Teaching Quantum Mechanics

THE HEISENBERG PRINCIPLE

'Close Encounters' Approach


Lorenzo Galante

Heisenberg main page

H7 - THE HEISENBERG PRINCIPLE (under construction)

We have come to an Uncertainty Relation between the pair of variables (x, k):

Δx Δk = constant

This relation tells us that the more a system is dispersed in space the less is dispersed in the domain of the other variable.

Now we want to better understand what k represents. So we move a step towards the quantum world in search for the physical meaning of k for a quantum system.

Firstly we have to say that in 1927 was carried out an experiment showing that the momentum p of electrons could be expressed through a wavelength λ (Davisson & Germer experiment). The relation being:

p = h / λ

The experiment was performed sending an electron beam against a Nickel crystal and observing diffraction patterns generated by the scattered electrons.

According to this relation p is proportional to k:

p α h k


THE PHOTON

A clue about the validity of the relation between k and p comes from the photon. We know that the Energy E of a photon is related to a frequency f:

E = h f

Thus with c = 1

E = h / λ α h k

From Special relativity we also have the beautiful relation among the Energy E, the momentum p and the mass m of a system (written with c=1):

E2 = m2 + p2

that for massless systems, as the photon, becomes:

E = p

Connecting the second equation with the last one we discover that the variable k is strictly related to the momentum p of a quantum system

p α h k

THE HEISENBERG UNCERTAINTY PRINCIPLE

As a conclusion we may say that for a quantum system the relation

Δx Δk = constant

transforms into

Δx Δp = constant

Which is what we call the Heisenberg Uncertainty Principle:

THE MORE A QUANTUM SYSTEM IS DISPERSED IN SPACE THE LESS IS DISPERSED IN MOMENTUM

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