A free Gaussian can be thought of as the ground state of a simple harmonic oscillator with the frequency adjusted to give the initial variance of the Gaussian. The free Hamiltonian, when expressed in terms of the raising and lowering operators, can be thought of as a squeezing operator acting on the SHO ground state. We show how this mapping can be used to understand the increasing variance of the Gaussian as it evolves in time. In cases where time evolution is taught after the operator method solution of the SHO, then one can explain free expansion in this alternative framework. One can even perform the calculation using only operator manipulations, instead of solving the time-dependent Schroedinger equation. We sketch how this is done. It requires the Weyl form of the BCH formula, the Hadamard lemma, and the exponential disentangling identity. This work was an REU project.

Free expansion of a Gaussian via squeezed harmonic oscillator states

Quantum mechanics instruction has remained unchanged for at least 75 years, following a coordinate-space-based formalism that requires significant class time for instruction on the mathematical background for the Frobenius method, delta functions, Fourier transforms, etc. This mathematics instruction, greatly limits the amount of physics that can be included. In this talk, I will tell you how to reverse this trend. In Fall 2020, I taught a one-semester quantum mechanics course at Georgetown that worked within a representation-independent formalism (emphasizing operators, not wavefunctions). It is mathematically simpler and frees up time for discussing conceptual and physical ideas. I discuss important experiments such as Stern-Gerlach, delayed choice, EPR, Bell inequality tests, Hong-Ou-Mandel, Pickering-Fowler lines, discovery of deuterium, proton radius, electron momentum spectroscopy, time of flight, hyperfine interactions and radio astronomy, cyclotron resonance and MRI, single-photon detection, homodyne detection, and LIGO. Come see how you can adopt this for your class too.

Bringing the physics back into the undergraduate quantum classroom

B9 - Upper Division Undergraduate

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AAPT WM 22

PS.E-MO-11: Upper Division Undergraduate

Join AAPT and Physics Educators worldwide in developing and strengthening your professional connections by attending the 2021 AAPT Virtual Summer Meeting. AAPT meetings are a great way to reflect on your teaching, recommit to best practices and re-energize your career with new research, products, and techniques.

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James Freericks

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Free expansion of a Gaussian via squeezed harmonic oscillator states

Bringing the physics back into the undergraduate quantum classroom

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