@glS

You might like this question

a quick comment: shouldn't you have $\psi(r_1+c_1,r_2+c_2)$ in the equation after the translation?

No I only applied the translation operator of the 2nd particle the position of the

wavefunction

mmh ok but then each $r_1$ and $r_2$ in the wavefunctions should also be a vector no?

@glS Yea sloppy notation

I'll fix it

Done!

You think it could do with a quantum information tag?

I was originally planning on posting it here

@MoreAnonymous sorry bit busy right now. I'll have a better look later today

@glS No worries someone else answered it

:)

I think I'm still confused after reading the answer (see the comments as to why)

@MoreAnonymous I'm struggling to visualise the physical meaning of this sort of potential. You are considering a potential that depends on the relative direction between the particles? That's weird

also, I don't understand what the translation $\vec c$ mean in the potential. You are saying that the interaction between the particles depends on their relative position, plus a constant translation? So the particles interact in a way that depends on how $V$ would make other particles interact when one of them is translated by $\vec c$. I can't think of anything that would represent

it feels like you are just artifically redefining the potential to act on "fictitious particles" whose positions are the same as the original one, except for one of them being translated by a constant factor

and then sure, if you just consider a wavefunction/state where you "translate back" on the particles, all of the physics will be the same

@glS I'm thinking of a potential that acts on the relative displacement between particles. For a simple example think of a couple's harmonic oscillator with potential $k(x_1 - x_2)^2$

@MoreAnonymous yes, but usually you'd want the potential to depend on the distance between them, not also the relative direction

@glS Ah i see what you mean now. I think mine just carries more information. Instead of writing $V((\vec r_1 - vec r_2 )^2)$ I've written: $V(\vec r_1 - vec r_2 )$

Also both situations seem to be different (see the comments of the answer)

Like I could have a potential of the form $\vec \lambda \cdot (\vec r_1 - \vec r_2)$ as well

btw

Where lambda could be say a constant electric field

(Also I gotta bounce feel free to message me I'll respond when I can)

Ignore " I could have a potential of the form $\vec \lambda \cdot (\vec r_1 - \vec r_2)$ as well"

@MoreAnonymous right, ok. Then, that aside, sure you can just consider a translated wavefunction and describe everything as you would with a potential without displacement. Still, I struggle to imagine what could a displacement in the argument of the potential represent, it's pretty weird. It's not the same thing as something like $\vec \lambda\cdot(\vec r_1-\vec r_2)$ at all

also what do you mean with "uncertainty principle for the approximation"?

or "uncertainty principle for the translation"

@gls I have 2 "equivalent" ways of describing the situation one is by saying the Hamiltonian suddenly changed the other is by saying the distance between the wavefunction was suddenly reduced ...

In can of the Hamiltonian we know how to proceed to construct a method when this approximation is valid. On the other hand if the wavefunction suddenly changed then how do we construct a way of knowing when the approximation is valid aka uncertainty principle?

@glS Instead of $V(\vec r_1 - \vec r_2)$ think of it as $V((\vec r_1 - \vec r_2)^2)$ if that helps

@glS Also an example of a potential with $\vec \lambda \cdot (\vec r_1 - \vec r_2)$ would be the dipole moment 's moment potential energy