Question

Explain the Heisenberg Uncertainty Principle, and how the interpretation of it changes from the quantum to classical limits?

Answer 1

The Heisenberg Uncertainty Principle states that if the uncertainty in the position of a wave-particle is small, the uncertainty in its momentum is large.

`DeltaxDeltap_x >= ℏ//2`

Mathematically, we could see that in the limit as `Deltax -> 0`, `Deltap -> oo`, so that `DeltaxDeltap_x >= ℏ//2`.

Note that this only applies to quantum objects like electrons and photons. Anything ordinarily-sized has an undetectable wave characteristic.

If that doesn't make sense, consider the following:

In `A`:

• The wavelengths are well-defined; we can see them clearly.
• The momentum is obtained from the wavelength, and so the momentum is well-defined.
• `Deltax` is large and `Deltap_x` is small (the latter expected from the second bullet point).

As we go from `A` to `D`, `Deltax darr` and `Deltap_x uarr`. That is, the position becomes more well-known and the momentum becomes less well-known.

This is because in order to be more sure about the position, we built a wave out of many superpositions, taking advantage of something called the correspondence principle.

That principle states that in the limit as we stack more and more waves on top of each other, eventually the wavelengths become so small that the energy spectrum becomes continuous and we reach the classical limit, in which position is well-known.

However, by doing that, we've layered so many waves that we've muddled them up amongst each other---we end up having an exceedingly difficult time discerning the momentum via the wavelength.

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If this still doesn't make sense, I found that this is an excellent explanation: