The content of the Central Forces paradigms has been divided into several units. Each unit is described for the students via an introductory handout. These handouts are accumulated below to help a course adopter understand the important features of each unit.
Unit: Classical Mechanics Orbits
In this unit, you will explore the classical mechanics of central forces, especially gravitational orbits like the earth going around the sun.
Motivating Questions
- What shapes can the orbits have?
- What are Kepler's laws and why are they true?
- What is an effective potential diagram and how can it be used to predict the shape of an orbit?
Key Activities/Problems
Unit Learning Outcomes
At the end of this unit, you should be able to:
- List the properties that define a central force system.
- Calculate a reduced mass for a two-body system and describe why it is important.
- Use the solution (algebraic or geometric) to a reduced mass system to describe the motion of the original system.
- Describe the role that conservation of energy and angular momentum play in a central force system. In particular, where do these properties appear in the solutions of the equations of motion?
- Use an effective potential diagram to predict the possible orbits in a central force system: which orbits are bound or unbound? which are closed or open? where will the turning points be?
Equation Sheet for This Unit
Unit: Quantum Mechanics on a Ring
In this unit, you will explore the quantum mechanics of a simple system: a particle confined to a one-dimensional ring.
Motivating Questions
- What are the energy eigenstates, i.e. eigenstates of the Hamiltonian?
- What physical properties of the energy eigenstates can be measured?
- What other states are possible and what are their physical properties?
- How do the states if this system and their physical properties depend on time?
Key Activities/Problems
Unit Learning Outcomes
At the end of this unit, you should be able to:
- Describe the energy eigenstates for the ring system algebraically and graphically.
- List the physical measurables for the system and give expressions for the corresponding operators in bra/ket, matrix, and position representations.
- Give the possible quantum numbers for the quantum ring system and describe any degeneracies.
- For a given state, use the inner product in bra/ket, matrix, and position representations, to find the probability of making any physically relevant measurement, including states with degeneracy.
- Use an expansion in energy eigenstates to find the time dependence of a given state.
Equation Sheet for This Unit
Unit: Partial Differential Equations
In this unit, you will explore the using the separation of variables procedure for solving partial differential equations (PDEs).
Motivating Questions
- How are Partial Differential Equations (PDEs) different from Ordinary Differential Equations (ODEs)?
- What new kinds of physics can we learn from solving partial differential equations?
- What can we learn about physics and geometry from the separation of varibles procedure?
Key Activities/Problems
Unit Learning Outcomes
At the end of this unit, you should be able to:
- Identify and classify several common partial differential equations.
- Solve simple partial differential equations through the separation of variables procedure.
- Identify the conditions where separation of variables is appropriate and useful.
Unit: Power Series Solutions of ODEs
In this unit, you will learn a common technique for solving linear, homogeneous ordinary differential equations that is used when the equation is more complicated than those with constant coefficients. You will start with some simplified pure math examples. Then you will apply the technique to several ordinary differential equations that come up in the solution of the (unperturbed) quantum hydrogen atom (e.g. Legendre's equation, LaGuerre's equation).
Motivating Questions
- When and how can you use a power series to solve an ordinary differential equation that you cannot solve in other ways?
- How do you find the coefficients of a power series solution
?
- When is an ordinary differential equation an eigenvalue equation and how does that affect the solutions?
- How do the boundary conditions that come from a physical setting affect the power series solutions?
Key Activities/Problems
Unit Learning Outcomes
At the end of this unit, for a linear, homogeneous ODE, you should be able to:
- Use the form of the ordinary differential equation to predict the form and number of power series solutions and their region of convergence.
- Write out the power series solutions for a given ODE in correct form, explaining the meaning of each of the terms.
- For examples, derive a recurrence relation from the ODE.
- For examples, use a recurrence relation to calculate the coefficients of a power series.
- In the case of eigenvalue equations, use the recurrence relation to find eigenvalues that will cause the series to terminate (i.e. have only a finite number of non-zero terms), resulting in polynomial solutions.
- Use a (truncated) power series to approximate the solution(s) to an ODE and discuss where the approximation is valid.
Unit: Angular Momentum in Quantum Mechanics
In this unit, you will derive (with help) and explore the energy eigenstates for a particle confined to a sphere (the rigid rotor problem). Then we will learn how to calculate probabilities and understand physicial quantities on the system.
Motivating Questions
- What is an expansion in Legendre polynomials and how do you calculate it?
- What are the quantum numbers for the rigid rotor (especially the energy)?
- What shapes do the probability densities for the energy eigenstates of the rigid rotor have?
- What is the algebraic form for the energy eigenstates of the rigid rotor?
Key Activities/Problems
Unit Learning Outcomes
At the end of this unit, you should be able to:
- Calculate the Legendre polynomial expansion for a function and describe where it is valid.
- Write the energy eigenstates for the rigid rotor in wave function form using a table of spherical harmonics.
- Describe how the values of the quantum numbers for the rigid rotor are related to each other and remember the limits on their ranges.
- For a state of the rigid rotor, given in bra/ket, matrix, or wave function notation, calculate the probabilities associated with energy, total angular momentum (squared), \(z\)-component of angular momentum, or position.
- For a state of the rigid rotor, given in bra/ket, matrix, or wave function notation, determine how it evolves with time.
- Relate the wave function and bra/ket notations for stationary states of the rigid rotor to their graphs.
Equation Sheet for This Unit
Unit: Quantum Mechanics of the (Unperturbed) Hydrogen Atom
In this unit, you will derive (with help) and explore the energy eigenstates for the hydrogen atom. Then we will explore how to do calculations and finally what these real world wavefunctions look like and how they represent what we have seen in chemistry courses.
Motivating Questions
- What are the quantum numbers for the hydrogen atom (especially the energy)?
- What shapes do the probability densities for the energy eigenstates of the hydrogen atom have?
- What is the algebraic form for the energy eigenstates of the hydrogen atom?
Key Activities/Problems
Unit Learning Outcomes
At the end of this unit, you should be able to:
- Write the energy eigenstates for the hydrogen atom in wave function form using a table of radial solutions and a table of spherical harmonics.
- Describe how the values of the quantum numbers for the hydrogen atom are related to each other and remember the limits on their ranges.
- For a state of the hydrogen atom, given in bra/ket, matrix, or wave function notation, calculate the probabilities associated with energy, total angular momentum (squared), \(z\)-component of angular momentum, or position.
- For a state of the hydrogen atom, given in bra/ket, matrix, or wave function notation, determine how it evolves with time.
- Relate the wave function and bra/ket notations for stationary states of the hydrogen atom to their graphs.
Equation Sheets for This Unit