Activity: Compare & Contrast Kets & Wavefunctions

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
• This activity is used in the following sequences
What students learn
• To translate entities and calculations from Dirac bra-ket notation to wavefunction notation.

Quantum States

The wavefunction is related to writing the state in the position basis (or the position “representation”) by inserting a completeness relation for the position basis.:

\begin{align*} \left|{\psi}\right\rangle &= (1) \left|{\psi}\right\rangle \\ &=\left(\int_{\tiny\mbox{all space}} \left|{x}\right\rangle \left\langle {x}\right|dx\right)\left|{\psi}\right\rangle \\ &= \int_{\tiny\mbox{all space}} \left|{x}\right\rangle \left\langle {x}\middle|{\psi}\right\rangle dx \\ &= \int_{\tiny\mbox{all space}} \underbrace{\left\langle {x}\middle|{\psi}\right\rangle }_{\mbox{wavefunction}}\left|{x}\right\rangle dx \\ &= \int_{\tiny\mbox{all space}} \underbrace{\psi(x)}_{\mbox{wavefunction}} \left|{x}\right\rangle dx \\ \end{align*}

Bra-ket Notation: $\left|{\psi}\right\rangle \doteq \left\langle {x}\middle|{\psi}\right\rangle$

Wavefunction Notation: $\left|{\psi}\right\rangle \doteq \psi(x)$

The dot equals sign indicates that the right hand side is an expression of the state in a specific but unwritten basis.

The Hermitian Adjoint of a Quantum State:

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

Bra-ket Notation: $\left\langle {\psi}\right| \doteq \left\langle {\psi}\middle|{x}\right\rangle$

Wavefunction Notation: $\left\langle {\psi}\right| \doteq \psi^*(x)$

Calculating a Probability Amplitude

For example, for an energy measurement: Bra-ket Notation: \begin{align*} c_{n} = \left\langle {E_n}\middle|{\psi}\right\rangle \end{align*}

Wavefunction Notation:

Go to wavefunction notation by inserting a completeness relation for the position basis.

\begin{align*} c_{n} &= \left\langle {E_n}\right|1\left|{\psi}\right\rangle \\ &= \left\langle {E_n}\right| \left( \int_{\tiny\mbox{all space}} \left|{x}\right\rangle \left\langle {x}\right| dx \right)\left|{\psi}\right\rangle \\ &= \int_{\tiny\mbox{all space}} \left\langle {E_n}\middle|{x}\right\rangle \left\langle {x}\middle|{\psi}\right\rangle dx\\ &= \int_{\tiny\mbox{all space}} E_n^*(x) \, \psi(x)dx\\ \end{align*}

Calculating a Probability

Bra-ket Notation: \begin{align*} \mathcal{P}(E_n) &= | c_n |^2 \\ &= |\left\langle {E_n}\middle|{\psi}\right\rangle |^2 \\ \end{align*}

Wavefunction Notation:

Using the result for the probability amplitude in wavefunction notation:

\begin{align*} \mathcal{P}(E_n) &= | c_n |^2 \\ &= \left| \int_{\tiny\mbox{all space}} E_n^*(x) \, \psi(x) dx \right|^2\\ \end{align*} Notice the norm square is outside of the integral!

Normalizing a State

Bra-ket Notation: \begin{align*} 1 &= \left\langle {\psi}\middle|{\psi}\right\rangle \\ \end{align*}

Wavefunction Notation:

Go to wavefunction notation by inserting a completeness relation for the position basis.

\begin{align*} 1 &= \left\langle {\psi}\right|1\left|{\psi}\right\rangle \\ &= \left\langle {\psi}\right|\left( \int_{\tiny\mbox{all space}} \left|{x}\right\rangle \left\langle {x}\right| dx \right)\left|{\psi}\right\rangle \\ &= \int_{\tiny\mbox{all space}} \left\langle {\psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\psi}\right\rangle dx \\ &= \int_{\tiny\mbox{all space}} \psi^*(x)\, \psi(x) dx \\ &= \int_{\tiny\mbox{all space}} |\psi(x)|^2 dx \\ \end{align*} Notice norm squared inside the integral!

Finding the Probability for a Range of Values

Bra-ket Notation: \begin{align*} \mathcal{P}(a \le n \le b) &= \sum_{n=a}^b| c_n |^2 \\ \end{align*}

Wavefunction Notation:

Reinterpret $|\psi(x)|^2$ as a probability density: \begin{align*} \mathcal{P}(a \le x \le b) &= \int_{a}^b |\psi(x)|^2 dx \\ \end{align*} Notice norm squared inside the integral!

• group Going from Spin States to Wavefunctions

group Small Group Activity

60 min.

Going from Spin States to Wavefunctions
Quantum Fundamentals 2023 (2 years)

Arms Sequence for Complex Numbers and Quantum States

Completeness Relations

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.
• 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 Time Dependence for a Quantum Particle on a Ring Part 1

group Small Group Activity

30 min.

Time Dependence for a Quantum Particle on a Ring Part 1
Theoretical Mechanics (6 years)

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 Expectation Values for a Particle on a Ring

group Small Group Activity

30 min.

Expectation Values for a Particle on a Ring
Central Forces 2023 (3 years)

Quantum Ring Sequence

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
• group Hydrogen Probabilities in Matrix Notation

group Small Group Activity

30 min.

Hydrogen Probabilities in Matrix Notation
Central Forces 2023 (2 years)
• keyboard Sinusoidal basis set

keyboard Computational Activity

120 min.

Sinusoidal basis set
Computational Physics Lab II 2023 (2 years)

Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.
• assignment Dimensional Analysis of Kets

assignment Homework

Dimensional Analysis of Kets
dirac notation dimensions probability completeness relations

Completeness Relations

1. $\left\langle {\Psi}\middle|{\Psi}\right\rangle =1$ Identify and discuss the dimensions of $\left|{\Psi}\right\rangle$.
2. For a spin $\frac{1}{2}$ system, $\left\langle {\Psi}\middle|{+}\right\rangle \left\langle {+}\middle|{\Psi}\right\rangle + \left\langle {\Psi}\middle|{-}\right\rangle \left\langle {-}\middle|{\Psi}\right\rangle =1$. Identify and discuss the dimensions of $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$.
3. In the position basis $\int \left\langle {\Psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\Psi}\right\rangle dx = 1$. Identify and discuss the dimesions of $\left|{x}\right\rangle$.
• group $|\pm\rangle$ Forms an Orthonormal Basis

group Small Group Activity

30 min.

$|\pm\rangle$ Forms an Orthonormal Basis
Quantum Fundamentals 2023 (3 years)

Completeness Relations

Student explore the properties of an orthonormal basis using the Cartesian and $S_z$ bases as examples.
• assignment Wavefunctions

assignment Homework

Wavefunctions
Quantum Fundamentals 2023 (3 years)

Consider the following wave functions (over all space - not the infinite square well!):

$\psi_a(x) = A e^{-x^2/3}$

$\psi_b(x) = B \frac{1}{x^2+2}$

$\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)$ (“sech” is the hyperbolic secant function.)

In each case:

1. normalize the wave function,
2. plot the wave function using Mathematica or other computer plotting tool (be sure to include the code you used and label your plots/axes appropriately),
3. find the probability that the particle is measured to be in the range $0<x<1$.

• 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.

Learning Outcomes