Exploring student ideas on change of basis in quantum mechanics

A common task when problem solving in quantum mechanics, including in a spins-first curriculum, involves changing the basis of a given state. Our research in undergraduate quantum mechanics courses at three institutions explores student thinking about basis, basis expansion coefficients, and change of basis in the context of spin-$½$ systems. Our investigation is based on conceptual and computational written questions as well as student reasoning interviews. We identify student ideas about whether and how changing basis affects the state, examine how students perceive notation as indicative of choice of basis, explore students’ interpretations of the structure and meaning of a basis expansion, and identify the range of methods students employ when changing basis. For instance, we find a recurring idea that changing basis alters the physical system, and observe that some students chose to relabel the ket representing a quantum state vector after changing basis. Together, these results paint a broad, qualitative picture of a variety of ways that students grapple with basis and change of basis, with potential implications for instruction.

Figure 1

Q4 from Table 1, repeated here for convenience. When considering whether changing basis involves relabeling the state vector, students may either (a) not relabel; (b) relabel by adding a subscript to the ket; (c) relabel by using a new symbol inside the ket. The correct answer is (a).

Figure 2

Student work showing the start of the projection method of finding a change of basis. The student began by writing the inner products and expanding into matrix notation.

Figure 3

Student work showing an alternative correct method for finding the representation of a state in a different basis. In this method, the student substitutes the new basis states into the generic representation. Subsequent combining of like terms allows the student to compare the expression to the original state and solve for $a$ and $b$.

Figure 4

A student’s written response when asked to write the general form of a state in the $S_x$ basis. Here the student uses two unknown variables, $a$ and $b$, and connects the coefficients to the appropriate inner products.

Figure 5

A student’s written response when asked to write the general form of a state in the basis of $S_x$. Here the student equates a state $|\psi {⟩}_x$ to the $S_x$ matrix acting on a generic column vector.