AH 3.1.6 The Uncertainty Principle

Question 1

Describe and explain what is meant by Heisenberg’s uncertainty principle. Include a statement of the uncertainty relation in your answer.

Answer 1

Heisenberg’s Uncertainty Principle states that for a particle, if we decrease the uncertainty of our position information, then we increase the uncertainty in our knowledge of it’s momentum. This uncertainty is an intrinsic property of nature itself and is not related at all to the apparatus being used or experimental technique.

The uncertainty relation is given by

Δ_xΔp_x ≥ h/4π

where Δx = uncertainty in position of the particle in the x direction

Δp_x = uncertainty in momentum of particle in the x direction

Question 2

The uncertainty in an electron’s position relative to an axis is given as ±5.0 × 10^–12 m.
Calculate the least uncertainty in the simultaneous measurement of the electron’s momentum relative to the same axis.

Answer 2

ΔxΔp_x ≥ h/4π

Δp_x = ±1.1 × 10^–23 kg m s^–1

Question 3

Heisenberg’s Uncertainty Principle may be stated thus:
Δ_xΔp_x ≥ h/4π

Quantum mechanics shows that there are other pairs of what are called conjugate variables which are subject to similar uncertainty relations, in particular, Energy and time. Write down the uncertainly relation involving energy and time and define its terms.

Answer 3

ΔEΔt ≥ h/4π

where ΔE = uncertainty in energy
Δt = uncertainty in time h = Planck’s constant

Question 4

An electron spends approximately 1.0 ns in an excited state.
Calculate the uncertainty in the energy of the electron in this excited state.

Answer 4

ΔEΔt ≥ h/4π

ΔE = ±5.3 × 10–26 J

Question 5

Explain the concept of quantum tunnelling as a consequence of the uncertainty principle.

Answer 5

Quantum tunnelling says that a quantum particle can exist in a position that, according to classical physics, it has insufficient energy to occupy.

This follows from the uncertainty principle which states that there is always some uncertainty in the particle’s position, meaning that there must be a low but finite probability that the particle can occupy a position of higher energy than classical physics would allow.

Question 6

a) Quantum tunnelling is essential in the nuclear fusion taking place in the sun. True or false?
b) Explain why.

Answer 6

a) TRUE!
b) The temperature in the stellar core is generally insufficient to allow atomic nuclei to overcome the Coulomb Barrier and achieve thermonuclear fusion. Quantum tunnelling increases the probability of penetrating this barrier. Though this probability is still low, the extremely large number of nuclei in the core of a star is sufficient to sustain a steady fusion reaction