Supporting physics teaching with research-based resources

1. Apply conservation of energy and thermal physics principles to real-world thermal systems, such as home heating and climate change.
2. Apply knowledge of work and Newton's laws to calculate basic dynamics and energy consumption of common transportation systems (cars, bicycles etc.)
3. Qualitatively explain how electricity is generated in various types of power plants and the --life cycle‖ of electricity from production through transmission to consumption, and calculate power consumption for various common circuits.
4. Use algebra to solve simple equations.
5. Appreciate that physics is relevant to the real world and is a useful tool for solving problems.
6. Develop the inclination and ability to apply problem solving techniques to simplify "real world" problems in terms of simple physics concepts and to compute or estimate solutions.
7. Recognize that scientific conclusions - whether from an outside source or from your own calculations - may be incorrect, and develop the ability to check these conclusions with simple calculations, 3rd party information, and/or common sense.

Week 1. Introduction, Course Outline, discussion of techniques, Measurement, Units, Data Analysis, Experimental Error

A student should be able to:

• appreciate why active engagement techniques are being used in this class
• access the course website to find more information
• summarize the expectations for course content and student workload
• apply their knowledge of measurement error to determine what an appropriate number of sig. figures is for a particular measurement device
• describe in simple terms the general nature of a scientific theory and its relationship to the experimental data
• explain the nature of a scientific model
• use and convert between various units

Week 2. Intro to Conservation Laws, Conservation of Energy, Dynamic Equilibrium

A student should be able to:

• identify correct system boundaries needed to apply conservation laws
• explain how energy can remain at a constant level even when there is a flux of energy into or out of the system
• give examples of a physical system in dynamic equilibrium
• calculate kinetic and potential energies in mechanical systems
• recognize assumptions needed to apply conservation of mechanical energy
• explain the effect of friction and air drag on the energy of a mechanical system

Week 13. Electricity Generation and Distribution, and Environmental Impacts

A student should be able to:

• compare the cost of electricity and GHG production for various forms of power generation systems (hydro, wind, coal, nuclear, solar)
• discuss the advantages and disadvantages of these power generation systems
• calculate power losses in electricity transmission lines and explain why we use high voltage transmission lines
• estimate the energy available in wind, hydro and fossil fuels.
• discuss the energy transformations for various forms of power generation systems (hydro, wind, coal, solar)

A. Consensus Themes (not SLOs)

1. Students should come away with a positive attitude about experimental physics.
2. Students should understand that physics is an experimental science.
3. Students should have a set-like vs a point-like understanding of measurements and uncertainty. (understand mean, stdev, distributions)
4. Communication of scientific process and result (especially using graphs and oral argumentation) is important.
5. Learning how to write lab reports and notebooks is not a goal for the course.
6. Experimental design is not a goal for the course, but small design components (or experimental choices) could be added in the later stages of the course.
7. Students should not be allowed to choose their own experiments and all students in the room should be doing the same experiments.
8. Any implementation should recognize the limitations of the skills/time of the TAs.
9. The new course should be sustainable even when many different faculty members rotate though the course.
10. This course must remain an “experimental experience” to satisfy ABET accreditation.
11. The physics topics covered in the lab should include material from both PHYS 1110 and PHYS1120 (the introductory physics sequence).

B. Consensus Learning Goals and Assessment Instruments

1. Students’ epistemology (i.e., theory of the nature of physics knowledge) of experimental physics should align with the expert view
   1. Alternative Definition: Student’s beliefs about the nature of experimental physics should align with expert physicists. This includes understanding that physics is an experimental science, what makes for a valid measurement, and how knowledge is gained through experiments.
   2. Assessment: Colorado Learning Attitudes about Science Survey for Experimental Physics (E-CLASS) epistemology items
2. Students should have a positive attitude about the course
   • Assessments: Student course evaluations (FCQ’s) with additional tailored questions
3. Students should have a positive attitude about experimental physics
   • Assessments: E-CLASS affect items + new questions about exp. physics
4. Students should be able to make a presentation quality graph showing a model and data
   • Assessments: course artifacts
5. Students should demonstrate a set-like reasoning when evaluating measurements
   • Alternative Definition: Students should understand that a measurement has an associated uncertainty and is not the “true” value. They should understand that repeated measurements form a distribution with a mean and a standard deviation.

1. Assessment: Physics Measurement Questionnaire (validated assessment developed at Cape Town)
2. Possible sub-goals for this SLO:
   1. Low-level of achievement
      1. Compute mean
      2. Compute standard deviation
      3. Understand that there is inherent distribution when measuring
      4. Count significant figures
      5. Report results properly (mean +- sig/sqrt(N) with proper significant figures
      6. Know when there is discrepancy between two results
   2. Medium-level of achievement
      1. Understand what sigma means
      2. Understand the difference between Sigma vs. Sigma/sqrt(N)
      3. Know how to treat outliers and when to remove data
   3. High level of achievement
      1. Propagate error
      2. Understand the difference between systematic vs. statistical uncertainty
      3. Know how to reduce systematic uncertainty – choosing the right measurement to improve, proposing causes

C. Additional guiding principles (things that are not our expectations of students)

1. Students will not write full formal lab reports.
2. Students will not choose which experiments they do each week.
3. Students will not be required to learn the details of coding in Mathematica, Matlab, etc.
4. Students will not be required to learn to derive differential propagation of errors formalism.
5. Students will be required to put in effort appropriate for a one-credit course, which includes no more than three hours outside of scheduled class time.

From lab #1 (“skills”)

After completing this lab, you should be able to:

• Use OneNote on your tablet computer as a digital lab notebook.
• Embed an Excel spreadsheet into your OneNote notebook.
• Turn in your lab notebook to Canvas at the end of the lab for credit.
• Recognize some of the lab equipment to be used in PHYS 1140.

From Lab #2 (“skills”)

After completing this week’s lab, you should be able to:

• Create a plot using Excel that includes labels, a trendline, and a caption.

From Lab #3 (“skills”)

After completing this week’s lab, you should be able to:

• Describe how the number of measurements affects the values calculated for standard deviation (σx) and standard deviation of the mean (σxbar)
• Explain the difference between (σx) and (σxbar) and how to use each when reporting measurements.
• Decide how well two data sets agree with each other using x and σx from each set.

From lab #4 (Mechanics)

This week’s lab reinforces the learning goals of statistics of Lab 3. In addition, after completing this lab, you should be able to:

• Decide whether to use the standard deviation, σx, or the standard error of the mean, σxbar, when comparing measurements.
• Decide if two measurements likely agree with one another.
• Report a measurement using the proper number of significant figures and value for the uncertainty.

From lab #5 (Mechanics)

This week’s lab reinforces the learning goals of Labs 3 and 4. In particular, we are emphasizing:

• Understanding what σx means for the probability of a single measurement of x.
• Deciding when to use σx versus when to use σxbar.

From lab #6 (Mechanics)

After completing this lab, you should be able to:

• Use the uncertainty in a measured quantity to determine the uncertainty in a calculated quantity.
• Propagate uncertainty in two different ways.

From lab #7 (E&M)

After completing this lab, you should be able to:

• Identify and address a systematic effect.
• Modify the physical model to take a systematic effect into account.

From lab #8 (E&M)

After completing this lab, you should be able to:

• Explain the operating principles of a capacitive sensor.
• Use the results of a linear fit to make predictions.

From lab #9 (E&M)

After completing this week’s lab, you should be able to:

• Calibrate an LED/photodetector system to produce any specific color

From lab #10 (Optics)

After completing this lab, you should be able to:

• Compare measurements to determine the probability of agreement
• Make decisions about whether two sets of data represent the same underlying distribution

From lab #11 (Optics)

After completing this lab, you should be able to:

• Make a plot showing multiple data series together
• Linearize data to perform a linear fit and extract a parameter’s value

From lab #12 (Optics)

After completing this week’s lab, you should be able to:

• Determine the best colors and filters for transmitting and detecting single-frequency signals through an optical fiber.
• Transmit and demultiplex multiple audio signals through an optical fiber.

