- Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency
*assignment*Effective Potentials: Graphical Version*assignment*Homework##### Effective Potentials: Graphical Version

Central Forces 2023 (3 years)Consider a mass \(\mu\) in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is \(\ell\ne 0\) for a given fixed value of \(\ell\).

- Give the definition of a central force system and briefly explain why this situation qualifies.
- Make a sketch of the graph of the effective potential for this situation.
- How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
- BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.

*group*Hydrogen emission*group*Small Group Activity30 min.

##### Hydrogen emission

Contemporary Challenges 2021 (5 years) In this activity students work out energy level transitions in hydrogen that lead to visible light.*assignment*Gibbs sum for a two level system*assignment*Homework##### Gibbs sum for a two level system

Gibbs sum Microstate Thermal average energy Thermal and Statistical Physics 2020Consider a system that may be unoccupied with energy zero, or occupied by one particle in either of two states, one of energy zero and one of energy \(\varepsilon\). Find the Gibbs sum for this system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\). Note that the system can hold a maximum of one particle.

Solve for the thermal average occupancy of the system in terms of \(\lambda\).

Show that the thermal average occupancy of the state at energy \(\varepsilon\) is \begin{align} \langle N(\varepsilon)\rangle = \frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}} \end{align}

Find an expression for the thermal average energy of the system.

Allow the possibility that the orbitals at \(0\) and at \(\varepsilon\) may each be occupied each by one particle at the same time; Show that \begin{align} \mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} + \lambda^2 e^{-\frac{\varepsilon}{kT}} \\ &= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right) \end{align} Because \(\mathcal{Z}\) can be factored as shown, we have in effect two independent systems.

*group*Wavefunctions on a Quantum Ring*group*Representations of the Infinite Square Well*group*Small Group Activity120 min.

##### Representations of the Infinite Square Well

Quantum Fundamentals 2023 (3 years)*accessibility_new*Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)*accessibility_new*Kinesthetic30 min.

##### Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

Quantum Fundamentals 2021 Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.*assignment*Potential vs. Potential Energy*assignment*Homework##### Potential vs. Potential Energy

Static Fields 2023 (6 years)In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like \(\frac{1}{r}\), so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

- Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
- Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
- Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

*group*Hydrogen Probabilities in Matrix Notation*group*Small Group Activity30 min.

##### Hydrogen Probabilities in Matrix Notation

Central Forces 2023 (2 years)*accessibility_new*Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems*accessibility_new*Kinesthetic10 min.

##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

Quantum Fundamentals 2023 (2 years)quantum states complex numbers arms Bloch sphere relative phase overall phase

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).*group*Using Tinker Toys to Represent Spin 1/2 Quantum Systems*group*Small Group Activity10 min.

##### Using Tinker Toys to Represent Spin 1/2 Quantum Systems

spin 1/2 eigenstates quantum states

Arms Sequence for Complex Numbers and Quantum States

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.- Quantum Fundamentals 2023 (3 years) 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.