This Curricular Arc describes a collection of activities & problems that scaffolds students' understanding of what a graph of a 1-D wavefunction tells about a quantum system, starting with probability histograms of spin-1/2 measurements and ending with a time-evolving superposition state of a particle in a 1-D box. This Arc assumes taking a spins-first approach to teaching quantum mechanics.

Our Quantum Fundamentals course provides an example of how to order the activities & problems in this Arc.

1. Probability Histogram: A probability histogram records the results of quantum experiments where probability (i.e., the fraction particles of incident particles in an output counter) is plotted on the vertical axis and the measurement value is plotted on the horizontal axis. This representation is strongly connected to the results of experiments. For Spin-1/2 systems, the probability histograms for spin component measured along all three spatial axes uniquely determine the quantum state.

   

   Problems & Activities that feature this representation:

   • Histogram (problem)
   • General State (problem)
   • Expectation Value (problem)
   • Going from Spins to Wavefunctions (activity)
   • Representations of Probabilities on the Ring (problem)
   • Angular Momentum Probability (problem)

2. Arms Representation of Spin-1/2 State: Quantum states are written algebraically as linear combinations of basis vectors with complex expansion coefficients called probability amplitudes. The norm square of the probability amplitudes yield probabilities. In the Arms representation, students work in pairs to represent a quantum state where each student represents a complex probability amplitude with their arm (i.e., a kinesthetic Argand diagram) for one of the basis states. This representation is one step removed from experimentally determined probabilities and is a step closer to algebraic representations of a quantum state written in terms of probability amplitudes.

   

   Problems & Activities that feature this representation:

   • Spin 1/2 with Arms
   • Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
   • Using Arms to Represent Time Dependence in Spin 1/2 Systems
   • Going from Spin States to Wavefunctions
   • Arms Sequence for Complex Numbers and Quantum States

3. Histogram of Probability Amplitudes: This representation is a combination of histogram and Argand diagram, where the horizontal axis is the value measured and at each value there is a perpendicular complex plane containing the corresponding probability amplitude. This representation is a pen & paper representation similar to the Arms representation.

   

   Problems & Activities that feature this representation:

   • Multiple Representations of a Quantum State
   • Going from Spin States to Wavefunctions

4. Graph of Wavefunction: A graph of wavefunctions is the continuous version of the histogram of probability amplitudes: it is a plot of the probability density amplitude over a region of space (akin to how the charge density describes the charge smeared over a region of space).

   

   Problems & Activities that feature this representation:

   • Going from Spin States to Wavefunctions
   • Time Evolution of the ISW
   • Wavefunctions
   • Mean position
   • Kinetic energy
   • Visualization of Quantum Probabilities for a Particle Confined to a Ring
   • Ring Function

5. Graph of Probability Density: A graph of probability density is circling back to the idea behind the probability histogram. Just like the norm squared of the probability amplitudes are probabilities, when the using the position basis to represent the quantum state, the norm squared of the probability density amplitude is the probability density.

   

   Problems & Activities that feature this representation:

   • Time Evolution of the ISW
   • Mean position
   • Visualization of Quantum Probabilities for a Particle Confined to a Ring
   • Ring Function
   • Wavefunctions on a Quantum Ring

Quantum Mechanics Multiple Representations Wavefunction Arms representation Quantum States