Improving students’ understanding of quantum measurement. II. Development of research-based learning tools

We describe the development and implementation of research-based learning tools such as the Quantum Interactive Learning Tutorials and peer-instruction tools to reduce students’ common difficulties with issues related to measurement in quantum mechanics. A preliminary evaluation shows that these learning tools are effective in improving students’ understanding of concepts related to quantum measurement.

Figure 1

Pictorial question in the warm up testing students’ understanding of the energy eigenstates for a 1D infinite square well. (I) and (II) are energy eigenstates but their superposition (III) is not.

Figure 2

Simulation program of the energy measurement on a superposition state [as shown in (a)] with two energy eigenstate components [as shown in (b) and (c)]. The vertical axis is the absolute value of the wave function (not the probability density for position measurement) and the horizontal axis is the position $x$.

Figure 3

Simulation program of the energy measurement on a superposition state [as shown in (a)] with nine energy eigenstate components. (b),(c) Examples of absolute values in position space of two basis energy eigenstates of the superposition state.

Figure 4

Simulation program showing one possible outcome of the position measurement on an energy eigenstate (in position space). (a) Before position measurement and (b) after position measurement.

Figure 5

Time evolution on a superposition state with two energy eigenstate components. (a) The absolute value of the initial state wave function ($t=0$). (b),(c) The absolute values of the wave function at different times [(b) $t=2.50\mathrm{\text{units}}$, (c) $t=5.34\mathrm{\text{units}}$]. The phases of the wave function are represented by different colors.

Figure 6

Energy measurement on a superposition state with two energy eigenstate components after time $t>0$. (a) The superposition state before the energy measurement. (b),(c) The two possible outcomes of the quantum measurement.

Figure 7

Time evolution of the position eigenfunction. (a) The position eigenfunction right after a position measurement. (b),(c) The wave function of the system at later times after the position measurement.

Figure 8

Energy measurement after a position measurement of the initial state with only two energy eigenstate components $n=1$ and $n=2$ as shown in (a). Following the position measurement in (b), the energy measurement yields the energy eigenvalue with $n=5$ as shown in (c).

Figure 9

Consecutive position measurement in quick succession (a),(b) and after waiting for some time (c),(d).