## Frequency

• Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency
• assignment Effective Potentials: Graphical Version

assignment Homework

##### Effective Potentials: Graphical Version
Central Forces 2023 (2 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 $\ell$.

1. Give the definition of a central force system and briefly explain why this situation qualifies.
2. Make a sketch of the graph of the effective potential for this situation.
3. 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.)
4. 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 Activity

30 min.

##### Hydrogen emission
Contemporary Challenges 2022 (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 2020
1. Consider 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.

2. Solve for the thermal average occupancy of the system in terms of $\lambda$.

3. 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}

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

5. 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 Small Group Activity

30 min.

##### Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)
• group Representations of the Infinite Square Well

group Small Group Activity

120 min.

##### Representations of the Infinite Square Well
Quantum Fundamentals 2022 (3 years)

Warm-Up

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

accessibility_new Kinesthetic

30 min.

##### Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
Quantum Fundamentals 2021

Arms Sequence for Complex Numbers and Quantum States

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.
• group Hydrogen Probabilities in Matrix Notation

group Small Group Activity

30 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 Kinesthetic

10 min.

##### Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems
Quantum Fundamentals 2022 (2 years)

Arms Sequence for Complex Numbers and Quantum States

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 Activity

10 min.

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

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.
• keyboard Kinetic energy

keyboard Computational Activity

120 min.

##### Kinetic energy
Computational Physics Lab II 2022

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
• Quantum Fundamentals 2022 (2 years) Consider a two-state quantum system with a Hamiltonian $$\hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix}$$ Another physical observable $M$ is described by the operator $$\hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix}$$ 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.