## Activity: Operators & Functions

Quantum Fundamentals 2023 (3 years)
• 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

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

• assignment Quantum Particle in a 2-D Box

assignment Homework

##### Quantum Particle in a 2-D Box
Central Forces 2023 (3 years) 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. $$\psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y)$$ 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.
• keyboard Kinetic energy

keyboard Computational Activity

120 min.

##### Kinetic energy
Computational Physics Lab II 2022

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
• keyboard Position operator

keyboard Computational Activity

120 min.

##### Position operator
Computational Physics Lab II 2022

Students find matrix elements of the position operator $\hat x$ in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.
• 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 2023 (3 years)

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.
• assignment Phase 2

assignment Homework

##### Phase 2
quantum mechanics relative phase overall phase measurement probability Quantum Fundamentals 2023 (3 years) Consider the three quantum states: $\left\vert \psi_1\right\rangle = \frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle$ $\left\vert \psi_2\right\rangle = \frac{4}{5}\left\vert +\right\rangle- i\frac{3}{5} \left\vert -\right\rangle$ $\left\vert \psi_3\right\rangle = -\frac{4}{5}\left\vert +\right\rangle+ i\frac{3}{5} \left\vert -\right\rangle$
1. For each of the $\left|{\psi_i}\right\rangle$ above, calculate the probabilities of spin component measurements along the $x$, $y$, and $z$-axes.
2. Look For a Pattern (and Generalize): Use your results from $(a)$ to comment on the importance of the overall phase and of the relative phases of the quantum state vector.
• group Wavefunctions on a Quantum Ring

group Small Group Activity

30 min.

##### Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)
• group Fourier Transform of a Derivative

group Small Group Activity

10 min.

##### Fourier Transform of a Derivative
Periodic Systems 2022

Fourier Transforms and Wave Packets

• group Fourier Transform of a Gaussian

group Small Group Activity

10 min.

##### Fourier Transform of a Gaussian
Periodic Systems 2022

Fourier Transforms and Wave Packets

• 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 2023 (3 years)

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.

Author Information
Janet Tate
Keywords
Learning Outcomes