PHY 102: Quantum Physics
Topic 5
The Uncertainty Principle
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 couple of “thought experiments”, for
example the “photon picture” of single slit diffraction and the
“Heisenberg Microscope”
Single Slit Diffraction
“geometrical” picture breaks down when slit width becomes comparable
with wavelength
Position of dark fringes in single-slit diffraction
m
sin  
a
If, like the 2-slit treatment we assume small angles, sin ≈ tan  =ymin/R, then
ymin
Rm 

a
Positions of intensity
MINIMA of diffraction
pattern on screen,
measured from central
position.
Very similar to expression derived for 2-slit experiment:
n
ym  R
d
But remember, in this case ym are positions of MAXIMA
In interference pattern
Width of central maximum
•We can define the width of the central maximum to be the distance
between the m = +1 minimum and the m=-1 minimum:
R
R 2 R
y 


a
a
a
Intensity
distribution
image of diffraction
pattern
Ie, the narrower the slit,
the more the diffraction
pattern “spreads out”
Single Slit Diffraction: Photon Picture
Since θ is small:
sin  

a
 

a
P
Photons directed towards outer
part of central maximum have
momentum
Py
Px
p  px  py

h
h
py  px  px  px

a
px a a
ie, localizing photons in the y-direction to a slit of width a leads to
a spread of y-momenta of at least h/a.
Intensity
distribution
• So, the more we seek to localize a photon
(ie define its position) by shrinking the slit
width, a, the more spread (uncertainty) we
induce in its momentum:
• In this case, we have pyy ~ h
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)
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 experiments give us a rough estimate for the
uncertainties in position and momentum:
xp ~ h
“Formal” statement of the Heisenberg uncertainty principle:
 x p 

2
The Uncertainty Principle: wave picture
Consider a particle of known kinetic energy moving freely
through space.
Wave function:
 ( x, t )  A sin( kx  t )
Momentum is exactly defined, but position of particle
completely undefined......................
What happens if we combine waves of different wavelength?
Beats (from sound waves)
These occur from the superposition of 2 waves of close, but different frequency:
f beat  f a  f b  f

Δt
ft 1
Δt
•So, by combining waves of different wavelength, we
can produce localized “wave groups”
•The more different wavelengths we combine, the greater
the degree of localization of the wave group (ie particle postition
becomes more well-defined)
•We can obtain a totally localized wavefunction (x =0) only by
combining an infinite number of waves with different wavelength
•We thus lose all knowledge of the momentum of the particle...
Energy-time Uncertainty
•Uncertainty principle also applies to simultaneous
measurements of energy and time
Et  h
Stationary state
Zero energy spread
Decay to lower state with finite
lifetime t: Energy broadening E
(explains, for example “natural linewidth”
In atomic spectra)