Teaching Quantum Mechanics
THE HEISENBERG PRINCIPLE
'Close Encounters' Approach
Lorenzo Galante
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Heisenberg main page
H7 - THE HEISENBERG PRINCIPLE (under construction)
H7 - THE HEISENBERG PRINCIPLE (under construction)
We have come to an Uncertainty Relation between the pair of variables (x, k):
Δx Δk = constant
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
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
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
E = h f
Thus with c = 1
E = h / λ α h k
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
E2 = m2 + p2
that for massless systems, as the photon, becomes:
E = p
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
p α h k
THE HEISENBERG UNCERTAINTY PRINCIPLE
As a conclusion we may say that for a quantum system the relation
Δx Δk = constant
Δx Δk = constant
transforms into
Δx Δp = constant
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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