## Activity: Going from Spin States to Wavefunctions

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.
• This activity is used in the following sequences
What students learn
• The position basis is continuous and uncountably infinite.
• The histogram of a state written in the position basis is the wavefunction.
• The wavefunction is the probability amplitude density of the quantum state. Squaring the wavefunction gives you the probability density.
• To get a probability, you have to integrate the norm square of the wavefunction.
• The completeness relation for the position basis is $\int \left|{x}\right\rangle \left\langle {x}\right| dx = 1$.
• Position basis vectors have dimensions of $1/\sqrt{\mbox{length}}$.
• accessibility_new Spin 1/2 with Arms

accessibility_new Kinesthetic

10 min.

##### Spin 1/2 with Arms

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
• accessibility_new Using Arms to Represent Time Dependence in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Time Dependence in Spin 1/2 Systems

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.
• accessibility_new Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

accessibility_new Kinesthetic

10 min.

##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).
• accessibility_new Curvilinear Basis Vectors

accessibility_new Kinesthetic

10 min.

##### Curvilinear Basis Vectors
AIMS Maxwell AIMS 21 Central Forces Spring 2021 Static Fields Winter 2021

Curvilinear Coordinate Sequence

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).
• format_list_numbered Arms Sequence for Complex Numbers and Quantum States

format_list_numbered Sequence

##### Arms Sequence for Complex Numbers and Quantum States
“Arms” is an engaging representation of complex numbers in which students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram). The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence. These activities can be combined and sequenced in many different ways; see the Instructor's Guide for the second activity for ideas about how to introduce the Arms representation the first time you use it.
• accessibility_new Using Arms to Visualize Complex Numbers (MathBits)

accessibility_new Kinesthetic

10 min.

##### Using Arms to Visualize Complex Numbers (MathBits)

Arms Sequence for Complex Numbers and Quantum States

Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.
• group Energy and Angular Momentum for a Quantum Particle on a Ring

group Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
• group Superposition States for a Particle on a Ring

group Small Group Activity

30 min.

##### Superposition States for a Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
• group Time Dependence for a Quantum Particle on a Ring

group Small Group Activity

30 min.

##### Time Dependence for a Quantum Particle on a Ring
Central Forces Spring 2021

Quantum Ring Sequence

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
• group Using Tinker Toys to Represent Spin 1/2 Quantum Systems

group Small Group Activity

10 min.

##### Using Tinker Toys to Represent Spin 1/2 Quantum Systems

Arms Sequence for Complex Numbers and Quantum States

Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases ($x$, $y$, and $z$). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.

## Instructor's Guide

This activity is part of the Arms Sequence for Complex Numbers and Quantum States.

### Prerequisite Ideas

Students should be familiar with spin-1/2 and spin-1 systems, particularly represently quantum states with Dirac notation, histograms, matrices, and arms.

### Prompts:

1. Set-Up

• Ask for a pair of students to demonstrate how to represent a quantum state with arms.
• Each student represents one of the $S_z$ basis states:
person on the left $\rightarrow \left|{+}\right\rangle$
person on the right $\rightarrow \left|{-}\right\rangle$
• Each student uses their left arm to represent the complex probability amplitude for their basis ket.

2. Instructor writes a quantum state in Dirac notation on the board. “How would you represent this state using arms?”

The instructor then asks the pair of volunteers (volunteers should write their answer on the board):

• “What does this state look like as a histogram?”
• “What does this state look like in matrix notation?”

3. Instructor asks.“What would we have to do in order to represent a spin-1 state with arms?” Answer - add a student.

• Instructor asks for another volunteer
• Instructor writes a spin-1 state on the board and asks the student volunteers to represent it using: Arms, histogram, and matrix notations.

4. Instructor repeats exercise for a spin-3 system (7 basis kets)

5. Instructor asks the students to consider a new situation: “Now imagine that I have a particle and the position of the particle in an observable I'm interested in.”

• “How could we present a quantum state in the position basis with Arms?” Ask as many students as volunteer to join the Arms representation. Still not enough - we'd need an infinite number of people! And an infinite number of people in between each person.

• What would the histogram look like? Answer: A function of x. This is the wavefunction.

• What would a quantum state written in a 1-D position basis look like? Answer: An infinitely long column with each entry corresponding a position eigenstate. The set of all the entries is the wavefunction.

• “What would the basis be like?”
• (uncountably) infinite - even if the region is bounded. Between each person would be an infinite number of people.
• spikey in space (like delta functions). Optional - you can foreshadow the uncertainty principle
• in Dirac notation, label with the eigenvalue:

$\hat{X}\left|{x}\right\rangle =x\left|{x}\right\rangle$

• Complete - can write down a completion relation. The sum for a discrete basis, like spin, become an integral for a continuous basis:

$\sum_{s} \left|{s}\right\rangle \left\langle {s}\right| = 1$

$\int \left|{x}\right\rangle \left\langle {x}\right| dx = 1$

• Talk about the dimensions of $\left|{x}\right\rangle \rightarrow$ 1/$\sqrt{\mbox{length}}$

### Wrap-up

• The position basis is continuous and (uncountably) infinite.
• Position basis kets are like spikey delta functions with dimensions of $1/\sqrt{\mbox{length}}$.

### Extension

• Write a state in the position basis by applying a completeness relation:

\begin{align*} \left|{\psi}\right\rangle &= \left(\int \left|{x}\right\rangle \left\langle {x}\right| dx \right) \left|{psi}\right\rangle \\ &= \int \left|{x}\right\rangle \left\langle {x}\middle|{\psi}\right\rangle dx \\ &= \int \left\langle {x}\middle|{\psi}\right\rangle \left|{x}\right\rangle dx \\ &= \int \psi(x) \left|{x}\right\rangle dx \\ \end{align*}

Notice: $\left\langle {x}\middle|{\psi}\right\rangle$ is the wavefunction. Because of the integral, we can't just take the norm squared of the wavefunction at a particular x to get the probability of the particle being at a particular location. The wavefunction doesn't have the right dimensions! Instead, the wavefunction is a “probability amplitude density”. If you square the wavefunction, you get a probability density. You have to integrate the norm square of the wavefunction over a length to get a probability.

• In fact, asking the question “What is the probability of the particle being at a particular point?” is not a question we can answer with the formalism of quantum mechanics. We can only calculate the probability of a particle being in an infinitesimal region:

$\mathcal{P}(\mbox{between }x_0 \mbox{and }x_0+dx) = |\left\langle {x}\middle|{\psi}\right\rangle |^2 dx$

The multiplication by $dx$ makes the quantity dimensionless.

To calculate the probability of the particle being in a finite region:

$\mathcal{P}(\mbox{between }x=a \mbox{and }x=b) = \int |\left\langle {x}\middle|{\psi}\right\rangle |^2 dx$

Learning Outcomes