Small Group Activity: Operators & Functions

Quantum Fundamentals 2021
Students are asked to:
  • 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
    • 2289/spoperatorfunchand.tex
    • 2289/spoperatorfunchand.pdf

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?

Operators and Functions: Instructor's Guide

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