• list the basic postulates of relativity, and be able to describe some of the basic implications of these that go against our usual intuition (and explain how experimental evidence supports these)
• analyze simple dynamical processes using relativistic dynamics. This will include:
  • relating physical quantities measured by observers moving at some relative velocity and knowing which quantities observers at relative velocities will agree upon.
  • knowing the limits of applicability of elementary formulae from mechanics and know the more general formulae that replace these in situations where velocities are not a negligible fraction of the speed of light
• describe and predict basic behavior of light and electrons in terms of both classical mechanics and quantum mechanics descriptions, and be able to specify differences
• argue how the assertions of quantum mechanics are inferred from the experimental evidence
• calculate the characteristics of quantum states and probabilities for outcomes of measurements for a few very mathematically simple quantum systems, including ones that student has not seen before. This requires understanding the basic mathematical framework well enough to apply it to such novel systems.
• give qualitative predictions and explanations of the behavior of simple quantum systems, such as the distribution of electrons in atoms and the spectrum of light emitted and absorbed by atoms
• argue that physics goes beyond a collection of empirical laws, and involves a deeper conceptual framework that is inferred from experiment but is not at all obvious from our everyday experience
• better understand popular science articles on current research in physics and answer questions about modern physics from curious friends and relatives
• see value in achieving a deeper understanding of quantum mechanics and learning more about modern physics

Newton’s Laws and Relativity

• explain what is meant by “relativity”
• explain what is meant by a “frame of reference” and an “inertial frame”
• give examples of how relativity manifests itself in ordinary situations
• show that Newton’s Laws obey the principle of relativity
• argue why the equivalence of physical laws in different frames implies that it is impossible to set up an experiment to measure an absolute velocity

Puzzles from Electromagnetism

• show how the speed of light follows directly from Maxwell’s equations
• give simple examples from electricity and magnetism to show that either the principle of relativity or some basic notions of distance, time, and velocity must be abandoned

Einstein’s Resolution

• argue how the experimental evidence implies that the velocity of light is always independent of the of the motion of the source or of the observer
• state Einstein’s postulates of special relativity
• explain how an given observer can set up a coordinate system for making measurements of time and position
• be able to show how Einstein's postulates imply that observers at large relative velocities will not agree on distances, time intervals or whether two events are simultaneous
• describe qualitatively the meaning of length contraction and time dilation
• correctly calculate the lengths and times differences that an observer will measure, properly accounting for length contraction and/or time dilation.
• use Lorentz transformation formulae to relate the measurements of observers moving at relative velocities
• use the velocity transformation formula to calculate the observed velocity of an object in a new frame given the velocity in the old frame and the relative velocity of the two frames
• calculate the Doppler shift of light frequencies for an observer moving relative to the source and discuss the importance of the Doppler effect in astrophysics

Relativistic Invariants

• describe the meaning of spacelike separation, timelike separation, proper length, and proper time know how to calculate the proper length/time between two events and the time elapsed on the clock of an observer on some general trajectory
• represent graphically simple scenarios on a spacetime diagram
• use spacetime diagrams to analyze simple processes involving relativistic velocities and resolve apparent paradoxes (e.g., ladder passing through barn)
• Relativistic Energy and Momentum
• argue why classical formulae for momentum and energy must be modified
• know the relativistic formulae for energy and momentum
• analyze high-energy particle decay processes and scattering processes using energy and momentum conservation
• give evidence for and explain the most basic implications of the equivalence between energy and mass

Quantum mechanics

• explain how classical physics picture of light and electrons is in conflict with observations, using examples such as model of atom as electrons orbiting nucleus and spectra of light emitted by atoms

Light as a Particle

• qualitatively describe what is measured in the photoelectric effect experiment and explain how this implies a quantum picture of light, including explaining what results the classical interpretation of light would predict for this experiment
• quantitatively analyze photoelectric data to deduce the relationship between energy of photons and frequency of light

Properties of Quanta of Light ("Photons")

• argue why results of sending light through a polarizer combined with quantum interpretation from photoelectric effect must require a probabilistic (i.e., non-deterministic) behavior of photons
• define what is meant by an "eigenstate" for a given measurement in the context of photons passing through polarizers
• recognize that complex superposition is intimately related to classical superposition with relative amplitude and phase
• compare and contrast classical superposition and quantum superposition
• relate any classical polarization to a complex superposition of photon eigenstates
• describe what is meant by incompatible observables in the context of the quantum description of light passing through a polarizer
• predict the probability that a given photon will pass through a polarizer of given orientation calculate how the initial quantum state of a photon is changed by passing through a polarizer

Wave Properties of Electrons

• describe the Davisson-Germer experiment and argue why its results demonstrate wave properties of electrons
• state de Broglie’s relationship between wavelength and momentum of a particle
• describe the double-slit experiment for light and electrons and explain why the interference pattern is different depending on whether or not a measurement is made as to which slit the photon/electron passes through
• explain why the results of the double slit experiment imply that the initial electrons do not have well defined positions
• explain what is meant by position and momentum eigenstates of electron
• write down the mathematical description of the wavefunctions for position and momentum eigenstates
• explain how a wavefunction can be used to describe a general complex superposition of all position eigenstates
• for a one-dimensional case, calculate the probability for finding a particle in a given region of space from its wavefunction.
• qualitatively relate the shape of the wavefunction to the relative probability of finding the particle at different positions in space
• qualitatively relate the shape of the wavefunction to the relative probability of finding the particle in a given momentum range
• know the definition of an expectation value
• calculate the expectation value of position (or simple functions of position) from a wavefunction state that an arbitrary function can be described as a superposition of sinusoidal functions, and argue why this is plausible
• calculate the expectation value for momentum of a particle given its position space wavefunction
• know the definition of uncertainty (i.e., standard deviation) and be able to calculate the uncertainty in position given a position space wavefunction
• qualitatively describe the implications of the Heisenberg Uncertainty principle in terms of measurements of the position and momentum of a particle

The Schrödinger Equation

• qualitatively describe what a wavepacket is and explain the difference between phase and group velocities
• show that requiring the group velocity for wavepackets to agree with the electron velocity requires the frequency for momentum eigenstate wavefunctions to be proportional to electron kinetic energy
• write down the time-dependent wavefunction for a momentum eigenstate and demonstrate that it satisfies the Schrödinger equation
• know the interpretation of the various terms in the one-dimensional Schrödinger equation
• write down the potential for simple physical setups

Bound states and atomic spectra

• describe what is meant by an energy eigenstate
• know what is meant by the statement that energy eigenstates are --stationary
• describe what the time-independent Schrödinger equation is used for
• describe a simple physical setup that can be approximately described by an infinite square well potential
• solve the Schrödinger equation for an infinite square well to calculate the allowed values of energy and the corresponding wavefunctions
• state the possible energies that might be obtained in a measurement performed on a simple superposition of energy eigenstates
• predict probabilities for measurements of energy or position give for simple superpositions of the energy eigenstates
• know that general bound systems have discrete energy spectra and use this to explain the discrete nature of atomic spectra

Tunneling

• describe the behavior of the wavefunction in regions where the particle is classically forbidden describe the qualitative evolution of a wavefunction initially localized in one half of a double well
• explain properties of radioactive alpha decay

Special Topics

• be aware of various fascinating topics in modern physics and realize that most of them are built upon the material presented in this course
• feel inspired to learn more about statistical mechanics and condensed matter physics, nuclear and particle physics, general relativity and cosmology

