Acoustic models of:
Hydrogen Atom Hydrogen Molecule Lowering symmetry to lift degeneracy
Band gaps in semiconductors
Thermodynamic measurement of:
Velocity of sound in air and other gasses Velocity of sound as a function of temperature
Magnitude of the gas constant
Quantum Analogs is TeachSpin’s contribution to the teaching of wave mechanics. The tag line, “Acoustic Experiments Modeling Quantum Phenomena,” best sums up what these unique experiments are about. Anyone who has taught quantum mechanics knows how difficult it is for students to “wrap their heads around” the subtle predictions of this remarkable theory. The idea at the heart of this apparatus is the analogy between the mathematics of the Schrödinger wave equation, and the wave equations that describe the behavior of ordinary sound waves in air. The various parts of our acoustic apparatus will allow students to explore acoustic analogs to quantum-mechanical systems in one, and three, dimensions. One of the advantages of the ‘acoustic analog’ is that sound phenomena occur on a very human scale of length and time.
TeachSpin’s first international collaboration, Quantum Analogs was developed with Professor Rene Matzdorf of the University of Kassel. Although officially a “theorist”, Professor Matzdorf, as a true teacher, recognized that for many people, even theorists, the ability to manipulate concrete phenomena rather than just equations both enhances conceptual understanding and encourages exploration. Manipulating equations sequentially to see the effect of changing the location of a defect on the band gaps of a semiconductor is one thing. Being able to vary the location or shape of a physical object and observe the band-gap changes within moments, encourages a far wider range of experimentation and may even lead to a kind of “instinct” that is usually available only to those for whom equations have a life of their own.
Professor Matzdorf has created an extensive manual for the instrument. The manual begins with a mathematical analysis of the similarities between the equations for standing waves in a tube and the eigenstates of a particle in a box. Students can then proceed through a series of experiments in which the standing waves in spheres and segmented tubes become analogs for the hydrogen atom, the hydrogen molecule and semiconductors.
Initial explorations for each section of the experiment are done using a signal generator and oscilloscope. In addition, Professor Matzdorf has created software which allows students to use a computer to record up to four experimental spectra on the same screen. The effects of changes can be explored efficiently not only qualitatively but quantitatively as well. This software is free, but comes without any warranty or liability. For the users of this experiment, Professor Matzdorf has created a special web page which answers frequently asked questions and provides software updates. The page also offers several excellent visualization programs that users are welcome to download.
Quantum Analogs uses sound waves in cylinders and spheres to model the quantum states in semiconductors, hydrogen atoms, and hydrogen molecules. The apparatus includes precisely machined aluminum cylinders, hemispheres and irises. The controller facilitates interfacing the speakers and microphones that generate and detect the sound with the user supplied function generator, oscilloscope and computer used to generate and display the signals.
Linear Array: A one-dimensional system is used to create an analog of the particle in a box or a semi-conductor. The wooden base which supports the V-groove used to hold the experimental array, also houses all the extra parts for this system.
Students mount a selection of precisely machined 2” diameter aluminum cylinders, with or without intervening irises, in a specially designed V groove. A speaker is mounted at one end of the V-groove. Once the selected cylinders and irises are arranged, the microphone, mounted in a similar cylinder, is placed at the other end of the array and secured so that the tubes are held together firmly. The system includes three sets of aluminum tubes of lengths 1.25, 5.0, and 7.5 cm and three sets of irises.
In the photograph above, the central iris of a series has been removed in order to explore the effect of a “defect” in a lattice.
Hemispheres: The three-dimensional system uses 4 aluminum hemispheres to create analogs of the hydrogen atom and molecule.
The “active” hemispheres used to create the “atom” have both a speaker and microphone mounted in the lower hemisphere and a microphone mounted in the upper. A pair of spacer rings provides three different ways to elongate the “atom”.
In this photograph the system is arranged to create a model for a hydrogen “atom”. The BNC connectors indicate the locations of the microphones. The speaker is at the lower right in this photograph.
