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Highlights• Electroencephalogram (EEG) was concurrently recorded in a simulated classroom from groups of four students and a teacher.• Alpha-band (8-12Hz) brain-to-brain synchrony predicted students' performance in a delayed post-test.• Moment-to-moment variation in alpha-band brain-to-brain synchrony indicated what specific information was retained by students.• Whereas student-to-student brain synchrony best predicted learning at a zero time lag, student-to-teacher brain synchrony best predicted learning when adjusting for a ~200 millisecond lag in the students' brain activity relative to the teacher's brain activity. SummaryLittle is known about the brain mechanisms that underpin how humans learn while interacting with one another in ecologically-valid environments (1-3). This is because cognitive neuroscientists typically measure one participant at a time in a highly constrained environment (e.g., inside a brain scanner). In the past few years, researchers have begun comparing brain responses across individuals (4-6) demonstrating that brain-to-brain synchrony can predict subsequent memory retention (7-9). Yet previous research has been constrained to noninteracting individuals. Surprisingly, the one study that was conducted in a group setting found that brain synchrony between students in a classroom predicted how engaged the students were, but not how much information they retained (10). This is unexpected because brain-to-brain synchrony is hypothesized to be driven, at least partially, by shared attention (11,12), and shared attention has been shown to affect subsequent memory (13). Here we used EEG to 3 simultaneously record brain activity from groups of four students and a teacher in a simulated classroom to investigate whether brain-to-brain synchrony, both between students and between the students and the teacher, can predict learning outcomes (Fig. 1A). We found that brain-tobrain synchrony in the Alpha band (8-12Hz) predicted students' delayed memory retention.Further, moment-to-moment variation in alpha-band brain-to-brain synchrony discriminated between content that was retained or forgotten. Whereas student-to-student brain synchrony best predicted delayed memory retention at a zero time lag, student-to-teacher brain synchrony best predicted memory retention when adjusting for a ~200 millisecond lag in the students' brain activity relative to the teacher's brain activity. These findings provide key new evidence for the importance of brain data collected simultaneously from groups of individuals in ecologicallyvalid settings. Results and DiscussionBehavioral results. Students' content knowledge was assessed a week before the EEG session, immediately following each one of four mini-lectures, and one week later ( Fig. 1B; See Methods). As expected, students' content knowledge significantly increased from the pre-test (0.43±0.02; mean ± standard deviation of the mean) to the immediate post-test (0.73±0.02; F(1,30)=210.76; p<10 -13 ), and from the pre-test to the delayed post-test (0.64±0.02; F(1,30)=...

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The Feynman–Hellmann theorem is employed to derive simple expressions for the expectation values of particular powers of the radial coordinate for a class of central or hypercentral force problems. The results are applied to the Kratzer, the modified oscillator, and the hyperspherical oscillator potentials.

Much of human learning happens through interaction with other people, but little is known about how this process is reflected in the brains of students and teachers. Here, we concurrently recorded electroencephalography (EEG) data from nine groups, each of which contained four students and a teacher. All participants were young adults from the northeast United States. Alpha-band (8–12 Hz) brain-to-brain synchrony between students predicted both immediate and delayed posttest performance. Further, brain-to-brain synchrony was higher in specific lecture segments associated with questions that students answered correctly. Brain-to-brain synchrony between students and teachers predicted learning outcomes at an approximately 300-ms lag in the students’ brain activity relative to the teacher’s brain activity, which is consistent with the time course of spoken-language comprehension. These findings provide key new evidence for the importance of collecting brain data simultaneously from groups of learners in ecologically valid settings.