1. Math/physics connection: Students should be able to translate a physical description of a sophomore-level classical mechanics problem to a mathematical equation necessary to solve it. Students should be able to explain the physical meaning of the formal and/or mathematical formulation of and/or solution to a sophomore-level physics problem. Students should be able to achieve physical insight through the mathematics of a problem.
2. Visualize the problem: Students should be able to sketch the physical parameters of a problem including sketching the physical situation and the coordinates (e.g., equipotential lines, a resonance curve, a pendulum with its angle as the coordinate,) as appropriate for a particular problem.
3. Expecting and checking solution: When appropriate for a given problem, students should be able to articulate their expectations for the solution to a problem, such as direction of a force, dependence on coordinate variables, and behavior at large distances or long times. For all problems, students should be able to justify the reasonableness of a solution they have reached, by methods such as checking the symmetry of the solution, looking at limiting or special cases, relating to cases with known solutions, checking units, dimensional analysis, and/or checking the scale/order of magnitude of the answer.
4. Organized knowledge: Students should be able to articulate the big ideas from each chapter, section, and/or lecture, thus indicating that they have organized their content knowledge. They should be able to filter this knowledge to access the information that they need to apply to a particular physical problem, and make connections/links between different concepts.
5. Communication: Students should be able to justify and explain their thinking and/or approach to a problem or physical situation, in either written or oral form. Students should be able to write up problem solutions that are well-organized, clear, and easy to read.
6. Build on Earlier Material: Students can connect new material in this course to related topics from introductory physics.
7. Problem-solving techniques: Students should continue to develop their skills in choosing and applying the problem-solving technique that is appropriate to a particular problem. This indicates that they have learned the essential features of different problem-solving techniques (e.g.,solving differential equations with constant coefficients, using fourier series methods to solve PDEs with appropriate boundary conditions, etc). They should be able to apply these problem-solving approaches to novel contexts (i.e., to solve problems which do not map directly to those in the book), indicating that they understand the essential features of the technique rather than just the mechanics of its application. Students should move away from using templates. They should be able to justify their approach for solving a particular problem.
   1. Vectors and coordinate systems: Students should be able to compute dot and cross products and solve vector equations without reference to books or external materials, and they should demonstrate comfort with these mathematical tools. Students should recognize whether variables are scalars or vectors, and vector and scalar variables should be clearly distinguishable in students’ written work. Students should be able to project a given vector into components in multiple coordinate systems, and to choose the most appropriate coordinate system in order to solve a given problem. Students should be able compute surface and volume integrals in Cartesian, cylindrical, and spherical coordinate systems (i.e., know the expressions for dA or dV in these coordinate systems and how to apply them in a particular situation).
   2. Approximations and expansions: Students should be able to recognize when approximations are useful, and use them effectively (e.g., recognize when air resistance is a small effect). Students should be able to recognize when a series expansion is appropriate to approximate a solution, and expand a Taylor Series beyond zeroth order. Students should be able to indicate how many and which terms of a series solution must be retained to obtain a solution of a given order, and should be able to identify when a Taylor expansion is appropriate and what the expansion parameter is in a given problem.
   3. Orthogonality: Students should recognize that both vectors and functions can be orthogonal and that any function can be built from a complete orthonormal basis. Students should be able to expand functions in an orthonormal basis (e.g., find the coefficients for a Fourier series) and interpret the coefficients physically. Students should be able to determine from the even or odd symmetry of a function which terms in the expansion are zero. Students should be able to define the terms complete and orthonormal in the context of an orthonormal basis.
   4. Differential equations: Given a physical situation, students should be able to write down the required ordinary differential equation, identify the method of solution, and correctly calculate the answer. Students should be able to identify the type of differential equation (homogeneous, linear vs. nonlinear, constant vs. variable coefficients, 1st, 2nd, or higher order, etc.) and choose the correct method to solve that type of ODE. Students should be able to explain how the type of differential equation helps determine which methods of solution will be applicable.
   5. Superposition: Students should recognize that – in a linear system – the solutions may be formed by superposition of components.
8. Problem-solving strategy: Students should be able to draw upon an organized set of content knowledge (LG#3), and apply problem-solving techniques (LG#4) to that knowledge in order to organize and carry out long analyses of physical problems. They should be able to connect the pieces of a problem to reach the final solution. They should recognize that wrong turns are valuable in learning the material, be able to recover from their mistakes, and persist in working to the solution even though they don’t necessarily see the path to the solution when they begin the problem. Students should be able to articulate what it is that needs to be solved in a particular problem and know when they have solved it.
9. Intellectual maturity: Students should accept responsibility for their own learning. They should be aware of what they do and don’t understand about physical phenomena and classes of problem. This is evidenced by asking sophisticated, specific questions; being able to articulate where in a problem they experienced difficulty; and take action to move beyond that difficulty.

Credit: Steven Pollock, CU Boulder.

A student should be able to:

Vectors, curvilinear coordinate systems and kinematics:

• compute dot and cross products and solve vector equations without reference to books, and they should demonstrate comfort with these mathematical tools.
• recognize whether variables are scalars or vectors, and vector and scalar variables should be clearly distinguishable in students’ written work.
• project a given vector into components in multiple coordinate systems.
• compute simple surface and volume integrals in Cartesian, cylindrical, and spherical coordinate systems (i.e., know the expressions for dA or dV in these coordinate systems and how to apply them in a particular situation).
• recognize that x-dot is the same as dx/dt, and identify these mathematical terms with the physical idea of rate of change of position with time.
• recognize that v-dot is the same as dv/dt, and identify these mathematical terms with the physical idea of rate of change of velocity with time.
• recognize that x-dot = v and v-dot = a.
• explain the physical meaning of position, velocity, and acceleration and describe how they are related to each other.
• break a vector equation into three equations – one for each component.
• solve problems in plane polar and spherical coordinates. They should be able to draw r-hat, phi-hat, and theta-hat for a given point.
• take time derivatives of unit vectors, and use this to rewrite Newton's 2nd law in other coordinate systems (e.g., plane polar, cylindrical, or spherical)
• identify and describe the physical meaning of the various terms which appear when Newton's 2nd law is written in other coordinate systems

Ordinary differential equations (ODEs) and Newton’s Laws:

• use Newton’s laws to write down the required ordinary differential equation, identify the method of solution, and correctly calculate the answer
• identify the type of differential equation (homogeneous, linear vs. nonlinear, constant vs. variable coefficients, 1st, 2nd, or higher order, etc) and choose the correct method to solve that type of ODE.
• use initial conditions as part of their solutions to ODEs.
• identify (and draw) an appropriate coordinate system before beginning to write down an ODE from a physical situation.
• make physical sense of the mathematical solution to a differential equation, which includes testing limiting cases and sketching the function.
• determine if an ODE is separable, be able to separate the equation if possible, and solve separable ODEs.
• decide if a given differential equation can be solved analytically. If not, they should be able to use Mathematica to find a solution, and recognize if Mathematica has returned a bogus result.
• Motion with air drag: (a good concrete example of the above learning goals)
  • translate a given physical situation for an object moving with air resistance to a differential equation, including the correct sign for each term of the equation.
  • predict the direction of drag when given the velocity of a moving object.
  • describe the concept of terminal velocity and be able to calculate it for a given object and form of the drag force (i.e., quadratic, linear, or both terms).
  • explain what quadratic and linear drag are and that they are limiting cases of the full equation for drag which is a combination of both.
  • list the variables that air resistance depends on (e.g., velocity of the object, viscosity of the liquid, shape of object, etc.) and be able to predict whether and how the air resistance changes when these variables are changed.
  • sketch the qualitative path of an object given the differential equation involving air drag and the initial conditions.

