Matter Waves and the Uncertainty Principle _________________________

1
Matter Waves and the Uncertainty
Principle_________________________

• Anu Venugopalan
• Centre for Philosophy and Foundations of Science
• New Delhi

2
The Birth of Quantum Mechanics___________________
________

• At the turn of the last century, there were
  several experimental observations which could not
  be explained by the established laws of classical
  physics and called for a radically different way
  of thinking
• This led to the development of Quantum Mechanics
  which is today regarded as the fundamental theory
  of Nature.

3
Some key events/observations that led to the
development of quantum mechanics________________
_________________

• Black body radiation spectrum (Planck, 1901)
• Photoelectric effect (Einstein, 1905)
• Model of the atom (Rutherford, 1911)
• Quantum Theory of Spectra (Bohr, 1913)

4
Some key events/observations that led to the
development of quantum mechanics________________
_________________

• Scattering of photons off electrons (Compton,
  1922)
• Exclusion Principle (Pauli, 1922)
• Matter Waves (de Broglie 1925)
• Experimental test of matter waves (Davisson and
  Germer, 1927)

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Rutherford
Planck
Einstein
Bohr
Compton
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Pauli
de Broglie
Davisson and Germer
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The nature of light.___________________________
_

• The birth of quantum mechanics is intimately
  linked with the theories and discoveries relating
  to the nature of light
• Is the nature of light that of a wave or a
  particle???

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The story of light..__________________________
_____

• Corpuscular theory (Newton)
• Wave nature (Huygens)
• Double-slit interference experiment (Young)

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The story of light (contd.).._________________
______________

• Light is an electromagnetic wave (Maxwell)
• Photoelectric effect existence of light quanta
  photons (Einstein)
• Photons have momentum (Compton)..

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Newton
Huygens
Young
Compton
Maxwell
Einstein
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Light has a dual nature__________________________
_

• Wave (electromagnetic) - Interference
• -
  Diffraction
• Particle (photons) - Photoelectric effect
• - Compton effect
• Wave - Particle Duality for light

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What about Matter?______________________________
_

• If light, which was traditionally understood
  as a wave also turns out to have a particle
  nature, might matter, which is traditionally
  understood as particles, also have a wave nature?
• Yes!

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Louis de Broglies hypothesis__________________
__________

• The dual nature of matter
• A particle with momentum p has a matter wave
  associated with it, whose wavelength is given by

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The connecting link Plancks
constant_______________________________

• Dual Nature
• Radiation
• Matter

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Why isnt the wave nature of matter more
apparent to us?_________________________________
__

• Plancks constant is so small that we dont
  observe the wave behaviour of ordinary objects
  their de Broglie wavelengths could be many orders
  of magnitude smaller than the size of a nucleus!

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Particle______________________________

• Our traditional understanding of a particle
• Localized - definite position, momentum,
  confined in space

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Wave____________________________

• Our traditional understanding of a wave.
• de-localized spread out in space and time

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How do we associate a wave nature to a
particle?___________________________________

• What could represent both wave and particle?
• Find a description of a particle which is
  consistent with our notion of both particles and
  waves
• Fits the wave description
• Localized in space

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____________________________________

• A Wave Packet

How do you construct a wave packet?
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What happens when you add up waves?_____________
___________________
The Superposition principle
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Adding up waves of different frequencies.._____
_______________________________
22
Constructing a wave packet by adding up several
waves ___________________________________
If several waves of different wavelengths
(frequencies) and phases are superposed together,
one would get a resultant which is a localized
wave packet
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A wave packet describes a particle_______________
_____________

• A wave packet is a group of waves with slightly
  different wavelengths interfering with one
  another in a way that the amplitude of the group
  (envelope) is non-zero only in the neighbourhood
  of the particle
• A wave packet is localized a good
  representation for a particle!

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Wave packet, phase velocity and group velocity
____________________________

• The spread of wave packet in wavelength depends
  on the required degree of localization in space
  the central wavelength is given by
• What is the velocity of the wave packet?

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Wave packet, phase velocity and group velocity
________________________________

• The velocities of the individual waves which
  superpose to produce the wave packet representing
  the particle are different - the wave packet as a
  whole has a different velocity from the waves
  that comprise it
• Phase velocity The rate at which the phase of
  the wave propagates in space
• Group velocity The rate at which the envelope
  of the wave packet propagates

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Wave packet, phase velocity and group velocity
________________________________

• Phase velocity
• Group velocity
• Here is the velocity of light and
  is the velocity of the particle

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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!

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Heisenberg's Uncertainty Principle______________
____________________

• Uncertainty in Position
• Uncertainty in Momentum

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Heisenberg's Uncertainty Principle - applies
to all conjugate variables_____________________
______________

• Position momentum
• Energy time

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Uncertainty Principle and the Wave
Packet___________________________________
If is large, is small
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Some consequences of the Uncertainty
Principle ___________________________________

• 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.

32
Why isnt the uncertainty principle apparent to
us in our ordinary experience? Plancks
constant, again!! ________________________________
___

• Plancks 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

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Summary___________________________________

• Matter and radiation have a dual nature of both
  wave and particle
• The matter wave associated with a particle has a
  de Broglie wavelength given by

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Summary (contd.)________________________________
___

• A (localized) particle can be represented by a
  group of waves called a wave packet
• The group velocity of the wave packet is
• The phase velocity of the wave packet is

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Summary (contd.)_______________________________
____

• Heisenbergs uncertainty principle is an
  important consequence of the wave-particle
  duality of matter and radiation and is inherent
  to the quantum description of nature
• It applies to all conjugate variables and also to
  the notion of the wave packet

36
Are matter waves for real?! ____________________
______________

• In 1927 Davisson and Germer showed that electrons
  can diffract they act like waves
• Big application Electron Microscopes

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Are matter waves for real?! ____________________
______________

• Double slit experiment with electrons (1989)
• (www.hqrd.hitachi.co.jp/em/doubleslit.cfm)

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Are matter waves for real?! ____________________
______________

• Today, advances in technology have led to matter
  wave interference experiments being demonstrated
  successfully not only with electrons but
  neutrons, atoms, and big and small molecules!
• Infact, the largest molecule showing
  interference has almost a 100 atoms!

40
Are matter waves for real?! ____________________
______________

• C60 molecules (Fullerenes or Bucky Balls) have a
  wave nature! (A. Zeilinger et al, Vienna, 1999)
• Biomolecules have it too!
• Porphyrin
• (2003)

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Are matter waves for real?! ____________________
______________

• What next? Matter wave interference for a
  Virus?!

42
Matter waves are real. ________________________
_______

• If there is wave, there must be a wave
  equation
• The Schrödinger Equation