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Quantum Ring Sequence
Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.

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Completeness Relations
This set of activities explores completeness relations in quantum mechanics and how they eventually support understanding of wavefunctions and what it means to be in the position or momentum representatations.

face Lecture

5 min.

Unit Learning Outcomes: Quantum Mechanics on a Ring
  • Found in: Central Forces course(s)

assignment_ind Small White Board Question

30 min.

Magnetic Moment & Stern-Gerlach Experiments
Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.

None

Spin Matrix
The Pauli spin matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) are defined by: \[\sigma_x= \begin{pmatrix} 0&1\\ 1&0\\ \end{pmatrix} \hspace{2em} \sigma_y= \begin{pmatrix} 0&-i\\ i&0\\ \end{pmatrix} \hspace{2em} \sigma_z= \begin{pmatrix} 1&0\\ 0&-1\\ \end{pmatrix} \] These matrices are related to angular momentum in quantum mechanics.
  1. By drawing pictures, convince yourself that the arbitrary unit vector \(\hat n\) can be written as: \[\hat n=\sin\theta\cos\phi\, \hat x +\sin\theta\sin\phi\,\hat y+\cos\theta\,\hat z\] where \(\theta\) and \(\phi\) are the parameters used to describe spherical coordinates.
  2. Find the entries of the matrix \(\hat n\cdot\vec \sigma\) where the “matrix-valued-vector” \(\vec \sigma\) is given in terms of the Pauli spin matrices by \[\vec\sigma=\sigma_x\, \hat x + \sigma_y\, \hat y+\sigma_z\, \hat z\] and \(\hat n\) is given in part (a) above.
  • Found in: Quantum Fundamentals course(s)

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Arms Sequence for Complex Numbers and Quantum States

“Arms” is an engaging representation of complex numbers. Students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram).

The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence.

These activities can be combined and sequenced in many different ways; see the Instructor's Guides for how to introduce the Arms representation the first time you use it.

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Fourier Transforms and Wave Packets
This is a unit that introduces the Fourier transform and its properties and then applies the Fourier transform to free particle wave packets in non-relativistic quantum mechanics. The activities and homework are listed here. Appropriate text materials for mini-lectures can be found in the chapter Fourier Transforms and Wave Packets in the free online textbook The Geometry of Mathematical Methods.

assignment_ind Small White Board Question

5 min.

Angular Momentum SWBQ
  • Found in: Central Forces course(s)

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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.

Sequential Stern-Gerlach Experiments

group Small Group Activity

30 min.

Expectation Values for a Particle on a Ring
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.

face Lecture

5 min.

Central Forces Introduction Lecture Notes
  • Found in: Central Forces course(s)

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.

assignment_ind Small White Board Question

5 min.

Normalization of the Gaussian for Wavefunctions
Students find a wavefunction that corresponds to a Gaussian probability density.
  • Found in: Periodic Systems course(s) Found in: Fourier Transforms and Wave Packets sequence(s)

group Small Group Activity

30 min.

Superposition States for a Particle on a Ring
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.

None

Quantum concentration
Consider one particle confined to a cube of side \(L\); the concentration in effect is \(n=L^{-3}\). Find the kinetic energy of the particle when in the ground state. There will be a value of the concentration for which this zero-point quantum kinetic energy is equal to the temperature \(kT\). (At this concentration the occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a higher occupancy than any other orbital.) Show that the concentration \(n_0\) thus defined is equal to the quantum concentration \(n_Q\) defined by (63): \begin{equation} n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32} \end{equation} within a factor of the order of unity.
  • Found in: Thermal and Statistical Physics course(s)

None

Eigenvectors of Pauli Matrices
  1. Find the eigenvalues and normalized eigenvectors of the Pauli matrices \(\sigma_x\), \(\sigma_y\), and \(\sigma_z\) (see the Spins Reference Sheet posted on the course website).
  • Found in: Quantum Fundamentals course(s)

group 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.

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)

accessibility_new Kinesthetic

10 min.

Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).