Initial state with specific momentum

Question

Hi,

I was wondering if it is possible to prepare as initial state for the real time evolution a state with one (or more) free particles (let's say fermions) with specific momenta (k1, k2, etc). For instance, I would like to study the evolution of a state with two particles initially at large distance with opposite momenta.

Thanks a lot,
Giuseppe

Comments

Hi Giuseppe, good question but could you provide us with a bit more information? The basic answer I'm planning to give (which I can give in more detail soon) is that you can construct a creation operator as an MPO and use that.

But could you be more specific about the initial state? As you know, states with well-defined momentum have an ill-defined position, so it's not exactly possible to put them at a large distance apart and have precisely specified momenta. So perhaps you are wanted to make wave packets such as Gaussians which are peaked in both momentum and position space?

Miles

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An initial suggestion I would have is that you can think about the creation operator of a momentum state, for example:

$$
a_k^{\dagger} = \sum_x e^{-ikx} a_x^{\dagger}
$$

This operator can be written as an MPO in ITensor of small bond dimension (I think just bond dimension 2). Then you can apply that operator to a product state MPS (for example, a product state of zero occupation). You can also make operators of different momenta and apply them to the same state to make multiple particles.

This would make entirely delocalized particles, so you may need to think about using other momentum states for a particle in a box or mixtures of momentum states to make localized wavepackets.

Let me know if that makes sense and if you have any questions about the procedure.

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Hi Miles, thank you very much. Sorry, you are completely right. Actually I meant wave packets (such as Gaussians). For instance, two wave packets at large distance with opposite momenta.

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Hi MattFishman, thank you very much for your useful answer. As I wrote to Miles in the previous comment, I actually meant localised wave packets (e.g. Gaussian). Do you think the method you suggested can be used also in this case? Thank you.

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I believe you could just create a wavepacket with a creation operator:

$$
a^{\dagger} = \sum_x e^{-(x-x_0)^2/\sigma} a_x^{\dagger}
$$

(centered about @@x_0@@ with a width set by @@\sigma@@). Again, you could represent it as a small bond dimension MPO and would apply it to a product state MPS.