17. Heisenberg’s uncertainty principle - both forms: position-momentum and time-energy explanation. (4p.)

Question 1

Heisenberg’s uncertainty principle - both forms: position-momentum and time-energy explanation.

Answer 1

Heinsenberg proposed some relationships defining limits of precision of measurements when we are dealing with very small objects.

• It is not possible to determine at the same time both the position and the momentum of a particle with unlimited precision:

∆ 𝑝 ∆𝑥 ≥ ℎ = h/2𝜋

• It is not possible to determine at the same time both the energy and the time coordinate of a particle with unlimited precision:

∆𝐸∆𝑡 ≥ ℎ = h/2𝜋

Deltas indicate a range of our uncertainty in measured values. It is a fundamental restriction imposed on us by nature.

If we try to reduce the uncertainty of one element, the uncertainty of the other will increase. E.g. if we try to reduce ∆𝑥 by using light of a shorter wavelength, the momentum of the photon increases, and so does its uncertainty.

As far as the second Heisenberg’s principle is concerned, to minimize the uncertainty in measuring the energy of a system, one must observe it for as long as possible. So the time uncertainty increases.

Once can measure either one element or the other precisely, but cannot measure both simultaneously.