Mixed Momentum-Position eigenstates ?

[Widdekind]

Is it possible to construct 'mixed' momentum-position eigenstates, of the form:

 

[math]\Psi \propto e^{i k x} \delta(y) \delta(z)[/math]

? Such a 'perfect pencil' of radiation would have to be perfectly monochromatic. When you put a photon into a state of monochromaticity, does its wave function "inflate" out to infinite extent, in that direction ??

[swansont]

What about that wave function is mixed? It's in terms of position. Do the Fourier transform and you get momentum.

 

Even if you had a monochromatic photon, does that mean you know where it is?

[Widdekind]

I must have mis-spoken. The Wave Function I offered, was a momentum eigenstate in one direction, and position eigenstates in the other two. Was that Wave Function quantum mechanically allowed ?

[ajb]

Think about the commutators of the position and the momenta.

[Widdekind]

Don't position operators, of some spatial direction, commute, with momentum operators, of other-and-orthogonal directions ? Does that permit, or deny, the above-proposed "spaghetti-strand wave function" ?

[ajb]

[math][x^{i}, p_{j}] = i \hbar \delta^{i }_{\:\: j}[/math]. (all others are zero)

 

So in particular [math][x, p_{y}]=0 [/math] etc. This means you can simultaneously diagonalise the operators [math]x[/math] and [math]p_{y}[/math] and thus have eigenvectors for both operators. So I think such a wave-function is ok.

[Widdekind]

Thank you very much for the response

[ajb]

We might have to be careful as such a wave-function is not square integrable (I think), we do not have a genuine Hilbert space. One needs to extend the notion to a rigged Hilbert space. That is one has to include distributions and not just square integrable functions. But this is probably more technical than you need.

[Widdekind]

I was wondering whether, with a sufficiently slowly generated ("slowly spun off, slowly woven") photon, one could exploit Quantum non-locality to "stretch" a photon's wave packet, all the way across the cosmos, in some super-luminal way.

 

[math]\Delta x \Delta p \approx \hbar[/math]

 

[math]\Delta E \approx \frac{c \hbar}{\Delta x}[/math]

For [math]\Delta x[/math] = 1 Gpc, [math]\Delta E \approx 10^{-32} eV[/math], an ultra-pure, nearly mono-chromatic, light source. And, that would require a generation time, of [math]\Delta t \approx \hbar / \Delta E \approx \Delta x / c[/math]. To wit, [math]\Delta x \approx c \Delta t[/math], and no "sneaky super-luminality" appears to be possible upon this particular path. What would be "weird", would be that the Wave Function would physically span the space between source & destination, being a "bridge" of photon WF. But there are probably a slew of reasons why such a suggestion is Quantum physically impractical.