From Cartesian coordinates to Hilbert space: Supporting student understanding of basis in quantum mechanics

We conducted a multiyear project across three institutions to develop an instructional tutorial that supports student understanding of change of basis in quantum mechanics. Building from our previous work, we identified learning goals to guide activity development. The tutorial makes an analogy between spin-$1/2$ states and a Cartesian coordinate system. This paper details the iterative development process including reports of observations from classroom implementations and the resulting modifications to the activity. Further, we report preliminary findings on the success of the activity in improving students’ ability to correctly change basis and their articulation that change of basis is a choice of representation, not a change to the physical system.

Figure 1

Student work depicting $|u⟩$ as a vector on a Cartesian coordinate plane. Here the individual components are labeled with inner products in Dirac notation.

Figure 2

Student work depicting all vectors on the same graph. The vectors $|i⟩$ and $|j⟩$ are represented by the horizontal and vertical axes, respectively. The new basis vectors, $|v_1⟩$ and $|v_2⟩$, effectively form a rotated set of axes. Students confirm that the vector $|u⟩$ remains unchanged regardless of which set of axes it is plotted against (so long as the appropriate components are used for each basis).

Figure 3

Timeline for research, tutorial development, and evaluation. The three main phases of the project are identified in the column header and the different years of the project are shown with brackets above the table. Each quiz, midterm, and final included a change-of-basis question (similar to the one in Fig. 4). The tutorial was not repeated at University A in year 3 due to a change in instructor. All data collection and tutorial implementation was completed prior to the 2020 pandemic.

Figure 4

One version of the quiz and exam change-of-basis question. Versions of this question were administered to students at each of the three institutions on several for-credit quizzes or exams in multiple semesters. Different versions of the question used different numerical coefficients and/or used varied language for the hint.

Figure 5

Survey 1: Three of the questions given to students as a pre-test for the quantum basis tutorial. Questions are numbered here for reference. Correct answers are bolded and starred. In year 3, question 2 was used again on the quiz, and question 3 was incorporated into survey 2.

Figure 6

Student solution to a standard change-of-basis problem. This student uses the projection method by taking the inner products of the state with the new basis vectors. The short calculation correctly and efficiently provides the coefficients in the new basis representation.

Figure 7

The start of a student’s solution to a change-of-basis question. This student employs the substitution method, where they insert the basis conversions into the generic basis representation. The subsequent steps (not shown) involve rearranging terms, setting each coefficient equal to the coefficient of the given state, and solving a system of equations for $a$ and $b$. Reproduced from Ref. [2].

Figure 8

An image of the given basis states (top) and a student incorrectly determining the $z$-basis representation by swapping subscripts (bottom). The student then substitutes these into the given state to change basis.

Figure 9

Portion of a student’s solution where they used probability calculations to find the expansion coefficients in a new basis representation. They started a probability calculation, $|{}_x⟨+|\psi ⟩{|}^2$, but rather than complete the complex square, they just took the result of the inner product from inside the complex square.

Figure 10

Student’s graph of the vector in the new basis. Inclusion of the grid allowed students to create their own scale and graph the vectors more accurately.

Figure 11

Student work where both sets of basis vectors are drawn as the horizontal and vertical axes. Here the student incorrectly equates the unit vectors.