Consider a particle with mass (m) confined to a 1D box, so that it is impossible

Question

Consider a particle with mass (m) confined to a 1D box, so that it is impossible to find the particle outside 0≤x≤L. Other than the confinement, the particle is not interacting with anything, so its energy is purely kinetic energy. Now say that you have a particle in the lowest energy level, the ground state, with energy E1 = h2/2mL2

This means that you can calculate the particle’s momentum:

Solving for p, we get

But notice that since h and L ar known exactly, therefore p is known exactly. And this means there is no uncertainty in your knowledge of the momentum: ∆p = 0. But since the particle is confined, ∆x ≈ L; in any case, ∆x ˂ ∞. Therefore

The Uncertainty Principle is violated! But this can’t be right. Find the error in reasoning above. Some options for you to consider:

Maybe ∆x = ∞ because the particle can quantum tunnel outside the box.

Maybe h is not known exactly, so its uncertainty needs to be taken into account.

Maybe ∆p ˃ 0 because there is a subtle mistake in the calculation.

Hint: Remember that momentum is a vector!

Explanation / Answer

In this case we have assumed that the particle to be confined, but it is actually not confined. In a 1 D box tunnelling of electrons occurs accross the tunnel-walls,and hence its position remains undefined,thus validating uncertainity principle.

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