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Lecture 4
Uncertainty Principle
References :
1.Concept of Modern Physics
by Arthur Beiser
2. Modern Physics
by Kenneth Krane
A localized wave or wave packet:
A moving particle in quantum theory
Spread in position
Spread in momentum
Superposition of waves
of different wavelengths
to make a packet
Narrower the packet , more the spread in momentum
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Basis of Uncertainty Principle
Heisenberg's Uncertainty Principle
___________________________________


The Uncertainty Principle is an important
consequence of the wave-particle duality of
matter and radiation and is inherent to the
quantum description of nature
Simply stated, it is impossible to know both the
exact position and the exact momentum of an
object simultaneously
A fact of Nature!
Heisenberg's Uncertainty Principle
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Uncertainty in Position :
Uncertainty in Momentum:
x
p x
h
xp x 
2
Heisenberg's Uncertainty Principle
- applies to all “conjugate variables”
___________________________________
Position & momentum
Energy & time
h
xp x 
2
h
E  t 
2
Uncertainty Principle and the Wave Packet
___________________________________
xp x
h

2
h

p
x
  p


p
Some consequences of the Uncertainty Principle
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•


The path of a particle (trajectory) is not well-defined in
quantum mechanics
Electrons cannot exist inside a nucleus
Atomic oscillators possess a certain amount of energy
known as the zero-point energy, even at absolute zero.
Why is n’t the uncertainty principle apparent to
us in our ordinary experience…?
Planck’s constant, again!!
___________________________________
h
xp x 
2

34
h6
.6
x
10J.s
Planck’s constant is so small that the
uncertainties implied by the principle are also
too small to be observed. They are only
significant in the domain of microscopic
systems
Heisenberg Uncertainty Principle
The uncertainty principle states that the position
and momentum cannot both be measured,
exactly, at the same time.
x p  h
Historic importance
or
h
or h
2
For
For
Numerical Applications
The more accurately you
know the position (i.e., the
smaller x is) , the less
accurately you know the
momentum (i.e., the larger
p is); and vice versa
Where h (6.6 x 10-34) is called Planck’s constant. As ‘h’ is so small, these
uncertainties are not observable in normal everyday situations
p is less
p is more
Increasing levels of wavepacket localization, meaning the
particle has a more localized position.
In the limit ħ → 0, the particle's
position and momentum become known
exactly. This is equivalent to the
classical particle.
Heisenberg Uncertainty Principle
•
•
•
•
The wave nature to particle means a particle is a wave packet,
the composite of many waves
Many waves = many momentums, observation makes one
momentum out of many.
Principle of complementarity: The moving electron will
behave as a particle or as a wave, but we can not observe both
aspects of its behavior simultaneously. It states that
complete description of a physical entity such as a
photon or an electron can not be made in terms of
only particle properties or only wave properties, but
that both aspects of its behavior must be considered.
Exact knowledge of complementarities pairs (position, energy,
time) is impossible.
•
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Example
• A pitcher throws a 0.1-kg baseball at 40 m/s
So momentum is 0.1 x 40 = 4 kg m/s
• Suppose the momentum is measured to an accuracy of 1 % , i.e.,
p = 0.01 p = 4 x 10-2 kg m/s
• The uncertainty in position is then
No wonder one does not observe the effects of the uncertainty principle in
everyday life!
• Same situation, but baseball replaced by an electron which has mass 9.11
x 10-31 kg
So momentum= 3.6 x 10-29 kg m/s and p = 3.6 x 10-31 kg m/s
• The uncertainty in position is then
Example: A free 10eV electron moves in the x-direction with a speed of
1.88106 m/s. assume that you can measure this speed to precision of 1%.
With what precision can you simultaneously measure its position?
the momentum of e- is
px = m vx = 9.1110-31 kg 1.88106 m/s
= 1.71 10-24 kg m/s
The uncertainty  px in momentum is 1%
x  h/4  px =3.1 n m
Example: The speed of an electron is measured to be 5.00X10E3 m/s to an
accuracy of .003%. Find the uncertainty in determining the position of this
electron.
Calculations yields x  3.91 10-4 m.
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Applications of uncertainty principle
1. Non-existence of electrons in the Nucleus
Assume that the electron is present in the nucleus. The radius of
the nucleus of any atom is of the order 5 fermi (1 fermi = 10-15 m). For
the existence of electron in the nucleus, the uncertainty x in its
position would be at least equal to the radius of the nucleus, i.e.
uncertainty in the position
x  R  5 1015 m
According to the uncertainty principle.
p  h/4x= 1.05410-20 kg-m/sec
If this is the uncertainty in momentum of the electron then the
momentum of the electron must be at least of the order of its
magnitude, that is , p  1.05410-20 kg-m/sec, an electron having so
much momentum should have a velocity comparable to the velocity
of light. Hence, its energy should be calculated by the relativistic
formula
E2=p2c2+mo2c4
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Applications of uncertainty principle
pc= 1.05410-20 3108 = 20 MeV
The rest energy of electron 0.51 MeV, is very small as compared to
pc. Hence second term in relativistic equation can be neglected.
Thus, if the electron is the constituents of the nucleus, it should
have an energy of the order of 20 MeV.
But the experiment shows that no electron in the atom
possesses kinetic energy more than 4 MeV. Therefore, it is
confirmed that electrons do not reside inside the nucleus.
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Experimental illustration of Uncertainty Principle: Single slit
diffraction.
To see more clearly into the nature of uncertainty, we consider
electrons passing through a slit:
We apply the condition of
minima from single slit
Momentum
diffraction,
uncertainty in the
y component
sin   n
and postulate that λ is the de
Broglie wavelength.
Px=h/λ
Since the electron can pass the slit
through anywhere over the width ω,
the uncertainty in the y position of the
electron is y=ω.
py  h

sin =

for small  sin   tan 
p y p y
tan  

px
h/

p y

  p y   h
h/ 
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p y y  h
which is in agreement with the uncertainty principle. If we try to
improve the accuracy of the position by decreasing the width of the slit,
the diffraction pattern will be widened. This means that the
uncertainty in momentum will increase.
The uncertainty principle is applicable to all material particles,
from electrons to large bodies occurring in mechanics. In case of large
bodies, however , the uncertainties are negligibly small compared to
the ordinary experimental errors.
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