1. Heisenberg's Uncertainty Principle
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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!
2. Heisenberg realised that ...
In the world of very small particles, one cannot
measure any property of a particle without
interacting with it in some way
This introduces an unavoidable uncertainty into
the result
One can never measure all the
properties exactly
Werner Heisenberg (1901-1976)
3. Measuring the position and momentum
of an electron
Shine light on electron and detect reflected
light using a microscope
Minimum uncertainty in position
is given by the wavelength of the
light
So to determine the position
accurately, it is necessary to use
light with a short wavelength
4. In order to see the electron, at least one photon must
bounce off it
During this interaction, momentum is transferred
from the photon to the electron
Therefore, the light that allows you to accurately
locate the electron changes the momentum of the
electron
5. Fundamental Trade Off …
Use light with short wavelength:
accurate measurement of position but not
momentum
Use light with long wavelength:
accurate measurement of momentum but not
position
6. Heisenberg’s Uncertainty Principle
If a measurement of position of a particle is made with precision Δx
and a simultaneous measurement of linear momentum is made with
precision Δp then the product of the two uncertainties can never be
smaller than h/4
The more accurately you know the position (i.e.,
the smaller Dx is) , the less accurately you know the momentum
(i.e., the larger Dp is); and vice versa
7. Implications
It is impossible to know both the position and
momentum exactly, i.e., Dx=0 and Dp=0
These uncertainties are inherent in the physical world
and have nothing to do with the skill of the observer
Because h is so small, these uncertainties are not
observable in normal everyday situations
9. Derivation for uncertainty in energy and time
position momentum relation
multiply and divide above eqn. by m & p
From Einstein’s equation E = p2/2m (since E = ½ mv2 and p = mv)
∆ E = ∆ p 2p/2m
= ∆ p (p/m)
Further ∆ x (m/p) = ∆ x / v = ∆ t since p = mv
Substituting for ∆ x (m/p) and ∆ p (p/m) in position –
momentum equation we get
4
h
p
x x
D
D
4
h
m
p
p
p
m
x x
D
D
4
h
t
E
D
D
11. 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.
12. Applications of the uncertainty principle
The non-existence of electron in the nucleus can be
proved
The binding energy of the hydrogen atom can be
calculated
Strength of the nuclear force can be estimated