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Matter Waves and the Uncertainty Principle _________________________

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Title: 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)

5
Rutherford
Planck
Einstein
Bohr
Compton
6
Pauli
de Broglie
Davisson and Germer
7
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???

8
The story of light..__________________________
_____
  • Corpuscular theory (Newton)
  • Wave nature (Huygens)
  • Double-slit interference experiment (Young)

9
The story of light (contd.).._________________
______________
  • Light is an electromagnetic wave (Maxwell)
  • Photoelectric effect existence of light quanta
    photons (Einstein)
  • Photons have momentum (Compton)..

10
Newton
Huygens
Young
Compton
Maxwell
Einstein
11
Light has a dual nature__________________________
_
  • Wave (electromagnetic) - Interference
  • -
    Diffraction
  • Particle (photons) - Photoelectric effect
  • - Compton effect
  • Wave - Particle Duality for light

12
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!

13
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

14
The connecting link Plancks
constant_______________________________
  • Dual Nature
  • Radiation
  • Matter

15
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!

16
Particle______________________________
  • Our traditional understanding of a particle
  • Localized - definite position, momentum,
    confined in space

17
Wave____________________________
  • Our traditional understanding of a wave.
  • de-localized spread out in space and time

18
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

19
____________________________________
  • A Wave Packet

How do you construct a wave packet?
20
What happens when you add up waves?_____________
___________________
The Superposition principle
21
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
23
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!

24
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?

25
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

26
Wave packet, phase velocity and group velocity
________________________________
  • Phase velocity
  • Group velocity
  • Here is the velocity of light and
    is the velocity of the particle

27
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!

28
Heisenberg's Uncertainty Principle______________
____________________
  • Uncertainty in Position
  • Uncertainty in Momentum

29
Heisenberg's Uncertainty Principle - applies
to all conjugate variables_____________________
______________
  • Position momentum
  • Energy time

30
Uncertainty Principle and the Wave
Packet___________________________________
If is large, is small
31
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

33
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

34
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

35
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

37
(No Transcript)
38
Are matter waves for real?! ____________________
______________
  • Double slit experiment with electrons (1989)
  • (www.hqrd.hitachi.co.jp/em/doubleslit.cfm)

39
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)

41
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
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