Taylor Expansion:

• recognize when approximations are useful, and use them effectively (e.g., recognize when air resistance is a small effect). Students should be able to indicate how many terms of a series solution must be retained to obtain a physically significant answer, and should be able to identify when Taylor expansion is appropriate and what the variable of expansion is in a given problem.
• take the Taylor series expansion around zero for common functions (cos, sin, exp, 1/(1+x), sqrt(1+x) , ln(1+x)) and express the solution both as a list of terms and using summation notation.
• recognize that Taylor series are used when a variable is <<1 and be able to choose an appropriate variable (or combination of variables) to expand in for a given situation.
• explain when a Taylor expansion is exact, a good approximation, or not a good approximation (e.g., near the point of expansion the Taylor expansion will be a good approximation with only a few terms, but father away more terms are needed for a good approximation of the actual value. To be exact at any point, one may need an infinite number of terms.)

Conservation of momentum:

• state the principle of conservation of momentum, know what the limitations are (i.e., that it applies only to systems with no external force, systems where all constituents are accounted for)
• connect the principle to Newton's 3rd law, and to the non-acceleration of the center of mass of a system with no external force.
• compute momentum of systems of pointlike particles (including using summation and vector notation), and use conservation of momentum to solve for motion of the center of mass.
• compute the center of mass of systems with continuous distribution of matter.
• set up, and be able to solve (given appropriate simplifying assumptions) the rocket equation, including cases with a given external force added.

Gradient Operator:

• compute the gradient of a function in Cartesian, polar, and/or spherical coordinates.
• explain the physical meaning of the gradient, predict relative direction and magnitude for several points given equipotential lines, and relate the gradient to the 1-d idea of slope.

Line integrals:

• take the line integral of F dot dr and should be able to explain this sum physically as the sum of dot products along a line.
• predict the magnitude (zero, non-zero) of line integrals for a given path in a drawn vector field.

Conservative forces:

• calculate the curl of a function in Cartesian, cylindrical and spherical coordinates
• determine from an equation or a drawn vector field whether the underlying force is conservative.
• explain both conceptually and mathematically how force (F) and potential (U) are related and when this relation is applicable.
• determine the relative magnitude and a direction of a force at several points on a set of drawn equipotential lines (or vice versa)
• determine the direction of a force at a point based on a plot of U vs. position (in 1-d ), recognize equilibrium points in the plot and be able to determine if these points are stable given the function U(x).

Gravitation:

• find the total mass for a given rho(r) using volume integration.
• translate the physical situation into an appropriate integral to calculate the gravitational force and/or gravitational potential at a particular point away from some simple mass distributions (a half-infinite line of charge, a ring of charge while on the axis of the ring, etc.)
• relate the gravitational potential to the gravitational force, using the standard relation F = -grad(U)
• apply Gauss’s law in the context of a gravitation problem
• explain what criteria must be satisfied for Gauss’s Law to be useful, and be able to identify mass distributions whose gravitational field can and cannot be determined using Gauss’s law.

Simple Harmonic Oscillator, including damping and driving:

• solve second order linear ODEs with constant coefficients.
• write down Hooke’s Law and explain what each variable means, and explain the physical meaning of this equation.
• explain, using the idea of a Taylor series, why Hooke’s law is a good approximation for the force near any stable equilibrium., and give a physical example of a system that obeys Hooke’s law that is not a spring.
• write the differential equation for harmonic motion with & without damping or driving forces, and be able to explain the physical meaning of each term.
• draw a phase diagram for a physically or mathematically given oscillation and vice-versa.
• explain the concept of resonance both conceptually and mathematically.
• sketch the motion of simple harmonic oscillators, with and without damping and driving forces on the same set of axes, with qualitatively correct relative amplitudes and frequencies.
• use Fourier Series to solve for the motion of a harmonic oscillator driven with a given arbitrary periodic forcing.

Complex Numbers:

• write a complex number in the following representations: x+iy, (x,y), r e^(i theta), and on a set of two axes. Students should be able to fluently move between these different representations and to choose the most appropriate representation to solve a given problem.
• treat a complex number equation as two equations – one for imaginary and one for real

Fourier Series:

• expand functions in an orthonormal basis (e.g., find the coefficients for a Fourier series) and interpret the coefficients physically.
• determine from the even or odd symmetry of a function which terms in the expansion are zero.
• define the terms complete and orthonormal in the context of an orthonormal basis.

Fourier Transforms:

• Qualitatively match graphs of functions to graphs of their Fourier transforms.

PDEs/Separation of Variables:

• explain the difference between ODEs and PDEs and give examples of physical situations that lead to each.
• derive the relevant separated ODEs in Cartesian coordinates from Laplace’s equation.
• use boundary conditions to solve Laplace’s equation using separation of variables in 2-D Cartesian and plane polar coordinates. For plane polar coordinates, students may refer to a book for the relevant separated ODEs.

1. Math/physics connection: Students should be able to translate a physical description of an upper-division electromagnetism problem to a mathematical equation necessary to solve it. Students should be able to explain the physical meaning of the formal and/or mathematical formulation of and/or solution to an upper-division electromagnetism problem. Students should be able to achieve physical insight through the mathematics of a problem.
2. Visualize the problem: Students should be able to sketch the physical parameters of a problem (e.g., E or B field, distribution of charges, polarization), as appropriate for a particular problem.
3. Organized knowledge: Students should be able to articulate the big ideas from each chapter, section, and/or lecture, thus indicating that they have organized their content knowledge. They should be able to filter this knowledge to access the information that they need to apply to a particular physical problem, and make connections/links between different concepts.
4. Communication: Students should be able to justify and explain their thinking and/or approach to a problem or physical situation, in either written or oral form.
5. Problem-solving techniques: Students should be able to choose and apply the problem-solving technique that is appropriate to a particular problem. This indicates that they have learned the essential features of different problem-solving techniques (e.g., separation of variables, method of images, direct integration). They should be able to apply these problem-solving approaches to novel contexts (i.e., to solve problems which do not map directly to those in the book), indicating that they understand the essential features of the technique rather than just the mechanics of its application. They should be able to justify their approach for solving a particular problem.
   1. Approximations: Students should be able to recognize when approximations are useful, and use them effectively (e.g., when the observer is very far away from or very close to the source). Students should be able to indicate how many terms of a series solution must be retained to obtain a solution of a given order.
   2. Series expansions: Students should be able to recognize when a series expansion is appropriate to approximate a solution, and complete a Taylor Series to two terms.
   3. Symmetries: Students should be able to recognize symmetries and be able to take advantage of them in order to choose the appropriate method for solving a problem (e.g., when to use Gauss’ Law, when to use separation of variables in a particular coordinate system).
   4. Integration: Given a physical situation, students should be able to write down the required partial differential equation, or line, surface or volume integral, and correctly calculate the answer.
   5. Superposition: Students should recognize that – in a linear system – the solutions may be formed by superposition of components.
6. Problem-solving strategy: Students should be able to draw upon an organized set of content knowledge (LG#3), and apply problem-solving techniques (LG#4) to that knowledge in order to organize and carry out long analyses of physical problems. They should be able to connect the pieces of a problem to reach the final solution. They should recognize that wrong turns are valuable in learning the material, be able to recover from their mistakes, and persist in working to the solution even though they don’t necessarily see the path to the solution when they begin the problem. Students should be able to articulate what it is that needs to be solved in a particular problem and know when they have solved it.
7. Expecting and checking solution: When appropriate for a given problem, students should be able to articulate their expectations for the solution to a problem, such as direction of the field, dependence on coordinate variables, and behavior at large distances. For all problems, students should be able to justify the reasonableness of a solution they have reached, by methods such as checking the symmetry of the solution, looking at limiting or special cases, relating to cases with known solutions, checking units, dimensional analysis, and/or checking the scale/order of magnitude of the answer.
8. Intellectual maturity: Students should accept responsibility for their own learning. They should be aware of what they do and don’t understand about physical phenomena and classes of problem. This is evidenced by asking sophisticated, specific questions; being able to articulate where in a problem they experienced difficulty; and take action to move beyond that difficulty.
9. Maxwell’s Equations: Students should see the various laws in the course as part of the coherent field theory of electromagnetism; i.e., Maxwell’s equations.
10. Build on Earlier Material: Students can connect new material in this course to related topics from introductory physics.

