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Activities

Small Group Activity

30 min.

##### Representations for Finding Components
In this small group activity, students draw components of a vector in Cartesian and polar bases. Students then write the components of the vector in these bases as both dot products with unit vectors and as bra/kets with basis bras.
• Found in: Quantum Fundamentals, Static Fields, None course(s)

Problem

##### Derivative of Fermi-Dirac function
Derivative of Fermi-Dirac function Show that the magnitude of the slope of the Fermi-Direc function $f$ evaluated at the Fermi level $\varepsilon =\mu$ is inversely proportional to its temperature. This means that at lower temperatures the Fermi-Dirac function becomes dramatically steeper.
• Found in: Thermal and Statistical Physics course(s)

Small Group Activity

30 min.

##### Finding Matrix Elements
In this small group activity, students multiply a general 3x3 matrix with standard basis row/column vectors to pick out individual matrix elements. Students generate the expressions for the matrix elements in bra/ket notation.
• Found in: Quantum Fundamentals course(s)

Small Group Activity

30 min.

##### Working with Representations on the Ring
This activity acts as a reintroduction to doing quantum calculations while also introducing the matrix representation on the ring, allowing students to discover how to index and form a column vector representing the given quantum state. In addition, this activity introduces degenerate measurements on the quantum ring and examines the state after measuring both degenerate and non-degenerate eigenvalues for the state.
• Found in: Central Forces course(s)

Problem

5 min.

##### Dirac Practice
For this problem, use the vectors $|a\rangle = 4 |1\rangle - 3 |2\rangle$ and $|b\rangle = -i |1\rangle + |2\rangle$.
1. Find $\langle a | b \rangle$ and $\langle b | a \rangle$. Discuss how these two inner products are related to each other.
2. For $\hat{Q}\doteq \begin{pmatrix} 2 & i \\ -i & -2 \end{pmatrix}$, calculate $\langle1|\hat{Q}|2\rangle$, $\langle2|\hat{Q}|1\rangle$, $\langle a|\hat{Q}| b \rangle$ and $\langle b|\hat{Q}|a \rangle$.
3. What kind of mathematical object is $|a\rangle\langle b|$? What is the result if you multiply a ket (for example, $| a\rangle$ or $|1\rangle$) by this expression? What if you multiply this expression by a bra?
• Found in: Quantum Fundamentals course(s)

Problem

30 min.

##### Completeness Relation Change of Basis
Students practice using inner products to find the components of the cartesian basis vectors in the polar basis and vice versa. Then, students use a completeness relation to change bases or cartesian/polar bases and for different spin bases.
• Found in: Quantum Fundamentals course(s) Found in: Completeness Relations sequence(s)

Problem

5 min.

##### Dimensional Analysis of Kets
Students consider the dimensions of spin-state kets and position-basis kets.
• Found in: Completeness Relations sequence(s)

Problem

10 min.

##### Coefficients in Wavefunction notation
Students use the completeness relation for the position basis to re-express expressions in bra/ket notation in wavefunction notation.

Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring
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.
• Found in: Theoretical Mechanics course(s) Found in: Quantum Ring Sequence sequence(s)

Kinesthetic

10 min.

##### Spin 1/2 with Arms
Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.
• Found in: Quantum Fundamentals course(s) Found in: Arms Sequence for Complex Numbers and Quantum States sequence(s)

Small Group Activity

60 min.

##### Multiple Representations of a Quantum State
Students re-represent a state given in Dirac notation in matrix notation
• Found in: Quantum Fundamentals course(s)

Small Group Activity

5 min.

##### Fourier Transform of the Delta Function
Students calculate the Fourier transform of the Dirac delta function.
• Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)