Activities
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.
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.
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.
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.
For this problem, use the vectors \(|a\rangle = 4 |1\rangle - 3 |2\rangle\) and \(|b\rangle = -i |1\rangle + |2\rangle\).
- Find \(\langle a | b \rangle\) and \(\langle b | a \rangle\). Discuss how these two inner products are related to each other.
- 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 \(\langle b|\hat{Q}|a \rangle\).
- 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?
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.
Students consider the dimensions of spin-state kets and position-basis kets.
Students use the completeness relation for the position basis to re-express expressions in bra/ket notation in wavefunction notation.
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.
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.
Students re-represent a state given in Dirac notation in matrix notation
Students calculate the Fourier transform of the Dirac delta function.