# Does a particle with infinite energy escape an infinite well?

Currently, my modern physics class is going over particles in finite and infinite wells, general quantum formalism, and tunneling.

What happens to a particle as it gains an infinite amount of energy? Does it stay inside of the infinite well? Does it escape? Can it not be determined? Does it depend?

Are there any issues with this question? Is it valid? Is there anything I need to define or presume before I ask it? Do I need to define the rates at which the potential of the walls go to infinity, or the rate at which the particle's energy goes to infinity?

• What happens when an unstoppable force meets an immovable object? – Jacob Dec 14 '18 at 3:02
• Which infinities are you using? Countable and uncountable ones? Larger cardinals? – Alec Rhea Dec 14 '18 at 3:09

Are there any issues with this question? Is it valid? Is there anything I need to define or presume before I ask it? Do I need to define the rates at which the potential of the walls go to infinity, or the rate at which the particle's energy goes to infinity?

Your question is valid, but only if you take an infinite well as a purely hypothetical idea, unachievable in practice. It's a device to illustrate how we go about getting solutions to (equally hypothetical) one dimensional problems.

It's an infinite well, so it's use is to demonstrate a basic, but actually impossible to achieve, cutoff to the probability of tunnelling.

Unless an infinite potential is used, there is always the possibility of the solution to it being incorrect.

What happens to a particle as it gains an infinite amount of energy? Does it stay inside of the infinite well? Does it escape? Can it not be determined? Does it depend?

The particle can't gain an infinite amount of energy in real life.

The maths breaks. As said in the comments, it's an "irresistable force meets immoveable object" paradox, and the mathematics concedes and concurs with its paradoxical nature by breaking.

As mentioned in the other comment, the energy wave functions (without normalization) are

$$\psi_n(x) = \sin\left(\frac{\pi n}{L} x\right)\ \mbox{inside the box}$$

If you take $$n = \infty$$, thus infinite energy, then you get $$\sin(\infty)$$. This is mathematical nonsense. Usually we extend the definition of a function when including $$\infty$$ by taking the limit as its input approaches that infinity, but

$$\lim_{x \rightarrow \infty} \sin(x)$$

does not exist.

This is the math puking its guts out, humbly choosing to offer its life than to dare pretend it is capable of offering a solution to this age-old philosophical conundrum and so suffer the tragedy and shame of hubris.

• Don't be anthropomorphic regarding math, it doesn't like that:) +1 – user214814 Dec 14 '18 at 21:36

Particle and potential wells are in the framework of quantum mechanics. In this framework one cannot be talking of potential wells arbitrarily changing the particle's energy, because the energy is strictly defined by the solution of the quantum mechanical equation for the given potential.

What happens to a particle as it gains an infinite amount of energy? Does it stay inside of the infinite well?

Here is an example with specific boundary conditions of an infinite potential well using the time independent Schrodinger equation for the solutions. The particle can be in one of these states where n can go to infinity . The energy is on the y axis . Taking a limit of n to infinity , a level exists at at each step, since the solution is a periodic function.

Does a particle with infinite energy escape an infinite well?

For this model, no. It will be caught in one specific value of n. There is no "outside" in this model.

The issue is to accept that particles and potential wells belong to the quantum mechanical regime and the models have to follow specific rules.

Do I need to define the rates at which the potential of the walls go to infinity, or the rate at which the particle's energy goes to infinity?

One may model infinite potential wells in different ways, also time dependent, BUT the possible energy states of the particle are defined by the potential well and the boundary conditions, one cannot change independently the energy of the particle.

The only way to properly define an infinite square well in a rigorous sense is do a finite square well and take the limit as $$V\rightarrow \infty$$. You could simultaneously take the limit that $$E \rightarrow \infty$$ but you would have to define how each approach the limit to have any chance of getting a answer to your question. The infinite square well is useful as a model in which $$V >> E$$ and as such it is not particularly useful to consider what happens as $$V >> 0$$ and $$E >> 0$$ which is sort of the question you are asking.