Course-level outcomes: Calculation and computation

A student should be able to:

• Compute gradient, divergence, curl, and Laplacian of a given function in cartesian, cylindrical, or spherical coordinates
• Evaluate line, surface, and volume integrals
• Apply the fundamental theorem for divergences (Gauss’ Theorem) in specific situations
• Apply the fundamental theorem for curls (Stoke’s Theorem) in specific situations
• Apply Coulomb’s Law and superposition principle to calculate electric field due to a continuous charge distribution (uniformly charged line segment, circular or square loop, sphere, etc.)
• Apply Gauss’ Law to compute electric field due to symmetric charge distribution
• Calculate electric field from electric potential and vice versa
• Compute the potential of a localized charge distribution
• Determine the surface charge distribution on a conductor in equilibrium
• Use method of images to determine the potential in a region
• Solve Laplace’s equation to determine the potential in a region given the potential or charge distribution at the boundary (Cartesian, spherical and cylindrical coordinates)
• Use multipole expansion to determine the leading contribution to the potential at large distances from a charge distribution
• Calculate the field of a polarized object
• Find the location and amount of all bound charges in a dielectric material
• Apply Biot-Savart Law and Ampere’s Law to compute magnetic field due to a current distribution
• Compute vector potential of a localized current distribution using multipole expansion
• Calculate magnetic field from the vector potential
• Calculate the field of a magnetized object
• Compute the bound surface and volume currents in a magnetized object

A student should be able to:

Vector analysis

• Evaluate the integral from negative infinity to infinity of the delta function, d(x)
• Evaluate the 3-dimensional divergence of 1/r² in the r-hat direction [4πd³(r)]
• Evaluate the integral of a function times the delta function
• Be able to evaluate the integral of 1/(x-r)^3/2dx
• Give a geometrical description of the divergence theorem, and fundamental theorem for curls.
• Change a multidimensional integral in Cartesian coordinates to one in another coordinate system using the Jacobian.

Electrostatics

Electric Field

• state Coulomb’s Law and use it to solve for E above a line of charge, a loop of charge, and a circular disk of charge.
• solve surface and line integrals in curvilinear coordinates (when given the appropriate formulas, as in the inner-front cover of Griffiths).

Divergence and Curl of E; Gauss’ Law

• recognize when Gauss’ Law is the appropriate way to solve a problem (by recognizing cases of symmetry; and by recognizing limiting cases, such as being very close to a charged body).
• recognize that E comes out of the Gaussian integral only if it is constant along the Gaussian surface.
• recognize Gauss’ Law in differential form and use it to solve for the charge density ρ given an electric field E.

Electric Potential

• Students should be able to state two ways of calculating the potential (via the charge distribution and via the E-field); indicate which is the appropriate formulation in different situations; and correctly evaluate it via the chosen formulation.
• calculate the electric field of a charge configuration or region of space when given its potential.
• state that potential is force per unit charge, and give a conceptual description of V and its relationship to energy.
• explain why we can define a vector potential V (because the curl of E is zero and E is a conservative field).
• defend the choice of a suitable reference point for evaluating V (often infinity or zero), and explain why we have the freedom to choose this reference point (because V is arbitrary with respect to a scalar – its slope is important, not its absolute value).

Work & Energy

• calculate the energy stored in a continuous charge distribution when given the appropriate formula
• explain in words what this energy represents.

Conductors

• sketch the induced charge distribution on a conductor placed in an electric field.
• explain what happens to a conductor when it is placed in an electric field, and sketch the E field inside and outside a conducting sphere placed in an electric field.
• explain how conductors shield electric fields, and describe the consequences of this fact in particular physical problems (e.g., conductors with cavities).
• state that conductors are equipotentials, that E=0 inside a conductor, that E is perpendicular to the surface of a conductor (just outside the conductor), and that all charge resides on the surface of a conductor.

Maxwell’s Equations

• interpret the first and second Maxwell’s equations for electrostatics and use them to describe electrostatics (i.e., Gauss’ Law is just one application of the first law).

Special techniques (Laplace’s equation, boundary conditions, method of images, separation of variables)

Laplace’s equation

• recognize that the solution to Laplace’s equation is unique.

Method of Images

• recognize when the method of images is applicable in a given problem and be able to solve simple cases.
• explain the difference between the physical situation (surface charges) and the mathematical setup (image charges).

Separation of variables/boundary value problems

• state the appropriate boundary conditions on V in electrostatics and be able to derive them from Maxwell’s equations.
• recognize where separation of variables is applicable and what coordinate system is appropriate to separate in.
• outline the general steps necessary for solving a problem using separation of variables.
• state what the basis sets are for separation of variables in Cartesian and spherical coordinates (i.e., exponentials, sin/cos, and Legendre polynomials.)
• apply the physics and symmetry of a problem to state appropriate boundary conditions.
• solve for the coefficients in the series solution for V, by expanding the potential or charge distribution in terms of special functions and using the completeness/orthogonality of the special functions, and express the final answer as a sum over these coefficients.

Multipole expansions

• explain when and why approximate potentials are useful.
• identify and calculate the lowest-order term in the monopole expansion (i.e., the first non-zero term).
• sketch the direction and calculate the dipole moment of a given charge distribution.

Electric fields in matter (polarization, dielectrics)

Polarization and dielectrics

• Transfer between two representations of dipoles – as point charges, and as generalized dipole vectors – for simple charge configurations.
• calculate the dipole moment of a simple charge distribution.
• describe similarities and differences between a conductor and a dielectric (both shield E, conductor shields E completely, dielectric shields via fixed dipoles, conductor shields via mobile electrons).
• predict whether a particular pattern of polarization will result in bound surface and/or volume charge
• explain the physical origin of bound charge at a macroscopic and microscopic level.

Field of a polarized object

• sketch the E field inside and outside a dielectric sphere placed in an electric field.
• explain what happens to a dielectric, when it is placed in an electric field.
• explain the difference between free and bound charge.
• identify the appropriate boundary conditions on D given its relationship to E and Q_f.

Electric displacement

• sketch the direction of D, P, and E for simple problems involving dielectrics
• calculate the E field inside a dielectric when given epsilon and the free charge on the dielectric.

Linear dielectrics

• articulate the difference between a linear and nonlinear dielectric.
• write down Maxwell’s equations (for electrostatics) in matter, when given the appropriate equations in vacuum.
• identify the appropriate boundary conditions on D, given its relationship to E.

Magnetostatics

Currents and charge density

• calculate current density J given the current I, and know the units for each.
• explain, in words, what the charge continuity equation means.
• state the vector form of Ohm’s Law and when it applies.
• calculate the current I, K and J in terms of the velocity of the particle or in terms of each other.

Magnetic fields and forces

• describe the trajectory of a charged particle in a given magnetic field.
• sketch the B field around a current distribution, and explain why any components of the field are zero.
• explain why the magnetic field does no work using concepts and mathematics from this course.

Biot-Savart Law

• state when the Biot-Savart Law applies (magnetostatics; steady currents, dp/dt=0).
• compare similarities and differences between the Biot-Savart law and Coulomb’s Law.
• choose when to use Biot-Savart Law versus Ampere’s Law to calculate B fields, and to complete the calculation in simple cases.

Divergence and curl of B (Ampere’s Law)

• draw appropriate Amperian loops for the cases in which symmetry allows for solution of the B field using Ampere’s Law (i.e., infinite wire, infinite plane, infinite solenoid, toroids), and calculate I_enc.
• Compare E and B in Maxwell’s equations

Magnetic vector potential

• explain why the potential A is a vector for magnetostatics, whereas it’s a scalar (V) in electrostatics, i.e., that the source of magnetic fields (the current) is a vector, whereas the source of electric fields (charge) is not.
• recognize that A does not have a physical interpretation similar to V, but be able to identify when it is useful for solving problems.

