The main theme of our research program is to examine methodologies that differ from the standard "Oppenheimer-Schiff-Bohm" approach to quantum mechanics, which is time-worn and does not work so well with students. This standard approach to quantum mechanics is based on a differential equation approach in the position-space basis. This was a natural starting point when the paradigm was developed in the 1940s as a senior graduate-level course, but since then instruction moved earlier and earlier in our curriculum (currently at the sophomore-level in the United States) and this approach needs to be rethought. Our research work has two main themes. The first is a focus on teaching undergraduate quantum mechanics based on the approach commonly called Schrödinger's factorization method. We prefer to call this approach Operator Mechanics as an alternative to matrix mechanics and wave mechanics. The other is based on Schrödinger's original solution to hydrogen, which employs the Laplace method for solving the Schrödinger equation. This latter method also allows for one to teach complex analysis techniques along with quantum mechanics. It is more appropriate for graduate instruction.
Our research work is primarily involved with investigating different forms of pedagogy. Currently, we are not involved in research on how well the new pedagogy works. While this is important, it is even more important to have new ideas for how to teach quantum mechanics in this representation-independent format. Hopefully, we will be able to evaluate the efficacy of this work in the future. Any physics education researcher interested in collaborating with us on this should contact us directly.
 Michael Rushka, Mark Esrick, Wesley N. Mathews Jr., and J. K. Freericks, Converting translation operators into plane polar and spherical coordinates and their use in determining quantum-mechanical wavefunctions in a representation-independent fashion, J. Math. Phys. 62, 072102 (2021)
 Eduardio Munguía-González, Sheldon Rego, and J. K. Freericks, Making squeezed-coherent states concrete by determining their wavefunction, Amer. J. Phys. (2021).
 Anna Galler, Jeremy Canfield and James K Freericks, Schrödinger's original quantum-mechanical solution for hydrogen, Eur. J. Phys. 42, 035405 (2021). Doi: 10.1088/1361-6404/abb9ff
 Tomasz Szymanski and J K Freericks, Algebraic derivation of Kramers-Pasternack relations based on the Schrodinger factorization method, Eur. J. Phys. 42, 025409 (2021). Doi: 10.1088/1361-6404/abd228
 M. Rushka, and J. K. Freericks, A completely algebraic solution of the simple harmonic oscillator, Am. J. Phys. 88, 976--985 (2020). Doi: 10.1119/10.0001702
 William F. Courtney, Lucas B. Vieira, Paul S. Julienne, and James K. Freericks, William F. Courtney, Lucas B. Vieira, Paul S. Julienne, and James K. Freericks, Incorporating the Stern-Gerlach delayed-choice quantum eraser into the undergraduate quantum mechanics curriculum, Am. J. Phys. 88, 298--307 (2020). Doi: 10.1119/10.0000519
 J. K. Freericks, D. Cutler, A. Kruse, and L. B. Vieira, Teaching quantum mechanics to over 28,000 nonscientists Physics Teacher 57, 326--329 (2019). Doi: 10.1119/1.5098924
 J. Alexander Jacoby, Maurice Curran, David R. Wolf, and James K. Freericks, Proving the existence of bound states for attractive potentials in one and two dimensions without calculus Eur. J. Phys. 40, 045404 (2019). Doi: 10.1088/1361-6404/ab21ed
 M. Weitzman and J. K. Freericks, Calculating spherical harmonics without derivatives, Condens. Matt. Phys. 21, 33002 (2018). Doi: 10.5488/CMP.21.33002
 J. K. Freericks and A. Y. Liu, The quantum world around us: teaching quantum and solid state physics to non-science majors, The Changing Role of Physics Department in Modern Universities: Proceedings of International Conference on Undergraduate Education, AIP Conference Proceedings 399 (American Institute of Physics, Woodbury, NY, 1997).
We are working on a number of projects related to quantum pedagogy. The main areas are (i) understanding the relationship between nonsingular and singular factorization chains and also single-shot factorization; (ii) using factorization methods to solve quantum-mechanical problems, especially the wavefunctions; (iii) developing methodologies for teaching Hilbert space properties and the rigged Hilbert space better than what is currently available; and (iv) developing the Laplace method to solve all problems that have wavefunctions that can be expressed in terms of confluent hypergeometric functions. We also are working on releasing the undergraduate course on edX and also to engineers.
