Heisenberg Uncertainty Principle

1
Heisenberg Uncertainty Principle

• Heisenberg (1926) thought about measuring
  simultaneously the position and momentum
  (velocity) of an electron. Realization ?
  measurement of both with precision is impossible
  and, in fact, the measurement process perturbs
  the system (electron, etc).

2
Heisenberg, Light and Electrons

• Measuring the position of a very small particle
  such as an e- requires the use of short
  wavelength light (high frequency, high momentum
  light (Compton effect)). High frequency light
  will alter the electrons momentum. Use low
  frequency (long ?) light and the position of the
  electron is not well determined.

3
Heisenberg Principle Equations

• The Heisenberg uncertainty principle has several
  forms. We denote the uncertainty in momentum, for
  example, by ?p. In one dimension we have
• ?px?x h
  
• If q denotes a three dimensional position
  coordinate then
• ?p?q h/4p

4
Heisenberg and Mass

• Since momentum p mu (u velocity) the
  Heisenberg expression can be written as
• ?u?q h/4pm
• This equation tells us that the product of the
  uncertainties in velocity and position decrease
  as the mass of the particle increases. Ignoring
  history, Heisenberg tell us that as the mass of a
  particle/system increases we move towards a
  deterministic description (Newton!).

5
Heisenberg and Spectroscopy

• The Heisenberg expression can be recast, in terms
  of energy and time, as
• ?E?t h/4p
• Relates in real life to widths of spectral
  lines. In electronic emission spectra ?t can be
  small (short lifetime in the excited state). This
  means that ?E will be large. The transition
  energy covers a range of values the observed
  spectral line is broadened.

6
de Broglie, Electrons and Waves

• Einsteins theory of relativity tells us that the
  energy of a particle having a momentum p and rest
  mass m is given by
• E2 p2c2 m2c4
• If our interest lies in photons (zero rest mass)
  then
• Ephoton pc p??

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De Broglie continued

• Using the familiar expression from Planck, E
  h?, we get for photons ( i.e., light or
  electromagnetic radiation).
• ? h/p
• De Broglie suggested that this last expression
  could apply to beams of particles with a finite
  rest mass, m, and a velocity u (momentum p mu).

8
De Broglie and Diffraction

• For finite rest mass particles de Broglie gives
  us
• ? h/p h/mu
• This expression does have meaning in our physical
  world. Wave properties of subatomic particles and
  light molecules have been demonstrated though ,
  for example, a variety of diffraction
  experiments.

9
de Broglie examples

• 1. Electrons are easily accelerated by an
  electric field. Find the de Broglie wavelength
  for an electron accelerated by a voltage of 12.5
  kilovolts.
• 2. Would the de Broglie picture be more useful in
  describing the behaviour of a neutron or an NFL
  lineman?

10
De Broglie and Molecular Structure

• Aside Highly accelerated and collimated
  electrons are diffracted when they move through a
  gas. Knowing the energy or de Broglie wavelength
  of the collimated electrons one can, for simple
  gas phase molecules, use the diffraction pattern
  obtained to determine a three dimensional
  structure for the molecules. (Why are protons and
  neutrons less useful for such experiments?).

11
de Broglie, Waves and Electrons

• The work of de Broglie and others showed that
  light and subatomic particles had something in
  common wave properties. This led Schrodinger,
  in particular, to wonder whether equations used
  to describe light waves could be modified to
  describe the behaviour of electrons. This led to
  quantum mechanics and a probabilistic description
  of the behaviour of electrons.

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Operators

• A brief class discussion of operators,
  eigenvalues, eigenfunctions and eigenvalue
  equations is needed before moving to, for
  example, the postulates of quantum mechanics.
  Some of this material may be familiar from
  mathematics courses. The eigenfunctions that
  are useful in describing particles with wave
  properties are of familiar form (and, in part,
  predictable?).

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Postulates of Quantum Mechanics

• The development of quantum mechanics depended on
  equations that are not, in the normal sense,
  derivable. This development was based on a small
  number of postulates. The reasonableness of these
  postulates will become clear through application
  of the postulates.

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Quantum Mechanics Postulates

• Postulate 1 A quantum mechanical system or
  particle can be completely described using a wave
  function, ?. (Wave functions were introduced
  briefly in Chemistry 1050).
• In different examples the problems of interest
  will have varying dimensionality.
  Correspondingly, one sees wave functions
  described in terms of one or more
  coordinates/variables.

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Postulate 1 Dimensionality

• Possible forms of wave functions
• One dimensional problems ?(x), ?(r), ?(?)
• Two dimensional problems ?(x,y) etc.
• Three dimensional problems ?(x,y,z), ?(r,?,f)
• If the time dependant evolution of the system
  needs to be treated (or, appears interesting!)
  the wave functions will have the form ?(x,y,z, t)
  or ?(r,?,f,t).

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Postulate 1 Wave function Properties

• The wave function ?(x,y,z) (for example) must be
  continuous, single valued (not a new
  requirement!) and square integrable. Both real
  and imaginary wave functions are encountered.
  Thus, if ?(x) is imaginary (for example, eimx ,
  where m is an integer) then we will need the
  complex conjugate wave function ?(x) e-imx.