Forbidden energy levels?

[space-time]

I noticed something about the normalized wave function solution to the particle in an infinite square well (which has the boundary conditions v(x)= ∞ at x = 0 and x = L , and v(x) = 0 in between those two).

The solution is: Ψ(x) = (2/L)^½ * sin(πnx/L) where n is an integer that signifies certain energy levels. Now here is what I noticed:

Let us say that L = 10, x = 1 and n= 10. If that were the case, then the argument of the sine function would simply reduce to π and sin(π) = 0. Therefore the wave function would equal 0 at the energy level n = 10.

Would this mean that a particle in this well just simply could not attain such an energy level? Perhaps there are a limited number of levels and it is simply impossible for a particle to reach a level that makes the wave function equal 0? If this is the case, does the particle just tunnel out of the well after reaching said energy level (or would reaching such energy be analogous to a hydrogen atom obtaining enough energy to lose its electron)?

Also, is a square well like this even possible in reality or is this example just used for teaching? I know that the energy of an electron on a hydrogen atom in its ground state is around the energy of the n = 1 energy eigenvalue for this particular wave function. However, I ask this because I am not sure of how something in real life would have boundaries of infinite potential energy. Even if you were to put a particle in an actual physical box, the particle could still gain enough penetration power to simply penetrate and break out of the box. If this well could exist, can you please give an example of a scenario where these boundary conditions would apply?

 

[Nugatory]

Let us say that L = 10, x = 1 and n= 10. If that were the case, then the argument of the sine function would simply reduce to π and sin(π) = 0. Therefore the wave function would equal 0 at the energy level n = 10.

If L=10 then x ranges from 0 to 10. What you've discovered is that at energy level n=10, the wave function equals zero at the point x=1 (in fact, all of the points x=0, x=1, x=2, ... x=9, x=10) so the particle will never be found at those points. At other points in the box the wave function is non-zero and there is some probability that the particle will be found there.

Try it for any other value of n and you'll find similar points, spaced at intervals of L/n.

 

[Nugatory]

Also, is a square well like this even possible in reality or is this example just used for teaching? I know that the energy of an electron on a hydrogen atom in its ground state is around the energy of the n = 1 energy eigenvalue for this particular wave function. However, I ask this because I am not sure of how something in real life would have boundaries of infinite potential energy. Even if you were to put a particle in an actual physical box, the particle could still gain enough penetration power to simply penetrate and break out of the box. If this well could exist, can you please give an example of a scenario where these boundary conditions would apply?

You're right, there's no such thing as an infinite square well in the real world. However, there are very deep nearly square wells, and the infinite square well is an excellent approximation for these situations.

(However, the infinite square well is not a good approximation for the hydrogen atom; that's a $1/r^2$ force so the well isn't square. Google for "Schrodinger equation hydrogen" and you'll understand why we prefer to use the infinite square well approximation whenever we can).