Using Computers to Visualize and Reason with Quantum Concepts

1
Using Computers to Visualize and Reason with
Quantum Concepts

• Peter Garik,1 Alan Crosby,2 Dan Dill,2 Alex
  Golger,2 Morton Z. Hoffman2
• 1) School of Education
• 2) Department of Chemistry
• Boston University
• Boston, Massachusetts 02215
• http//quantumconcepts.bu.edu

2
The Challenge-I

• Quantum Concepts are among the most challenging
  topics in general chemistry.
• Some instructors are uncomfortable about teaching
  that material.
• Our objective is to present the time-dependent
  interaction of light with matter to science and
  pre-medical general chemistry students (CH101).

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The Challenge-II

• Students are weak in their understanding of the
  energetics of waves and the nature of fields.
• They generally do not know that EM waves have
  electric and magnetic fields associated with
  them.
• They do not understand how light interacts with
  electrons.
• They do not associate the energy of emitted or
  absorbed light with a difference in electronic
  energy levels.

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Our Approach

• Learning cycle-based activities data collection,
  analysis, extension.
• Use of bridging analogies whenever possible with
  computer visualization and representation.
• Guided-inquiry approach and software
• Interactive graphical renderings of
  time-dependent atomic orbitals and their
  interaction with light without mathematics!
• Visualizations of the beats that correspond to
  dipole excitations of atoms.

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Learning Objectives

• Spatial and time dependence of normal modes.
• Spatial and time dependence of the complex
  wavefunction.
• The Planck relationship for electrons
• E h?.
• The superposition of wavefunctions and the
  resulting oscillation at the difference between
  the two Planck frequencies (beating).

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Guided Inquiry Software

• Used in conjunction with lecture demonstrations,
  lecture/discussion workshops, lab exercises, and
  homework.
• Project 1 spectroscopy of atomic hydrogen and
  hydrogen-like ions.
• Project 2 introduction to the normal modes of
  one- (cable) and two-dimensional (square and
  circular membranes) waves with analogy to the
  modes of a bound electron.
• Project 3 time-dependent behavior of electron
  orbitals and their interaction with light.

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Waves
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Harmonics of an Oscillating Cable
Students studied the harmonics of an oscillating
cable by measuring the amplitude, wavelength,
period, and frequency. The connection between the
spatial and temporal aspects of the wave was
emphasized through the use of two related graphs.
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Energy Density for an Oscillating Cable
The total energy density, as well as the kinetic
and potential energy densities at a point, are
connected by the interactive graph of
displacement and the energy histogram.
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Representations of a One-Dimensional Harmonics

• Eventually, students work with a visual
  representation in which the intensity is colored
  coded. The two representations are presented
  side-by-side in order to build familiarity.

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Two-Dimensional Normal Modes
For the 2-D modes, two representations are
initially provided. Each has its own advantage
for displaying the frequency of the mode. The 3-D
display of the 2-D mode can be rotated in space.
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Beats in Two-Dimensions
To understand the phenomena of beats, students
are asked to measure the frequency of superposed
harmonic modes. Visually this is difficult for
arbitrary modes however we restrict the activity
to those for which they can be successful.
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Harmonics of an Oscillating Disk

• The activities performed for the rectilinear
  geometries are now repeated but with cylindrical
  symmetry. The aim is to help students develop an
  understanding of rotational degeneracy.

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Harmonics of an Hydrogenic Electron
Students are now expected to extend their
understanding of harmonics to the normal modes of
hydrogenic orbitals. To assist them, a phasor
indicating the phase both by color and complex
number is provided.
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Beating of a Hydrogen Atom
In a lecture demonstration, the students heard
the beats of superposed sound waves. Now they
will strive to see the connection of emitted
light to superposed orbital frequencies.
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Assessing the Efficacy of the Approach

• Pre- and post-tests and student interviews were
  used.
• The sophistication of the questions demonstrated
  our level of expectation to the students.
• The test results from more than 500 general
  chemistry students suggest that they can master
  the concepts that underlie the modern quantum
  model of chemistry, spectroscopy, and
  nanotechnology.

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Energy Distribution in a Wave
Pre Test B (N304) Post Test B (N288)
a) The energy of vibration is uniformly distributed over the length of the string. At this instant, all the energy is kinetic energy of motion. 47.2 3.2
b) The energy is uniformly distributed over the length of the string. At this instant, all the energy is potential energy. 17.5 8.1
c) The energy is non-uniformly distributed over the length of the string. At this instant, all the energy is kinetic energy. 18.2 7.7
d) The energy is non-uniformly distributed over the length of the string. At this instant, all the energy is potential energy. 6.3 64.5
e) None of the above. 9.6 13.7
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Learning About the Complex Nature of the
Wavefunction

• On the post-test we asked the students the
  question, At a specific time, one lobe of a 2p
  orbital has a phase angle corresponding to 1 i.
  Which of the following complex numbers
  corresponds to the phase of the second lobe?
• This is a question which would have been
  meaningless on the pre-test.
• 58.3 of the general chemistry students (N
  550) selected -1 i.
• Since the complex nature of the wavefunction had
  not been discussed in lecture in the general
  chemistry course, this result suggests success
  for our visualization methods for the complex
  phase.

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Time Dependence of the Wavefunction

• 8.1 Select the best explanation for the
  time-dependence of an electron in an atomic
  orbital.
• a) The electron has a velocity that corresponds
  to its kinetic energy, which varies as it
  approaches or recedes from the nucleus.
• b) The time-dependence for an electron in an
  orbital corresponds to the uncertainty in its
  position and, therefore, its instantaneous
  velocity.
• c) An atomic orbital has complex values with
  periodic oscillation in its value with time.
• d) The square of the values of an atomic orbital
  corresponds to a probability density for finding
  an electron at a point in space this does not
  vary in time, so there is no time-dependence for
  the orbital.
• e) None of the above.

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Time Dependence of the Wavefunction

• c) An atomic orbital has complex values with
  periodic oscillation in its value with time.
• Pre-test A (N235) 18.7
• Post-test A (N201) 51.6
• Post-test B (N274) 62.4
• Physical chemistry Post-test (N22) 22.7

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Conclusions-I

• Our results support the conclusion that general
  chemistry students can learn about quantum
  concepts through the use of guided-inquiry
  interactive graphics and visualizations.

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Conclusions-II

• The vocabulary of time-dependent electron
  orbitals provides new insights for the students
  about the absorption and emission of
  electromagnetic radiation across the spectrum,
  van der Waals interactions, and London dispersion
  forces.

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Acknowledgements

• Peter Carr, Programmer
• Joshua Csehak and Lars Travers, Ace Coders
  Programming
• Judith Kelley and Russell Faux, Project
  Evaluators
• Funding, U.S. Department of Education Fund for
  the Improvement of Post Secondary Education
  (FIPSE Grant P116B020856)