skip to main content


Title: Time evolution in quantum systems: a closer look at student understanding
Abstract

Time evolution of quantum systems has been shown to be one of the most difficult components of a typical undergraduate quantum mechanics course. In this work, we examine the current literature, and then take a closer look at the process that students use to determine how the quantum state of a spin-1/2 particle evolves with time. We divide the process of writing a time-dependent state into five elements and use these to both directly probe student understanding and guide our coding of student responses. We focus on three elements of this process, including knowledge of the Hamiltonian, the energy eigenstates and eigenvalues, and what basis should be used when writing the state as a function of time using the phaseeiEnt/. Analysis of four exam questions given at three institutions suggests that knowledge of the energy eigenbasis and its importance for time evolution may be a weak point in student understanding.

 
more » « less
NSF-PAR ID:
10303217
Author(s) / Creator(s):
; ; ;
Publisher / Repository:
IOP Publishing
Date Published:
Journal Name:
European Journal of Physics
Volume:
41
Issue:
1
ISSN:
0143-0807
Page Range / eLocation ID:
Article No. 015705
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    We perform particle-in-cell simulations to elucidate the microphysics of relativistic weakly magnetized shocks loaded with electron-positron pairs. Various external magnetizationsσ≲ 10−4and pair-loading factorsZ±≲ 10 are studied, whereZ±is the number of loaded electrons and positrons per ion. We find the following: (1) The shock becomes mediated by the ion Larmor gyration in the mean field whenσexceeds a critical valueσLthat decreases withZ±. AtσσLthe shock is mediated by particle scattering in the self-generated microturbulent fields, the strength and scale of which decrease withZ±, leading to lowerσL. (2) The energy fraction carried by the post-shock pairs is robustly in the range between 20% and 50% of the upstream ion energy. The mean energy per post-shock electron scales asE¯eZ±+11. (3) Pair loading suppresses nonthermal ion acceleration at magnetizations as low asσ≈ 5 × 10−6. The ions then become essentially thermal with mean energyE¯i, while electrons form a nonthermal tail, extending fromEZ±+11E¯itoE¯i. Whenσ= 0, particle acceleration is enhanced by the formation of intense magnetic cavities that populate the precursor during the late stages of shock evolution. Here, the maximum energy of the nonthermal ions and electrons keeps growing over the duration of the simulation. Alongside the simulations, we develop theoretical estimates consistent with the numerical results. Our findings have important implications for models of early gamma-ray burst afterglows.

     
    more » « less
  2. Abstract

    We use ALMA observations of CO(2–1) in 13 massive (M*≳ 1011M) poststarburst galaxies atz∼ 0.6 to constrain the molecular gas content in galaxies shortly after they quench their major star-forming episode. The poststarburst galaxies in this study are selected from the Sloan Digital Sky Survey spectroscopic samples (Data Release 14) based on their spectral shapes, as part of the Studying QUenching at Intermediate-z Galaxies: Gas, anguLarmomentum, and Evolution (SQuIGGLE) program. Early results showed that two poststarburst galaxies host large H2reservoirs despite their low inferred star formation rates (SFRs). Here we expand this analysis to a larger statistical sample of 13 galaxies. Six of the primary targets (45%) are detected, withMH2109M. Given their high stellar masses, this mass limit corresponds to an average gas fraction offH2MH2/M*7%or ∼14% using lower stellar masses estimates derived from analytic, exponentially declining star formation histories. The gas fraction correlates with theDn4000 spectral index, suggesting that the cold gas reservoirs decrease with time since burst, as found in local K+A galaxies. Star formation histories derived from flexible stellar population synthesis modeling support this empirical finding: galaxies that quenched ≲150 Myr prior to observation host detectable CO(2–1) emission, while older poststarburst galaxies are undetected. The large H2reservoirs and low SFRs in the sample imply that the quenching of star formation precedes the disappearance of the cold gas reservoirs. However, within the following 100–200 Myr, theSQuIGGLEgalaxies require the additional and efficient heating or removal of cold gas to bring their low SFRs in line with standard H2scaling relations.