Separation of variables/boundary value problems

• state the appropriate boundary conditions on B in magnetostatics and be able to derive them from Maxwell’s equations

Maxwell’s Equations

• interpret the third and fourth Maxwell’s equations for electrostatics and use them to describe magnetostatics (i.e., Ampere’s Law and Biot-Savart law are just applications of these laws).

Magnetic fields in matter (magnetization, bound currents)

Magnetization

• calculate the torque on a magnetic dipole in a magnetic field.
• explain the difference between para, dia, and ferromagnets, and predict how they will behave in a magnetic field.

The Field of a magnetized object

• predict whether a particular magnetization will result in a bound surface and/or volume current, for simple magnetizations.
• give a physical interpretation of bound surface and volume current, using Stokes’ Theorem.

Auxiliary field H

• calculate H when given B or M
• recognize that H is a mathematical construction, whereas B and M are physical quantities.
• use H to calculate B when given J_f for an appropriately symmetric current distribution.
• articulate in which physical situations it is useful to use H.
• identify the appropriate boundary conditions on H given its relationship to M and K_f.

1. Build on earlier material: Students should deepen their understanding of introductory electromagnetism, upper-division E&M, and appropriate math skills (in particular, vector calculus and differential equations).
2. Maxwell’s equations: Students should see the various topics in the course as part of a coherent theory of electromagnetism; i.e., as a consequence of Maxwell’s equations.
3. Math/physics connection: Students should be able to translate a description of an upper-division E&M problem into the mathematical equation(s) necessary to solve it; explain the physical meaning of the final solution, including how this is reflected in its mathematical formulation; and be able to achieve physical insight through the mathematics of a problem.
4. Visualization: Students should be able to sketch the physical parameters of a problem (e.g., electric or magnetic fields, charge distributions). They should be able to use a computer program to graph physical parameters, create animations of time-dependent solutions, and compare analytic solutions with computations. Students should recognize when each of the two methods (by hand or computer) is most appropriate.
5. Organized knowledge: Students should be able to articulate the important ideas from each chapter, section, and/or lecture, thus indicating how they have organized their content knowledge. They should be able to filter this knowledge to access the information they’ll need to solve a particular physics problem, and make connections between different concepts.
6. Communication: Students should be able to justify and explain their thinking and/or approach to a problem or analysis of a physical situation, in either written or oral form. Students should be able to understand and summarize a significant portion of an appropriately difficult scientific paper (e.g., an AJP article) on a topic from electromagnetism; and have the necessary reference skills to electronically search for and retrieve a journal article.
7. Problem-solving techniques: Students should be able to choose and apply the problem-solving technique (e.g., separation of variables, method of images, direct integration) that is appropriate to a particular problem. They should be able to apply these methods to novel contexts (i.e., solve problems which do not map directly to examples found in the book), indicating how they understand the essential features of the technique, rather than just the mechanics of its application.
   1. Approximations: Students should be able to recognize when approximations are useful, and use them effectively (e.g., when the observer is far away from or very close to the source). Students should be able to indicate how many terms of a series solution must be retained to obtain a solution of a given order.
   2. Series expansions: Students should be able to recognize when a series expansion is an appropriate approximation of a solution, and be able to complete a Taylor Series to at least two terms.
   3. Symmetries: Students should be able to recognize symmetries and be able to take advantage of them in order to choose the appropriate method for solving a problem (e.g., correctly applying Gauss’ Law to calculate the electric field).
   4. Integration: Given a physical situation, students should be able to write down the required line, surface or volume integral, and correctly follow through with the integration.
   5. Superposition: Students should recognize that – in a linear system – a solution can be formed by a superposition of different components.
8. Problem-solving strategy: Students should be able to draw upon an organized set of content knowledge (LG#5), and apply problem-solving techniques (LG#7) with that knowledge in order to carry out lengthy analyses of physical problems. They should be able to connect all the pieces of a problem to reach a final solution. They should recognize that wrong turns are valuable in learning the material, be able to recover from their mistakes, and persist in working towards a solution even though they don’t necessarily see the path to that solution when they first begin the problem. Students should be able to articulate what it is that needs to be solved for in a particular situation, and know when they have found it.
9. Expecting and checking solutions: When appropriate for a given problem, students should be able to articulate their expectations for the solution, such as the magnitude or direction of a vector field, the dependence of the solution on coordinate variables, or its behavior at large distances. For all problems, students should be able to justify the reasonableness of a solution (e.g., by checking its symmetry, looking at limiting or special cases, relating to cases with known solutions, dimensional analysis, and/or checking the scale/order of magnitude of the answer).
10. Derivations/proofs: Students should recognize the utility and role of formal derivations and proofs in the learning, understanding, and application of physics. They should be able to identify the necessary elements of a formal derivation or proof; and be able to reproduce important ones, including an articulation of their logical progression. They should also have some facility in recognizing the range/limitations of a result based on the details of its derivation.
11. Intellectual maturity: Students should accept responsibility for their own learning. They should be aware of what they do and don’t understand about physical phenomena and classes of problems, be able to articulate where they are experiencing difficulty, and take action to move beyond that difficulty (e.g., by asking thoughtful, specific questions).

Credit: Steven Pollock, CU Boulder

A student should be able to:

Electrodynamics

• correctly apply Stokes’ Law and the Divergence theorem, and be able to use them to convert a variety of equations from differential to integral form (and vice-versa), as well as interpret the physical meaning of the resulting terms (e.g., what's a flux? what's a divergence? what does the curl tell you?)
• write down, and explain in words and pictures, the full set of Maxwell's Equations in vacuum. This includes recognizing which are relevant for solving a given problem, and using them to solve problems with sufficient symmetry. In particular, students should be able to use the integral form of Faraday’s Law to determine induced E-fields, and the Maxwell-Ampere law to determine induced B-fields.
• determine the total resistance or conductivity of a material for a given geometry (assuming the geometry has sufficient symmetry to make the problem tractable), and be able to apply Ohm’s Law (in the form J = sigma E) to relate the current density to the electric field, and calculate the total current for a given situation.
• state the continuity equation in both differential and integral form, explain how it is an expression of charge conservation, and understand the implications of this equation for quantities such as J and E in steady state situations.
• know how EMF is defined, and be able to compute EMF (either motional or Faraday-induced) for a variety of situations (typically with known or easily computed fields).
• relate mutual and self-inductance to magnetic flux and the resulting induced EMF and currents, and find the mutual and self-inductance for a situation with sufficient symmetry.
• set up and then solve the differential equations describing time-dependent circuit quantities (such as current, charge, voltage differences) in circuits with either AC or DC drivers (or no driver at all), and simple networks of resistors, capacitors, and inductors.
• work with complex exponentials as solutions to differential equations in circuit analysis, and be able to translate between complex (imaginary) solutions and physically real quantities.
• calculate the total energy contained in electromagnetic fields (from the energy densities), as well as compute the power and energy flow in AC circuits (by considering the energy stored in inductors and capacitors, and dissipated in resistors)
• translate between E- and B-fields and the auxiliary fields D & H, in terms of the polarization and magnetization of a material, and be able to set up appropriate boundary conditions to determine E, B, D & H at the interface of two different media.

Conservation laws in E&M:

• interpret the continuity equation, and manipulate it using the divergence theorem to express it in integral and differential form.
• recall how Poynting's theorem is derived (what assumptions go into it, and the basic mathematical manipulations), and to physically interpret and use S, along with the energy density, to solve problems involving the transfer of energy through electric and magnetic fields.
• correctly determine the direction of S in physical situations, and interpret the sign of the flux integral of S in terms of energy flow.
• qualitatively use the expression for momentum volume density, and explain its connection to conservation of momentum in EM systems (and similarly for angular momentum volume density).

