Here is a verbal description of the learning progression for the concept of wavefunction in a spins-first approach. In the end, we imagine our learning progressions to have a graphical presentation with tabular and web-like structures and be linked to corresponding curricular materials.

Lowest Anchor: Students begin our Quantum Fundamentals course with pretty robust ideas about vectors as arrows in space that can be expressed in different coordinate systems, density as an amount of something per unit volume or per unit length, and the deterministic nature of classical physics.

Level 1: Students start developing this understanding that the probabilistic nature of quantum mechanics is distinct from classical physics. In our spins-first course, we begin by focusing heavily on experimental evidence (through Stern-Gerlach experiments) that quantum mechanics is probabilistic rather than deterministic.

The students' ideas about vectors start to expand to include the properties of an orthonormal basis (orthogonal, normal, and complete) and that orthonormal bases do not have to be spatial but can be used to model things that are mutually exclusive (like primary colors or spin eigenstates).

Students begin to understand that quantum states are mathematically modeled as complex valued vectors and that the complex coefficients of the components (the probability amplitudes) of those vectors are related to the probabilities of the outcomes of quantum measurements. The key idea here is that the probability of an outcome is the norm square of the complex probability amplitude that corresponds to that outcome.

For spin systems, students consider spin component measurements along all the spatial axes and find that there is a complete orthonormal basis for the spin components along any one axis. Students begin to understand the completeness relation as a tool to change the basis of the quantum state vector. Completeness relations are related to another key idea that if you know all the components of the vector then you know the vector.

At this stage, students become familiar with various ways to represent spin states using bra/ket notation, matrix notation, Arms, and graphs of complex probability amplitude. All of these representations encode information about the complex coefficients of the quantum state vector. In our instruction we also highlight how each of these representations is related to histograms of probabilities.

Level 2: Students consider higher spins systems and generic quantum systems with states represented in a discrete basis. Students begin to understand that as the quantum system has more possible outcomes, the number of components in the quantum state vector increases.

Students also become more comfortable calculating probabilities for quantities other than spin-component value, like \(S^2\), energy, or a generic observable.

Level 3: Next, students start thinking about the continuous observable of position and the position basis in the context of particle-in-a-box. The position basis has infinitely many elements even for finite regions of space. The students start to see subtle differences with the spin systems they've become familiar with. While the completeness relation for a spin component basis was a sum, the completeness relation for the position basis is an integral over space. Instead of talking about complex probability amplitudes for a quantum state in a discrete basis, we talk about a complex probability density amplitudes for a quantum state in a continuous basis. Another name for this probability density amplitude is the wavefunction. To find the probability of the particle in a region, one integrates the norm squared of the wavefunction over that region.

Since the complex wavefunction tells you the components of the quantum state vector in the position basis, the wavefunction IS a representation of the vector. This is the position representation of the quantum state. The norm square of the wavefunction gives you a probability density function of position.

Students begin exploring how to re-represent all of the calculations the students know in bra/ket representation to wavefunction notation using a completeness relation for the position basis. The use of a completeness relation signals writing the state vectors in the specific basis.

Level 4: Students recognize that a system can have both continuous observables and discrete observables. For example, for the infinite square well, the position observable is continuous (but bounded) but the energy observable is discrete (but infinite). Furthermore, students encounter energy eigenstates written in the position basis. This gives rise to situations where the state is written in both the energy and position bases: one can then write a wavefunction like

\[\psi(x) = \frac{1}{2} \sqrt{\frac{2}{L}}\sin (\pi x/L) + \frac{\sqrt{3}}{2} \sqrt{\frac{2}{L}} \sin (2\pi x/L)\]

which is an energy eigenstate expansion written in the position representation.

Level 5: Later, students generalize to wavefunctions with 2 or 3 spatial dimensions.

Level 6: Students use a momentum completeness relation to change from a wavefunction in the position representation to a wavefunction in the momentum representation. Students begin to understand the Fourier transform as a change of basis.