Heisenberg's Uncertainty Principle states that the more precisely one property of a particle is measured, like position, the less precisely another related property, like momentum, can be known. This is due to the wave-particle duality of matter at the quantum scale. It is impossible to simultaneously know both the exact position and momentum of a particle. The principle also applies to energy and time - the more precisely energy is known, the less precisely time can be known and vice versa. This inherent uncertainty is a fundamental aspect of quantum mechanics.

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

8. Heisenberg’s Uncertainty Principle
involving energy and time
 The more accurately we know the energy of a body,
the less accurately we know how long it possessed
that energy

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

10. Heisenberg's Uncertainty Principle
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Position & momentum
Energy & time

4
h
p
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