Electromagnetic waves:

• construct general solutions to the wave equation, in 1-D to 3-D (including superposition solutions which might not look like the canonical "traveling wave"). For traveling waves, students should be comfortable working with the usual elements such as wavelength, wavenumber, frequency and angular frequency, period, phase, polarization, and velocity.
• derive the traveling wave equation (and thus its solutions) in free space and in matter, starting from Maxwell's Equations. This would include deriving and using the connections between E and B (and wave speed, wave vector, and polarization directions) as they arise from Maxwell's equations.
• interpret and work with plane wave solutions in complex notation, and move back and forth between this notation and the real (physical) wave formulas, as well as apply concepts from previous chapters (S, energy, momentum and angular momentum densities) in the context of EM plane waves.
• find the boundary conditions for EM waves in free space and in matter, starting from Maxwell's Equations, and apply the correct boundary conditions to solve for and interpret reflected and transmitted waves, and reflected and transmitted intensities (including understanding and working with the Brewster angle and Snell's Law).
• extend the previous item to include the case of EM waves normally incident on a conductor. This includes interpreting the complex wave-vector that arises, knowing the relevant approximations and assumptions we made to derive simple results, and being able to compute, interpret, and use the resulting formulas for reflection and transmission coefficients (even in the case where they are complex)

Potentials and fields with time dependence:

• compute time dependent fields, given scalar and vector potentials.
• qualitatively explain the concept and basic consequences of "gauge invariance"
• know/use the definitions of "Coulomb" and "Lorentz" gauges, and be able to state/show how they lead to solvable wave equation
• qualitatively interpret "retarded time", and compute and interpret potentials using the retarded time formalism in cases of very simple geometry.

EM Radiation:

• identify terms in expressions which correspond to "radiation fields"
• explain and justify both the math and physics of the "three approximations" involved in calculating electric or magnetic dipole radiation.
• use and explain the mathematical forms of E and B for electric or magnetic dipole radiation fields, including computation or representation of energy flow, intensity, and power, (with angular and radial dependence)
• derive and use "radiation resistance" of simple dipole radiators
• qualitatively describe, and use in straightforward (nonrelativistic) cases, the basic Larmor formula for radiation from an accelerating point charge.

Special relativity and electrodynamics:

• describe the two principles of relativity, and their basic consequences (including relativity of simultaneity, Lorentz contraction and time dilation)
• use Lorentz transformations and four-vector notation to convert relevant observables between inertial frames.
• Know what "invariant quantity" means in the context of relativity, distinguish it from "conserved quantities", and use both to compute, e.g., relativistic kinematics problems.
• interpret and sketch space-time diagrams, and connect them to mathematical formulas.
• use E and B transformations to convert fields between frames.

1. Math/physics connection: Students should be able to translate a physical description of an upper-division quantum mechanics problem to a mathematical equation necessary to solve it. Students should be able to explain the physical meaning of the formal and/or mathematical formulation of and/or solution to an upper-division QM problem. Students should be able to achieve physical insight through the mathematics of a problem.
2. Visualize the problem: Students should be able to sketch the physical parameters of a problem (e.g., psi, potential energy, probability distribution), as appropriate for a particular problem.
3. Organized knowledge: Students should be able to articulate the big ideas from each chapter, section, and/or lecture, thus indicating that they have organized their content knowledge. They should be able to filter this knowledge to access the information that they need to apply to a particular physical problem.
4. Communication: Students should be able to justify and explain their thinking and/or approach to a problem or physical situation, in either written or oral form.
5. Problem-solving techniques: Students should be able to choose and apply the problem-solving technique that is appropriate to a particular problem. This indicates that they have learned the essential features of different problem-solving techniques (e.g., separation of variables, power series solutions, operator methods). They should be able to apply these problem-solving approaches to novel contexts (i.e., to solve problems which do not map directly to those in the book), indicating that they understand the essential features of the technique rather than just the mechanics of its application. They should be able to justify their approach for solving a particular problem.
   1. Approximations: Students should be able to recognize when approximations are useful, and use them effectively (e.g., when the energy is very high, or barrier width very wide,). Students should be able to indicate how many terms of a series solution must be retained to obtain a solution of a given order.
   2. Symmetries: Students should be able to recognize symmetries and be able to take advantage of them in order to choose the appropriate method for solving a problem (e.g., when parity allows you to eliminate certain solutions).
   3. Matrix methods and Dirac notation: Given a physical situation, students should be able to interpret and compute using Dirac notation, write down the required matrix equation for Schrodinger's equation, and correctly calculate the answer.
   4. differential equation methods: Given a physical situation, students should be able to write down the required (Schrodinger) differential equation, and correctly calculate the answer.
   5. Metacognition: Students should be able to justify their choices in problem solving methods (see LG #4 above) verbally or in writing, and explicitly engage in discussion about their thinking and what helped them learn.
6. Problem-solving strategy: Students should be able to draw upon an organized set of content knowledge (LG#3), and apply problem-solving techniques (LG#4) to that knowledge in order to organize and carry out long analyses of physical problems. They should be able to connect the pieces of a problem to reach the final solution. They should recognize that wrong turns are valuable in learning the material, be able to recover from their mistakes, and persist in working to the solution even though they don’t necessarily see the path to the solution when they begin the problem. Students should be able to articulate what it is that needs to be solved in a particular problem and know when they have solved it.
7. Expecting and checking solution: When appropriate for a given problem, students should be able to articulate their expectations for the solution to a problem, such as general shape of the wave function, dependence on coordinate choice, and behavior at large distances. For all problems, students should be able to justify the reasonableness of a solution they have reached, by methods such as checking the symmetry of the solution, looking at limits, relating to cases with known solutions, checking units, dimensional analysis, and/or checking the scale/order of magnitude of the answer.
8. Intellectual maturity: Students should accept responsibility for their own learning. They should be aware of what they do and don’t understand about physical phenomena and classes of problem. This is evidenced by asking sophisticated, specific questions; being able to articulate where in a problem they experienced difficulty; and take action to move beyond that difficulty.
9. Build on Earlier Material: Students should deepen their understanding of modern physics material.

Credit: Steven Pollock, CU Boulder

Stern-Gerlach experiments and quantum states:

A student should be able to:

• Describe and derive the basic classical physics of the Stern-Gerlach device, including predicting qualitative motion of uncharged classical objects in a given Stern-Gerlach field
• predict the outcome probabilities and outcome "states" along any given paths of Chained-Stern-Gerlach devices of any arbitrary configuration, including ones oriented in X, Y, Z, or n (tilted by theta and phi) directions.
• work backwards along Chained-S-G devices, e.g., given outcomes, decide what S-G orientations were needed or what starting states we began with
• work with basic Dirac notation, including manipulating basic formal expressions (e.g., knowing when term order matters, or which expressions are meaningless), or how to distribute terms involving sums in the bra or ket.
• be able to convert kets to and from bras, normalize kets, find orthogonal kets, compute brackets, and handle the basic complex number manipulations associated with such problems.
• use Postulate 4 (the probability expression) to predict experimental outcomes. This includes working with the formal notation of Postulate 4 and being able to move back and forth between sections of Chained S-G's and the Postulate 4 probability formula.
• generalize to quantum systems beyond spin 1/2 (e.g., doing normalization or use Postulate 4 for spin-1 systems)
• use matrix notation to compute brackets and express bras and kets

Operators and measurement:

