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PHY 102: Waves & Quanta
Topic 15
Introduction to Quantum
Theory II
John Cockburn (j.cockburn@... Room E15)
The Uncertainty Principle
•One of the fundamental consequences of quantum
mechanics is that it is IMPOSSIBLE to SIMULTANEOUSLY
determine the POSITION and MOMENTUM of a particle
with COMPLETE PRECISION
•Can be illustrated by a “thought experiment” known as
Heisenberg’s Microscope, using radiation of wavelength λ to
“look at” the particle…….
Heisenberg Microscope
Suppose we have a particle, whose momentum
is, initially, precisely known. For convenience
assume initial p = 0.
“microscope”
From wave optics (Rayleigh Criterion, Lecture
8)
sin  
D

D
From our diagram:
y
2
x
x
sin  
y
2
2y
2 y
x 
D
Δx
Heisenberg Microscope
2 y
x 
D
Since this is a “thought experiment” we are free
from any practical constraints, and we can
locate the particle as precisely as we like by
using radiation of shorter and shorter
wavelengths.
“microscope”
D
y
2
But what are the consequences of this?
Δx
Heisenberg Microscope
In order to see the particle, a photon must
scatter off it and enter the microscope.
Thus process MUST involve some transfer of
momentum to the particle…….
“microscope”
BUT there is an intrinsic uncertainty in the Xcomponent of the momentum of the scattered
photon, since we only know that the photon
enters the microscope somewhere within a
cone of half angle :

Δp =2psin
p

p
By conservation of momentum, there must be
the same uncertainty in the momentum of the
observed particle……………
Heisenberg Microscope: Summary
Uncertainty in position of particle:
2 y
x 
D
Can reduce as much as we like by making λ small……
Uncertainty in momentum of particle:
2h D
p  2 p photon sin  
 2y
So, if we attempt to reduce uncertainty in position by decreasing λ, we
INCREASE the uncertainty in the momentum of the particle!!!!!!
Product of the uncertainties in position and momentum given by:
2 y Dh
xp 
 2h
D y
The Uncertainty Principle
Our microscope thought experiment gives us a rough estimate for the
uncertainties in position and momentum:
xp ~ h
“Formal” statement of the Heisenberg uncertainty principle:
 x p 

2
Circular apertures and resolving powers
Diffraction effects can, of course be observed with any shape of aperture
(see 2nd year optics course).
Take circular aperture, for example:
Angular radius of bright
central spot (Airy disc):
sin   1.22
1

D