Structural features of algebraic quantum notations

[This paper is part of the Focused Collection on Upper Division Physics Courses.] The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation. The varied features of these representations affect how computations are performed. For example, identifying probabilities of measurement outcomes for a state described in Dirac notation may involve identifying expansion coefficients by inspection, but if the state is described as a function, identifying those expansion coefficients often involves performing integrals. In this study, we focus on three notational systems: Dirac notation, algebraic wave-function notation, and matrix notation. These quantum notations must include information about basis states and their associated complex probability amplitudes. In this theory paper, we identify four structural features of quantum notations, which we term individuation, degree of externalization, compactness, and symbolic support for computational rules. We illustrate how student reasoning interacts with these structural features with episodes from interviews with advanced undergraduate physics majors reasoning about a superposition state of an infinite square well system. We find evidence of the students coordinating different notations through the use of Dirac notation, using an expression in Dirac notation to guide their work in another notation. These uses are supported by the high degree of individuation, compactness, and symbolic support for computation and the moderate degree of externalization provided by Dirac notation.

Figure 1

Prompted to represent the state in matrix notation, Carlton begins by writing an expression in Dirac notation, then writing a parallel expression with column vectors in place of kets.

Figure 2

Diego writes expressions with kets equal to wave functions in (a) the upper left of his whiteboard and (b) the lower right of his whiteboard. The first line in (a) was written first. Then the equations in (b) were written (in order from top to bottom). Finally, he wrote the second line in (a).

Figure 3

Seth first uses Dirac notation as a template for an expression in matrix notation.

Figure 4

(a) Calculation of energy expectation value in Dirac notation, but the student has written the Hamiltonian as a matrix and is attempting to perform matrix multiplication. In this picture, the student is multiplying to the $H_{11}$ and ${\psi }_{11}$ elements. (b) The student is continuing the “matrix multiplication” by multiplying the $H_{12}$ and ${\psi }_{12}$ elements.

Figure 5

(a) Beginning of discussion when student describes calculating the expectation value as a weighted average. (b) Full board when the student makes an initial attempt at writing the integral for calculating the energy expectation value (boxed in red). (c) Detail of initial expectation value integral with incorrect ordering of terms in integrand. (d) Detail of the student’s second attempt at writing down the integral, after attending to the Dirac bracket for the expectation value.