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

Abstract

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 phase <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="italic">E</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mi mathvariant="italic">ℏ</mml:mi> </mml:mrow> </mml:msup> </mml:math> . 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.