In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. \[\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix} \]

Students use completeness relations to write a matrix element of a spin component in a different basis.

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• How to translate a complicated wavefunction into eigenstates.
• Refresher on how to find expectation values and probabilities in a region.
• How to use the symmetry of the wavefunction to tell you something about measurements.

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.

In this small group activity, students solve for the time dependence of two quantum spin 1/2 particles under the influence of a Hamiltonian. Students determine, given a Hamiltonian, which states are stationary and under what circumstances measurement probabilities do change with time.

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.

Students work in small groups to use completeness relations to change the basis of quantum states.

• How to form a state as a column vector in matrix representation.
• How to do probability calculations on all three representations used for quantum systems in PH426.
• How to find probabilities for and the resultant state after measuring degenerate eigenvalues.

Students use their arms to act out two spin-1/2 quantum states and their inner product.

Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).

Students are asked to find eigenvalues, probabilities, and expectation values for \(H\), \(L^2\), and \(L_z\) for a superposition of \(\vert n \ell m \rangle\) states. This can be done on small whiteboards or with the students working in groups on large whiteboards.

Students then work together in small groups to find the matrices that correspond to \(H\), \(L^2\), and \(L_z\) and to redo \(\langle E\rangle\) in matrix notation.

Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.

Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.

Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]

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.

• to perform a magnetic vector potential calculation using the superposition principle;
• to decide which form of the superposition principle to use, depending on the dimensions of the current density;
• how to find current from total charge \(Q\), period \(T\), and the geometry of the problem, radius \(R\);
• to write the distance formula \(\vec{r}-\vec{r'}\) in both the numerator and denominator of the superposition principle in an appropriate mix of cylindrical coordinates and rectangular basis vectors;

• Vectors and their magnitudes are geometric quantities, independent of coordinates and choice of basis

• Recall the relationship between the sign of the dot product and the orientation of the vectors.
• Use graphical methods to estimate the value of a vector line integral.
• Lay the groundwork for thinking about conservative and non-conservative vector fields.

• Practice evaluating line integrals;
• Practice choosing appropriate coordinates and basis vectors;
• Introduction to the geometry behind conservative vector fields.

In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.

Students consider the dimensions of spin-state kets and position-basis kets.

• Work as area under curve in a \(pV\) plot
• Heat transfer as area under a curve in a \(TS\) plot
• Reminder that internal energy is a state function
• Reminder of First Law

In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.

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.

Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.