• Represent an operator in the Sz basis, given its eigenvalue equations and eigenvectors.
• diagonalize a 2x2 matrix (find the eigenvalues, and/or given an eigenvalue, find an eigenvector)
• Use the Sx, Sy, and Sz matrices and eigenvectors in problems involving S-G problems, probability predictions, or other typical situations.
• Use the Sn matrix, and eigenvectors to handle chained S-G problems involving N-hat oriented S-G's. This includes understanding how to relate "theta" in the n-hat picture to "theta/2" which appears in the |+/->n kets.
• Use postulate 4 to predict probabilities of measurements given any combination of spin 1/2 states and detectors, or work backwards (given the results, to draw conclusions about the measurement device's orientation)
• identify when a matrix is Hermitian, and form a hermitian conjugate of a 2x2 or 3x3 matrix.
• Know the definition of a "projection operator", Pn = |+n><+n| , and use it to project components of kets, as well as engage in basic formal dirac notation manipulations (such as what appears in Eq 2.54, or throughout section 2.2.4)
• Interpret both numerator and denominator of postulate 5 (projection/measurement postulate), including being able to compute either of those separately given a state and a measurement outcome, and interpret the meaning of Postulate 5 to predict outcome states from chained S-G setups.
• compute "expectation values" of any given operator, interpret this as a sum of possible results*probabilities (as given, e.g., in Eq 2.74, but for any operator).
• compute the RMS deviation (Delta) for an operator, given a state.
• construct or interpret probability distributions such as McIntyre's figure 2.8 for arbitrary states and measurements.
• compute the commutator of 2 given operators.
• connect commutation (or non-commutation) to measurements ("compatibility"), uncertainties, and whether 2 operators share common eigen-bases.
• Know the definition of S.S (the S^2 operator), and evaluate its effect on spin 1/2 states.
• apply the rules and postulates of quantum mechanics described above to spin 1 systems in simple cases. (e.g., normalizing spin 1 states, finding brackets, determining orthogonality, computing probabilities using Postulate 4, predicting output states using Postulate 5, or sketching probability distributions as in, e.g., Fig 2.13)
• Generalize to arbitrary spin (beyond spin 1).
• Apply the Uncertainty principle to arbitrary spin 1/2 operators and states.
• know (or quickly rederive) the commutation relations for angular momentum

Schrodinger Time evolution:

• construct a Hamiltonian operator (2x2 or 3x3 matrix) for a spin 1/2 (or 1) particle in an arbitrary B-field
• solve for the "cn" coefficients in the initial state in the |En> basis.
• write down the time dependent solution given initial conditions, and use this to compute probabilities or measurement outcomes or expectation values of any operators of interest at later times. (You should be able to do this even if the field points in a direction other than z-hat, and be able to handle spin-1 as well as spin 1/2)
• qualitatively describe "spin precession", for an arbitrary magnetic field direction.
• apply Rabi's formula to find "flip probability".
• extend the time-dependent formalism above to "non-spin" systems (neutrino oscillations being one good example, but any 2-state system would work the same)

Entanglement and EPR (“Quantum spookiness”):

• Use, interpret, and generate 2-particle states in the form used in McIntyre, e.g., his Eq 4.1
• Do standard Quantum calculations with 2-state systems including computing brackets and acting single-particle operators on a state (as, e.g., done in Eq 4.2 and 4.3)
• given an EPR gedanken experiment setup (like in Fig 4.1) predict how observations at A will correlate with observations at B.
• distinguish "hidden variable" from "quantum" predictions - e.g., be able to decide if a setup will or will not obey the Bell inequalities.

Quantized energies and “particle in a box”:

1. Interpret "probability histograms" (like in Fig 5.2), be able to go back and forth between that representation or an expansion of the form |psi> = Sum c_n |En> (where you should be able to compute c_n by evaluating <En|psi> and interpret |c_n|^2 as the probability of finding the energy En, as shown in that histogram.
2. know what the x and p operators are, in the position representation
3. Compute probability of finding a particle within dx of position x using the wave function (i.e., you should know that Prob density = |psi(x)|^2, and you should know the difference between "probability density" and "probability"
4. Set up and compute brackets of position-space wavefunctions by forming an integral, and interpreting the bracket as we have in earlier chapters to compute probabilities, or normalize wavefunctions.
5. Compute expectation values of given operators in the position space representation by integrating
6. write down the time-independent Schrodinger equation as a 2nd order ODE for a given potential function.
7. Solve the infinite square well (even if I make small changes, e.g., the location of the starting/ending points of the well, the width of the well, the value of the base of the well)
8. use the rules of QM we have developed to compute the time dependence of superposition states of "infinite well" |En> states, find the probabilities of measuring energy for such time dependent superposition states, and find the position probability density for such states.
9. know the standard definitions of wavelength, and wave number
10. know and apply the "boundary conditions" - namely, that wave functions are continuous everywhere, d/dx(wave function) is continuous unless V=infinity, in which case the wave function must vanish)
11. Be able to write the "general solution" to Schrodinger's equation in regions where V(x)=constant
12. solve the "finite square well" problem (even if I make small changes, e.g., the location of the starting/ending points of the well, the width of the well, the value of the base or top of the well)
13. Sketch qualitatively correct wave functions even when V(x) is not simple to solve exactly - including getting the sign of the curvature right, the behaviour in "allowed" or "forbidden" regions, application of boundary conditions, getting the relative wavelength right based on KE, and getting the relative amplitudes right based on classical arguments about where the particle spends the most time

Unbound 1-D systems:

• Solve the Schrodinger equation in position space for unbound particles (when V=constant): both the time-independent (exp[i kx]) solutions , knowing what "k" is, and also the time-dependent solutions.
• Know and use the deBroglie relation for plane waves to relate momentum, wavenumber k, and wavelength.
• know the eigenfunctions of momentum in position space (using the Dirac normalization)
• use the definitions to compute Fourier transform and Inverse fourier transform.
• Have a qualitative understanding of wave packets, including familiarity with terminology like "envelope", "phase velocity" and "group velocity", and "momentum representation"
• relate the width of wave packets in position and momentum space.
• Use the position-momentum Heisenberg uncertainty relation to make rough estimates of of energy or momentum
• solve the Schrodinger Equation in regions of space where V=constant , using "boundary conditions" on the wave function (continuity of psi and psi'(x)) to piece together solutions of Schrodinger equation for unbound particles hitting a potential well or barrier
• Interpret the coefficients of plane waves as a measure of "flux" or "flow", thus deciding which (if any) vanish
• Use formulas for T or R to make qualitative and quantitative predictions of reflection and transmission, in scattering and tunneling situations. This means you know and can interpret/evaluate the "k" values in the different regions, including forbidden ones.

Angular momentum:

• Recognize situations where you can use the general method of "separation of variables" to simplify partial differential equations, and do so.
• Separate the Center of mass from relative motion coordinates in central potential problems. (You should be able to simply write down the CM part of the wave function, which is a 3-D plane wave)
• know and use spherical coordinates including volume integrals
• be able to compute basic commutation relations involving angular momentum operators, including knowing which ones do and do not commute.
• be able to compute or predict experimental outcomes of measurements (including probabilities, and also the "final state") of L^2 or components of L (Lz, or Ly, or Lx) for states given in the usual form |l, m>
• solve "particle on a ring" problems, including knowing the energy formula, and being able to predict experimental outcomes of energy as well as Lz. You should also be able to add in "time dependence" and compute spatial probability densities for such states.
• solve "particle on a sphere" problems, including knowing the energy formula, and being able to predict experimental outcomes of energy as well as Lz or L^2. You should also be able to add in "time dependence" and compute spatial probability densities for such states.
• Know the solution for the Phi(phi) separated solution of wavefunctions (plane waves in phi), including normalization
• Know and interpret the standard notation of Legendre functions and Spherical harmonics
• Use the postulates of quantum mechanics to compute probabilities of measurements on angular momentum eigenstates in dirac notation |l,m>

Hydrogen atom:

• know and interpret the standard notation of general solutions to the hydrogen atom, psi_n,l,m
• use the postulates of quantum mechanics to compute probabilities of measurements of energy or angular momentum on hydrogen atom states
• use the standard "time evolution" methods from Ch. 3 on hydrogen atom wave functions.
• Connect this chapter’s formalism to the familiar “spectrum of Hydrogen” from modern physics. Calculate energy levels and resulting photon energies in hydrogen.