# Particle in a box as a quantum field

A common introductory quantum system is the single particle in a 1 dimensional square "box" or well with infinite potential walls.

Is there a reasonably introductory treatment of this system in the quantum field case? i.e. a quantum field in an infinite square well in a single particle state. I am interested in the correspondence between the formalisms and the predicted dynamics etc.

Update: I would most like to see a treatment in the context of the most widely used Heisenberg (or Interaction?) representation rather than the Schrödinger representation even though that might be more direct.

Related: Schrödinger functional representation and Is there any theorem that suggests that QM+SR has to be an operator theory?

• You can study the Klein-Gordon model in a spacetime with boundary. Of course you have to choose boundary conditions appropriately. The ones analogous to the particle in the infinite potential well are the Dirichlet $\phi = 0$ conditions. It's basically like studying a (quantum) clamped string. It's a nice exercise to work it all out in second quantization. I recommend it! – Ryan Thorngren Apr 18 '18 at 11:37
• Er ... doesn't the usual introductory second quantization recipe involve starting with box modes and then letting the box grow without bound? So that the "particles" of the theory are fully de-localized momentum eigenstates. Or something. – dmckee Apr 18 '18 at 15:16
• @dmckee Yes, that is exactly where I am starting from, although in the opposite direction: trying to understand that treatment is my end goal and I am hoping to look at its single-particle case, which is analogous to the basic QM scenario I already understand, to give me more insight into the general model. By the way, according to answers to my other question physics.stackexchange.com/questions/388270 , "particles" don't have to be de-localized, it seems that's a common misconception that I also had, as they are so frequently presented like that. – user183966 Apr 18 '18 at 19:18
• @Ryan Thorngren I think I'm not quite at the level you are suggesting. It sounds like you are explaining I need to apply the second quantization procedure in a specific case. Unfortunately I am not fully clear on the second quantization procedure, better understanding of it is actually the goal of this question. But hopefully I will be there soon! – user183966 Apr 18 '18 at 19:26
• @RyanThorngren at paragraph 3 and already like this presentation – user183966 Apr 18 '18 at 22:05