## 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.

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?

Author Information
Janet Tate
Keywords
Learning Outcomes