Using the two additional hemispheres supplied, students build a pair connecting spheres which become a model for a hydrogen molecule. At the junction between the upper and lower spheres in the “molecule”, irises of four different diameters can used to explore bonding-antibonding states.
Controller: The controller acts as an interface between the source of the signal and the experimental speakers as well as between the microphone and the oscilloscope or computer used to display the amplitude of the sound at the microphone’s location. The function of each part of the controller is displayed in the chart below.
Controller Label Function
Microphone Input provides a source of +5 V dc (for biasing of capacitor microphones), and accepts the ac signal placed atop that bias by the microphone
AC Amplifier provides a fixed gain, of order 100, from about 20 Hz to 20 kHz; ac-coupled at input
Attenuator 10-turn scale provides attenuation of amplified ac signal, by a 'transmission factor' given by (dial setting)/10. Example -- a dial setting of 0.1 turns implies a transmission of 0.1/10 = 0.01 or 1% of maximum gain
AC Monitor provides a direct view of the amplified ac signal at the attenuator's output
Envelope Detector a rectifier system, giving the amplitude of the sine-wave signal present at the AC Monitor output, on a cycle-by-cycle basis
Detector Output a dc-coupled positive voltage, the output of the envelope detector
Sine Wave Input provides the entry point for ac signals from signal generator or computer sound-card
Speaker Output directly coupled to Sine Wave Input below it on the panel; provides the point of attachment for 3.5-mm speaker plug
Frequency-to-Voltage Converter when toggled to On, this module derives a signal from Sine Wave Input, and converts its frequency to a voltage, at conversion ratio 1 Volt per kHz
DC Offset 10-turn dial, allowing the addition of a 0 to -10 Volt offset to the output of the Frequency-to-Voltage converter
DC Output the (possibly dc-offset) output voltage of the F-to-V converter module
Other Hardware and Software: Some explorations require a user supplied oscilloscope and signal generator. Professor Rene Matzdorf (Universitaet Kassel), with whom TeachSpin collaborated to build this apparatus, has created software which uses the soundcard of a computer both to generate the sound signal and to observe the amplitude of the resulting sound waves. The controller makes it possible to observe the sound wave amplitude simultaneously on both the computer and an oscilloscope. The software, both written and supported by Professor Matzdorf, makes it possible to record up and compare up to four different spectra.
Dedicated Website: Professor Matzdorf has created a dedicated website where he has posted a downloadable current copy of the program, a manual for the software and answers to frequently asked software questions. The page also offers several excellent visualization programs that users are welcome to download for your students.
The links provided in the section below this will take you to a Conceptual Introduction to Quantum Analogs and to newsletters describing many of the experiments that can be done with TeachSpin’s Quantum Analogs. Below, you will find the table of contents for the Student Version of the User Manual. It will give you an insight into the range of explorations possible with this apparatus.
Table of Contents
0. Introduction to the Apparatus
1. Standing Sound Waves in a Tube –
Analog to a Quantum Mechanical Particle in a Box
1.1 Initial set up and theory
1.2 Measure a spectrum in the tube using an oscilloscope
1.3 Measure a spectrum with the computer and compare
it to the spectrum found with the oscilloscope
2. Modeling a Hydrogen Atom with a Spherical Resonator
2.0 Theoretical Background
2.1 Measure resonances in the spherical resonator and
determine their quantum numbers
2.2 Measure spectra and wave functions in the spherical
resonator using the computer
3. Broken Symmetry in the Spherical Resonator and Modeling a Molecule
3.1 Lifting Degeneracy of states with different magnetic
3.2 Modeling a molecule
4. Modeling a One Dimensional Solid
4.1 From a free electron to an electron in a periodic potential
4.2 Atom – Molecule – Chain
4.3 Superstructures and unit cells with more than one atom
4.4 Defect States
Appendix 1 Controller Box – Technical Description
Appendix 2 Recognizing and Correcting Saturation
Appendix 3 Connecting to a Computer – Instructions & Troubleshooting