# Particle in a box, quantization of energy

I'm learning about how the energy of matter is quantized like how the energy of light is. My textbook illustrates the concept of quantization with the particle in a box:

"A particle of mass $m$ moves in one dimension as it bounces back and forth with speed $v$ between the ends of the box of length $L$. We'll assume that the collisions at the ends are perfectly elastic, so the particle's energy--entirely kinetic--never changes."

I don't understand how the particle's energy can be "entirely kinetic." When the particle hits the end of the the box and reverses its direction, isn't there an instance in time where the particle's speed is 0 which would imply that all of its energy is now potential energy and that it's kinetic energy is zero?

• An assumption of the 'particle in a box' model is that infinitely large forces repel the particle if it touches a wall of the box. – GodotMisogi Apr 29 '16 at 15:19
• Well, that would have been nice for the book to have mentioned. – user50420 Apr 29 '16 at 15:20
• True, as is the case with most introductory high school/college textbooks oversimplifying quantum mechanics. Which book is it? Also, would you like to see the Schrodinger equation solution of the 'particle in a box' model? – GodotMisogi Apr 29 '16 at 15:23
• It's called Physics for Scientists and Engineers (3e) by Knight. He just introduced the de Broglie wavelength and he won't be introducing the Schrodinger stuff until the next chapter, so I know nothing about Schrodinger yet. – user50420 Apr 29 '16 at 15:25

The walls do not really matter that much here. I would say that the only points that matter to understand that quote are:

1. The confining potential is conservative so that $K+U = constant$
2. The confining potential is virtually flat $U = 0$ almost everywhere inside the box (say 99% of the box) except very close to the walls

These two conditions are enough to say that the total energy of the particle (its value) is the kinetic energy it has in a big fraction of the box.

Here's an image of the 'particle in a box' model:

image http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/pbox1.gif

At the endpoints, you can see that there is an infinite potential outside the boundary of the box and zero potential inside. The unrealistic assumption in this model is that every time the particle reaches the boundary, an infinite force repels it to keep it in the box.

• One more step is needed: the impulse must be finite, so we have to assume that $F\rightarrow\infty$ and $\Delta t\rightarrow 0$ in such a way that the product $F\Delta t$ remains finite, and since $\Delta t$ is approaching zero, the particle spends "no time" with $v=0$ – garyp Apr 29 '16 at 15:55
• Yes, of course. I forgot to mention that the impulse must remain finite. – GodotMisogi Apr 29 '16 at 16:46