Our senior collaborators include Wesley Mathews from Georgetown, Mark Esrick from Georgetown, and ZuYao Teoh from Xiamen.
This work is supported by the National Science Foundation.
We have given a number of research talks on our pedagogical work. A collection of these talks appears below.
Undergraduates are well positioned to participate in developing quantum mechanics pedagogy. They know what it was like to not understand material in a particular subject and also are great at clarifying what helped them to eventually understand the material. We are parterning with undergraduates on a series of research projects. Any undergraduates interesting in joining the effort should contact us to discuss further.
Mike was a student at Georgetown University from 2016-2020. He did a senior research project on calculating wavefunctions using only operators. Two publications resulted from this work. This work led to Mike being awarded the Research Prize in Physics in 2020.
M. Rushka, and J. K. Freericks, A completely algebraic solution of the simple harmonic oscillator, Am. J. Phys. 88, 976--985 (2020). Doi: 10.1119/10.0001702
Michael Rushka, Mark Esrick, Wesley N. Mathews Jr., and J. K. Freericks, Converting translation operators into plane polar and spherical coordinates and their use in determining quantum-mechanical wavefunctions in a representation-independent fashion, J. Math. Phys. 62, 072102 (2021)
Harrison Cooley. Harrison is a student at Georgetown from 2018-2022. He is currently working on a research project on how to introduce the Franson interferometer within the undergraduate quantum mechanics curriculum. The Franson interferometer is an interesting device because it entangles photons that are separated in space by an entanglement in time. This can lead to many misinterpretations about how it operates and how to describe it. But, using simple rules, it is easy to understand how it works.
Hansen Lian. Hansen is a student at Georgetown from 2018-2022. He is currently working on a research project on how to properly visualize waveparticle duality using coherent and squeezed states. We hope to dispel the quite common myth that quantum objects "travel like waves but are detected like particles," which has a number of serious issues with it when one tries to reconcile it with causality and wavefunction collapse.
Sandro Orjuela. Sandro is a student at Bowdoin University. In the summer of 2021, he worked as an REU student at Georgetown. He is currently working on a research project on how to properly visualize waveparticle duality using operator methods to calculate the spreading of a Gaussian wave packet. This uses the Hadamard lemma and the exponential disentangling identity to compute the form of the wavepacket and how it spreads in time.
Since 2018, we have been engaged in what we call deep citizen science. This is a form of citizen science where the intent is to write a research paper with the citizen as the lead author from the outset of the project. One should think of this as similar to senior theses written with undergraduate students. Most citizen scientists have undergraduate degrees in a technical field. Those in the lifelong learner category tend to be retired professionals (engineers, doctors, etc.). Advanced high school students can also perform similar work for well designed projects. In all cases, it is important to develop problems that the citizen scientist can work on and to mentor them carefully to the completion of the project. We have found that quantum pedagogy projects often work well with citizen scientists. In the listing below, we include all Citizen Scientists who are high school students or whom we met through engagement with various MOOCs. Not all fall into the traditional citizen scientists category, but the point of citizen science is that it is always nontraditional!
If you are interested in becoming a deep citizen scientist, just contact us!
Bill is a retired engineer and an alumnus of Quantum Mechanics for Everyone. We wrote a paper for the American Journal of Physics on extending the delayed choice quantum-eraser experiments to Stern-Gerlach analyzer loops. We even described a possible way to perform such experiments using a generalized form of the Stern-Gerlach apparatus that works with alkaline earth atoms.
William F. Courtney, Lucas B. Vieira, Paul S. Julienne, and James K. Freericks, Incorporating the Stern-Gerlach delayed-choice quantum eraser into the undergraduate quantum mechanics curriculum, Am. J. Phys. 88, 298--307 (2020). Doi: 10.1119/10.0000519
We also wrote an article on the Deep Citizen Science paradigm. We had issues with the journal we were submitting it to due to changes in editorial leadership. We are weighing options for other places to send it. The paper is entitled Deep Citizen Science by Design.
