Undergraduate quantum mechanics focuses on teaching through a wavefunction approach in the position-space representation. This leads to a differential equation perspective for teaching the material. However, we know that abstract representation-independent approaches often work better with students, by comparing student reactions to learning the series solution of the harmonic oscillator versus the abstract operator method. Because one can teach all of the solvable quantum problems using a similar abstract method, it brings up the question, which is likely to lead to a better student understanding? In work at Georgetown University and with edX, we have been teaching a class focused on an operator-forward viewpoint, which we like to call operator mechanics. It teaches quantum mechanics in a representation-independent fashion and allows for most of the math to be algebraic, rather than based on differential equations. It relies on four fundamental operator identities—(i) the Leibniz rule for commutators; (ii) the Hadamard lemma; (iii) the Baker-Campbell-Hausdorff formula; and (iv) the exponential disentangling identity. These identities allow one to solve eigenvalues, eigenstates and wavefunctions for all analytically solvable problems (including some not often included in undergraduate curricula, such as the Morse potential or the P¨oschl-Teller potential). It also allows for more advanced concepts relevant for quantum sensing, such as squeezed states, to be introduced in a simpler format than is conventionally done. In this paper, we illustrate the three approaches of matrix mechanics, wave mechanics, and operator mechanics, we show how one organizes a class in this new format, we summarize the experiences we have had with teaching quantum mechanics in this fashion and we describe how it allows us to focus the quantum curriculum on more modern 21st century topics appropriate for the second quantum revolution.
We develop the theory for nonresonant Raman scattering of strongly correlated electrons in time-resolved pump- probe experiments. The electrons are initially pumped with an intense pulse of light and then a probe pulse measures the Raman response function after an adjustable time delay. We describe how the width of the probe pulse and the strength of the electron correlations affect the Raman cross section. The theory is developed for the case of B1g symmetry, where incident and scattered light are polarized perpendicular to each other. We illustrate the exact solution with the Falicov-Kimball model, which is solved via nonequilibrium dynamical mean-field theory.
From the early days of many-body physics, it was realized that the self-energy governs the relaxation or lifetime of the retarded Green’s function. So it seems reasonable to directly extend those results into the nonequilibrium domain. But experiments and calculations of the response of quantum materials to a pump show that the relationship between the relaxation and the self-energy only holds in special cases. Experimentally, the decay time for a population to relax back to equilibrium and the linewidth measured in a linear-response angle-resolved photoemission spectroscopy differ by large amounts. Theoretically, aside from the weak-coupling regime where the relationship holds, one also finds deviations and additionally one sees violations of Mathiessen’s rule. In this work, we examine whether looking at an effective transport relaxation time helps to analyze the decay times of excited populations as they relax back to equilibrium. We conclude that it may do a little better, but it has a fitting parameter for the overall scale which must be determined.
We present a theoretical description of time-resolved photoemission in charge-density-wave insulators that derive their ordering from electron nesting effects. In these pump/probe experiments, a large amplitude (but short duration) pump pulse excites the system into nonequilibrium and then a higher frequency low amplitude probe pulse photoexcites electrons, which are measured at the detector. We describe effects of electron correlations on the photoelectron spectroscopy and provide details for the theoretical techniques used to solve these problems. We also show how the gap fills in as the system is excited, even though the order parameter does not go to zero. The theory is developed for the Falicov-Kimball model, which can be solved exactly with nonequilibrium dynamical mean-field theory.
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