     
    more » « less
  3. Abstract

    We present a Keck/MOSFIRE rest-optical composite spectrum of 16 typical gravitationally lensed star-forming dwarf galaxies at 1.7 ≲z≲ 2.6 (zmean= 2.30), all chosen independent of emission-line strength. These galaxies have a median stellar mass oflog(M*/M)med=8.290.43+0.51and a median star formation rate ofSFRHαmed=2.251.26+2.15Myr1. We measure the faint electron-temperature-sensitive [Oiii]λ4363 emission line at 2.5σ(4.1σ) significance when considering a bootstrapped (statistical-only) uncertainty spectrum. This yields a direct-method oxygen abundance of12+log(O/H)direct=7.880.22+0.25(0.150.06+0.12Z). We investigate the applicability at highzof locally calibrated oxygen-based strong-line metallicity relations, finding that the local reference calibrations of Bian et al. best reproduce (≲0.12 dex) our composite metallicity at fixed strong-line ratio. At fixedM*, our composite is well represented by thez∼ 2.3 direct-method stellar mass—gas-phase metallicity relation (MZR) of Sanders et al. When comparing to predicted MZRs from the IllustrisTNG and FIRE simulations, having recalculated our stellar masses with more realistic nonparametric star formation histories(log(M*/M)med=8.920.22+0.31), we find excellent agreement with the FIRE MZR. Our composite is consistent with no metallicity evolution, at fixedM*and SFR, of the locally defined fundamental metallicity relation. We measure the doublet ratio [Oii]λ3729/[Oii]λ3726 = 1.56 ± 0.32 (1.51 ± 0.12) and a corresponding electron density ofne=10+215cm3(ne=10+74cm3) when considering the bootstrapped (statistical-only) error spectrum. This result suggests that lower-mass galaxies have lower densities than higher-mass galaxies atz∼ 2.

     
    more » « less
  4. Abstract

    We measure the thermal electron energization in 1D and 2D particle-in-cell simulations of quasi-perpendicular, low-beta (βp= 0.25) collisionless ion–electron shocks with mass ratiomi/me= 200, fast Mach numberMms=1–4, and upstream magnetic field angleθBn= 55°–85° from the shock normalnˆ. It is known that shock electron heating is described by an ambipolar,B-parallel electric potential jump, Δϕ, that scales roughly linearly with the electron temperature jump. Our simulations haveΔϕ/(0.5miush2)0.1–0.2 in units of the pre-shock ions’ bulk kinetic energy, in agreement with prior measurements and simulations. Different ways to measureϕ, including the use of de Hoffmann–Teller frame fields, agree to tens-of-percent accuracy. Neglecting off-diagonal electron pressure tensor terms can lead to a systematic underestimate ofϕin our low-βpshocks. We further focus on twoθBn= 65° shocks: aMs=4(MA=1.8) case with a long, 30diprecursor of whistler waves alongnˆ, and aMs=7(MA=3.2) case with a shorter, 5diprecursor of whistlers oblique to bothnˆandB;diis the ion skin depth. Within the precursors,ϕhas a secular rise toward the shock along multiple whistler wavelengths and also has localized spikes within magnetic troughs. In a 1D simulation of theMs=4,θBn= 65° case,ϕshows a weak dependence on the electron plasma-to-cyclotron frequency ratioωpece, andϕdecreases by a factor of 2 asmi/meis raised to the true proton–electron value of 1836.

     
    more » « less
  5. Abstract

    One of the cornerstone effects in spintronics is spin pumping by dynamical magnetization that is steadily precessing (around, for example, thez-axis) with frequencyω0due to absorption of low-power microwaves of frequencyω0under the resonance conditions and in the absence of any applied bias voltage. The two-decades-old ‘standard model’ of this effect, based on the scattering theory of adiabatic quantum pumping, predicts that componentISzof spin current vector(ISx(t),ISy(t),ISz)ω0is time-independent whileISx(t)andISy(t)oscillate harmonically in time with a single frequencyω0whereas pumped charge current is zeroI0in the same adiabaticω0limit. Here we employ more general approaches than the ‘standard model’, namely the time-dependent nonequilibrium Green’s function (NEGF) and the Floquet NEGF, to predict unforeseen features of spin pumping: namely precessing localized magnetic moments within a ferromagnetic metal (FM) or antiferromagnetic metal (AFM), whose conduction electrons are exposed to spin–orbit coupling (SOC) of either intrinsic or proximity origin, will pump both spinISα(t)and chargeI(t) currents. All four of these functions harmonically oscillate in time at both even and odd integer multiplesNω0of the driving frequencyω0. The cutoff order of such high harmonics increases with SOC strength, reachingNmax11in the one-dimensional FM or AFM models chosen for demonstration. A higher cutoffNmax25can be achieved in realistic two-dimensional (2D) FM models defined on a honeycomb lattice, and we provide a prescription of how to realize them using 2D magnets and their heterostructures.

     
    more » « less