## Activity: Operators & Functions

Quantum Fundamentals Winter 2021
• Test to see if one of the given functions is an eigenfunction of the given operator
• See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
What students learn
• In wavefunction notation, operators (differential operators) act on functions.
• Identifying eigenvalues relationships with functions, e.g., : $\hat{H}\phi_n(x) = E_n \phi_n(x)$
• Complex exponentials are eigenfunctions of both the linear momentum and the kinetic energy operators (in this case, the Hamiltonian for an infinite square well).
• Linear combinations of eigenfunctions are NOT eigenfunctions UNLESS the functions in the sum have the same eigenvalue.
• Media
• assignment Quantum Particle in a 2-D Box

assignment Homework

##### Quantum Particle in a 2-D Box
Central Forces Spring 2021

(2 points each)

You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length $L$ are $\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}$. If you want the eigenfunctions for a particle in a 2-D box, then you just multiply together the eigenfunctions for a 1-D box in each direction. (This is what the separation of variables procedure tells you to do.)

1. Find the normalized eigenfunctions for a particle in a 2-D box with sides of length $L_x$ in the $x$-direction and length $L_y$ in the $y$-direction.
2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
3. Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to $n=3$, $m=3$. Arrange the terms, conventionally, in terms of increasing energy.

You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}

4. Find a formula for the $c_{nm}$s in part (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.

• group Time Evolution of a Spin-1/2 System

group Small Group Activity

30 min.

##### Time Evolution of a Spin-1/2 System
Quantum Fundamentals Winter 2021

In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.
• computer Visualization of Quantum Probabilities for a Particle Confined to a Ring

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for a Particle Confined to a Ring
Central Forces Spring 2021

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
• 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 Mass is not Conserved

group Small Group Activity

30 min.

##### Mass is not Conserved
Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021

Groups are asked to analyze the following standard problem:

Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

• assignment Center of Mass for Two Uncoupled Particles

assignment Homework

##### Center of Mass for Two Uncoupled Particles
Central Forces Spring 2021

Consider two particles of equal mass $m$. The forces on the particles are $\vec F_1=0$ and $\vec F_2=F_0\hat{x}$. If the particles are initially at rest at the origin, find the position, velocity, and acceleration of the center of mass as functions of time. Solve this problem in two ways, with or without theorems about the center of mass motion. Write a short description comparing the two solutions.

• face Time Evolution Refresher (Mini-Lecture)

face Lecture

30 min.

##### Time Evolution Refresher (Mini-Lecture)
Central Forces Spring 2021

Quantum Ring Sequence

The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
• 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 Quantum Expectation Values

group Small Group Activity

30 min.

##### Quantum Expectation Values
Quantum Fundamentals Winter 2021
• assignment Matrix Elements and Completeness Relations

assignment Homework

##### Matrix Elements and Completeness Relations
Quantum Fundamentals Winter 2021

Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.

What if I want to calculate the matrix elements using a different basis??

The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: $\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y$

In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)

One way to do this to stick completeness relationships into the braket: \begin{eqnarray*} \left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle \end{eqnarray*}

where $I$ is the identity operator: $I=\left|{+}\right\rangle _y {}_y\left\langle {+}\right|+\left|{-}\right\rangle _y {}_y\left\langle {-}\right|$. This effectively rewrite the $\left|{+}\right\rangle$ in the $\left|{\pm}\right\rangle _y$ basis.

Find the top row matrix elements of the operator $\hat{S}_y$ in the $S_z$ basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)

Operators & Functions

For each of the following operators, test each function to see if it is an eigenfunction of the operator.

• If it is, what is the eigenvalue?
• If it is not, can you write it as a superposition of functions that are eigenfunctions of that operator?
1. $\hat{p} = -i\hbar\frac{d}{dx}$
• $\psi_1(x) = Ae^{-ikx}$

• $\psi_2(x) = Ae^{+ikx}$

• $\psi_3(x) = A\sin kx$

2. $\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$

• $\psi_1(x) = Ae^{-i\frac{p}{\hbar}x}$

• $\psi_2(x) = Ae^{+i\frac{p}{\hbar}x}$

• $\psi_3(x) = A\sin \frac{px}{\hbar}$

3. $\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}$

• $\psi_1(x) = A\sin kx$

• $\psi_2(x) = A\cos kx$

• $\psi_3(x) = Ae^{ikx}$

4. $\hat{S_z} \doteq \begin{bmatrix} \frac{\hbar}{2} & 0\\ 0 & -\frac{\hbar}{2}\end{bmatrix}$
• $\left|{\psi_1}\right\rangle \doteq \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

• $\left|{\psi_2}\right\rangle \doteq \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

• $\left|{\psi_3}\right\rangle \doteq \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 1 \end{bmatrix}$

Question: Are superpositions of eigenfunctions also eigenfunctions themselves?

## Prerequisites

• Some knowledge of 2-state systems (e.g. spin)
• An operational understanding of eigenstates/eigenfunctions

## Activity: Introduction

We usually start with lecture introducing differential operators - momentum and energy.

## Activity: Student Conversations

• Many students have trouble identifying eigenfunctions. They have to be reminded about the definition of an eigenstate/eigenfunction. Students also have difficulty recognizing eigenfunctions, especially if the eigenvalue is not 1. It helps to re-emphasize that they are checking to see if the eigenvalue equation holds for each function.
• Students also have difficulty expanding functions in terms of eigenfunctions. We build on the idea of expanding a function in terms of other functions in each paradigm (power series, Fourier series, complete sets of states, etc) and will continue to do so (spherical harmonics, Bessel functions, Legendre polynomials, Laguerre functions).

## Activity: Wrap-up

This is activity where the wrap-up is key: students will happily grind through the computations of this activity without reflecting on the existence of patterns. The wrap-up discussion should not only go over the answers so that groups can check if they've done the computations correctly, but should also, and more importantly, review how eigenfunctions can be identified, and highlight similarities/differences in the eigenfunctions for the momentum and energy operators. Some points to highlight:
• eigenvalues of the Hamiltonian should have dimensions of energy and eigenvalues of p should have dimension of momentum. Ask them to think about the dimension of $\hbar k$.
• degeneracy of energy eigenvalues
• connect to traveling waves -
• When addressing the momentum operator, emphasize that $A\sin(kx)$ is NOT an eigenfunction, but can be written as superposition of eigenfunctions.
• ask what you would actually measure for momentum for $\psi(x)=A\sin(kx)$: you will only measure $\hbar k$ and $-\hbar k$, but the average (expectation value) is zero.
• it can also be helpful to remind them of a spins example in bra-ket notation, in order to help them connect this to what they have already done.

## Extensions

Author Information
Janet Tate
Keywords
Learning Outcomes