a time-varying state.

 

[reilly]

He's working in the Schrödinger picture, though -- G(p) is a time-varying state.

Right you are, as is often the case.. Reilly