format_list_numbered Sequence

Curvilinear Coordinate Sequence
The curvilinear coordinate sequence introduces cylindrical and spherical coordinates (including inconsistencies between physicists' and mathematicians' notational conventions) and the basis vectors adapted to these coordinate systems.

group Small Group Activity

30 min.

\(|\pm\rangle\) Forms an Orthonormal Basis
Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.

None

Diagonalization Part II
First complete the problem Diagonalization. In that notation:
  1. Find the matrix \(S\) whose columns are \(|\alpha\rangle\) and \(|\beta\rangle\). Show that \(S^{\dagger}=S^{-1}\) by calculating \(S^{\dagger}\) and multiplying it by \(S\). (Does the order of multiplication matter?)
  2. Calculate \(B=S^{-1} C S\). How is the matrix \(E\) related to \(B\) and \(C\)? The transformation that you have just done is an example of a “change of basis”, sometimes called a “similarity transformation.” When the result of a change of basis is a diagonal matrix, the process is called diagonalization.
  • Found in: Quantum Fundamentals course(s)

group Small Group Activity

10 min.

Using Tinker Toys to Represent Spin 1/2 Quantum Systems
Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases (\(x\), \(y\), and \(z\)). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.

group Small Group Activity

60 min.

Going from Spin States to Wavefunctions
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 Small Group Activity

10 min.

Changing Spin Bases with a Completeness Relation
Students work in small groups to use completeness relations to change the basis of quantum states.

accessibility_new Kinesthetic

10 min.

Curvilinear Basis Vectors
Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).

None

Completeness Relation Change of Basis

  1. Given the polar basis kets written as a superposition of Cartesian kets \begin{eqnarray*} \left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\ \left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle \end{eqnarray*}

    Find the following quantities: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]

  2. Given a vector written in the polar basis \[\left|{\vec{v}}\right\rangle = a\left|{\hat{s}}\right\rangle + b\left|{\hat{\phi}}\right\rangle \] where \(a\) and \(b\) are known. Find coefficients \(c\) and \(d\) such that \[\left|{\vec{v}}\right\rangle = c\left|{\hat{x}}\right\rangle + d\left|{\hat{y}}\right\rangle \] Do this by using the completeness relation: \[\left|{\hat{x}}\right\rangle \left\langle {\hat{x}}\right| + \left|{\hat{y}}\right\rangle \left\langle {\hat{y}}\right| = 1\]
  3. Using a completeness relation, change the basis of the spin-1/2 state \[\left|{\Psi}\right\rangle = g\left|{+}\right\rangle + h\left|{-}\right\rangle \] into the \(S_y\) basis. In otherwords, find \(j\) and \(k\) such that \[\left|{\Psi}\right\rangle = j\left|{+}\right\rangle _y + k\left|{-}\right\rangle _y\]
  • Found in: Completeness Relations sequence(s) Found in: Quantum Fundamentals course(s)

None

Frequency
Consider a two-state quantum system with a Hamiltonian \begin{equation} \hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix} \end{equation} Another physical observable \(M\) is described by the operator \begin{equation} \hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix} \end{equation} where \(c\) is real and positive. Let the initial state of the system be \(\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle \), where \(\left|{m_1}\right\rangle \) is the eigenstate corresponding to the larger of the two possible eigenvalues of \(\hat{M}\). What is the frequency of oscillation of the expectation value of \(M\)? This frequency is the Bohr frequency.
  • Found in: Quantum Fundamentals course(s)

assignment_ind Small White Board Question

5 min.

Representations of Vectors
Students each recall a representation of vectors that they have seen before and record it on an individual whiteboard. The instructor uses these responses to generate a whole class discussion that compares and contrasts the features of the representations. If appropriate for the class, the instructor introduces bra/ket notation as a new, but valuable representation.

face Lecture

30 min.

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

group Small Group Activity

30 min.

Quantum Measurement Play
The instructor and students do a skit where students represent quantum states that are “measured” by the instructor resulting in a state collapse.

None

Dimensional Analysis of Kets
  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 \(\left|{x}\right\rangle \).
  • Found in: Completeness Relations sequence(s)

keyboard Computational Activity

120 min.

Sinusoidal basis set
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.

None

Orthogonal Brief
Consider the three quantum states: \[\left\vert \psi_1\right\rangle = \frac{1}{\sqrt{3}}\left\vert +\right\rangle+ i\frac{\sqrt{2}}{\sqrt{3}} \left\vert -\right\rangle\] \[\left\vert \psi_2\right\rangle = \frac{1}{\sqrt{5}}\left\vert +\right\rangle- \frac{2}{\sqrt{5}} \left\vert -\right\rangle\] \[\left\vert \psi_3\right\rangle = \frac{1}{\sqrt{2}}\left\vert +\right\rangle+ i\frac{e^{\frac{i\pi}{4}}}{\sqrt{2}} \left\vert -\right\rangle\] For each of the \(\vert \psi_i\rangle\) above, find the normalized vector \(\vert \phi_i\rangle\) that is orthogonal to it.
  • Found in: Quantum Fundamentals course(s)

group Small Group Activity

30 min.

Murder Mystery Method

group Small Group Activity

30 min.

Which Way is North?
Students construct two different rectangular coordinate systems and corresponding vector bases, then compare computations done with each.

accessibility_new 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.

keyboard Computational Activity

120 min.

Position operator
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.

None

Measurement Probabilities
A beam of spin-\(\frac{1}{2}\) particles is prepared in the initial state \[ \left\vert \psi\right\rangle = \sqrt{\frac{2}{5}}\; |+\rangle_x - \sqrt{\frac{3}{5}}\; |-\rangle_x \](Note: this state is written in the \(S_x\) basis!)
  1. What are the possible results of a measurement of \(S_x\), with what probabilities?

  2. Repeat part a for measurements of \(S_z\).

  3. Suppose you start with a particle in the state given above, measure \(S_x\), and happen to get \(+\hbar /2\). You then take that same particle and measure \(S_z\). What are the possible results and with what probability would you measure each possible result?
  • Found in: Quantum Fundamentals course(s)