Mark Weitzman. Mark Weitzman is a retired professional poker player with a Master's degree in physics from Caltech. He also takes many MOOCs and often serves as a teaching assistant on them. We wrote a paper for a special volume of Condensed Matter Physics (honoring Ihor Stasyuk) on how one computes spherical harmonics solely from operators (no differential equations necessary!). The approach uses an exponential disentangling identity that should be part of all undergraduate quantum curricula.
M. Weitzman and J.K. Freericks, Calculating spherical harmonics without derivatives, Condens. Matt. Phys. 21, 33002 (2018). Doi: 10.5488/CMP.21.33002
Alex Jacoby (left) and Maurice Curran (right)
Alex Jacoby, Maurice Curran and David Wolf.Alex started working on this project when he was a junior in high school; Maurice was a rising senior high school student; and David Wolf is a professor at Austin Community College in Austin, Texas. This work uses the Schrödinger factorization method to show that all one and two-dimensional attractive potentials always have at least one bound state, but in three dimensions, the potential must be sufficiently attractive. Again, this work uses no calculus. It also is an example of what we now call single-shot factorization. In our research work, we are working on the single-shot factorization methodology.
Alexander Jacoby, Maurice Curran, David R. Wolf and James K. Freericks, Proving the existence of bound states for attractive potentials in one and two dimensions without calculus Eur. J. Phys. 40, 045404 (2019). Doi: 10.1088/1361-6404/ab21ed
Sheldon Rego (left) and Eduardo Munguía-González (right).
Sheldon Rego and Eduardo Munguía-Gonz´lez. I met both Sheldon and Eduardo through the MITx MOOC, as well. They completed a paper on determining the wavefunction for squeezed states of the simple harmonic oscillator using three different methods: (i) A differential equation-based approach; (ii) a Fock-state-based approach, and (iii) a completely operator-based approach.
E. Munguia-Gonzalez, S. Rego, and J. K. Freericks, Making squeezed-coherent states concrete by determining their wavefunction, Amer. J. Phys. (2021).
Click to enlarge. Download Mathmatica notebook to generate animations of squeezed states.
Tomasz Szymanski. I also met Tomasz through the MITx Quantum MOOC. Tomasz is currently a Ph. D. student at the University of Wroclaw. We wrote a paper on calculating the so-called Kramers-Pasternack relations, which are a set of recurrence relations for the expectation values of powers of r in the energy eigenstates of hydrogen. This work was done entirely with operator methods.
Tomasz Szymanski and J K Freericks, Algebraic derivation of Kramers-Pasternack relations based on the Schrodinger factorization method, Eur. J. Phys. 42, 025409 (2021). Doi: 10.1088/1361-6404/abd228
Nilesh Goel. Nilesh is an undergraduate student at the Indian Institute of Technology, Dehli, who worked on generalizing quantum gates in an ion-trap quantum computer from one and two-qubit entangling gates to entangling gates over multiple qubits with the so-called i-Toffoli gate. This work is currently under review.
N. Goel and J. K. Freericks, Native multiqubit Toffoli gates on ion trap quantum computers (2021).
Apostolos Christou. Apostolos is an engineer from Greece. He is working on a project to examine the analogy of a Stern-Gerlach analyzer loop made from a trapped ion quantum simulator. This work is ongoing, but is likely nearing completion in 2021.
Xinliang (Bruce) Lyu
Xinliang (Bruce) Lyu. I met Bruce online from the MITx Quantum MOOC. At the time, Bruce was a Civil Engineering student, but he is now studying for his Ph.D. in Physics. Bruce worked on a new solution for the hydrogen atom using Cartesian operator formalism. This solution uses no calculus and works in both momentum space and position space. We should be submitting the paper in 2021.
Zuyao Teoh. I met Zuyao through the MITx MOOC as well. He is a Professor of Mathematics at Xiamen University in Malaysia. ZuYao is working with us on developing the existence theorems for singular and nonsingular factorization chains.