Unit 15: Modern Physics

Eight topics, eight places classical physics ran out

Every other unit in AP Physics 2 hands you a model and asks you to apply it. Unit 15 hands you the experiments that broke the earlier models, and the ideas physics had to adopt to survive them. That thread runs through all eight topics, and it is the fastest way to stop the unit feeling like eight unrelated facts.

Three topics are experiments about light. Blackbody radiation measured the light coming off hot objects and found a curve no classical calculation could produce. The photoelectric effect shone light on a metal and found that emission depends on the frequency of the light rather than on how much of it there is. Compton scattering bounced photons off free electrons and found the scattered light came back with a longer wavelength, the way a ball loses energy in a collision. Each forced the same conclusion from a different direction: light arrives in discrete packets carrying $E = hf$.

Two topics are about what quantization did to the atom. The Bohr model puts the electron in a circular orbit held by the electric force, then restricts which orbits are allowed. Emission and absorption spectra are what you actually see when that restriction is real: an element emits a fixed set of wavelengths and absorbs the same fixed set, because only certain energy differences exist inside it.

Topic 15.1 sits in front of all of them as the general claim, that a single object can show both particle-like and wave-like behavior, and it is the topic that runs the claim backwards: if light can act like a particle, matter can act like a wave, with a de Broglie wavelength $\lambda = h/p$.

The last two topics move inside the nucleus. Fission, fusion, and nuclear decay covers the conservation rules that constrain nuclear reactions and the statistics of radioactive decay, and types of radioactive decay names the specific processes: alpha, beta-minus, beta-plus, and gamma.

The CED's own framing, from the Developing Understanding section of the unit opener, is that Unit 15 lays the groundwork for the study of modern physics by resolving the conflicts and unanswered questions from Units 13 and 14, and that while Unit 15 introduces new models and representations such as energy level diagrams, students will make connections between this unit's content, the fundamental principles of physics, principles of conservation, and models and representations used earlier in the course. The opener adds that students will revisit the wave-particle duality of light through their investigations of phenomena such as the photoelectric effect.

What the CED says about Unit 15

The numbers, verified against the unit opener and the exam weighting table:

• Exam weighting: 12 to 15 percent of the multiple-choice section, the same band as Units 12, 13 and 14. Units 9, 10 and 11 carry 15 to 18 percent each.
• Suggested pacing: about 14 to 22 class periods. Pacing in the CED is based on 45-minute periods meeting five days a week.
• Eight topics, nine learning objectives. Every topic carries exactly one learning objective except Topic 15.7, which has two: 15.7.A on the properties that constrain interacting nuclei, and 15.7.B on the radioactive decay of a sample.
• Progress Check 15 is listed as roughly 24 multiple-choice questions plus 4 free-response questions.

The unit opener hangs the content on four essential questions: what are the benefits and dangers of radioactivity, how do we measure things we cannot see, why do infrared telescopes need to be cooled, and how does the ultraviolet catastrophe link thermodynamics and modern physics.

That last one repays a second look, because the phrase "ultraviolet catastrophe" appears in the essential question and nowhere in the unit's required course content. What the required content says instead, in 15.4.A.3.i, is that the distribution of the intensity of a blackbody's spectrum as a function of temperature cannot be modeled using only classical physics concepts, and that a blackbody's spectrum is described by Planck's law, which assumes that the energy of light is quantized. Neither the CED nor the equation sheet prints Planck's law itself.

The science practices the opener flags for the unit are 2.C, 2.D, 3.B and 3.C. Its guidance is that Unit 15 provides opportunities for students to compare physical quantities between scenarios or at different times in a single scenario (2.C), as well as determine new values of quantities using functional dependencies between variables (2.D), and that from there students can make and justify claims based on these physical principles and functional relationships (3.B, 3.C). The opener's worked case is the photoelectric effect: students could describe conceptually what happens to the maximum kinetic energy of ejected electrons from a metal plate if the plate is replaced by a plate with a higher work function, and then justify what impact that change will have on the required stopping potential. It closes by saying that by the end of the unit it is important for students to be comfortable with making claims about the reasonableness of their claims and justifications made with functional dependence, starting with first principles of physics.

On the exam, the opener points at the first free-response question, the Mathematical Routines question, and notes that while Unit 15 offers content perfect for practice, the Mathematical Routines question on the AP Physics 2 exam can pull content from any of the seven units of the course.

Topic 15.1: the claim the rest of the unit tests

Topic 15.1, Quantum Theory and Wave-Particle Duality, carries learning objective 15.1.A, describe the properties and behavior of an object that exhibits both particle-like and wave-like behavior.

Its opening statement, 15.1.A.1, is the unit's thesis: quantum theory was developed to explain observations of matter and energy that could not be explained using classical mechanics, and these phenomena include, but are not limited to, atomic spectra, blackbody radiation, and the photoelectric effect. Those three phenomena are Topics 15.3, 15.4 and 15.5. The unit is organised around its own opening sentence. Note the "but are not limited to": the CED is not claiming that list is exhaustive.

The topic then does two symmetrical jobs. It makes light particulate: 15.1.A.2 says light can be modeled both as a wave and as discrete particles, called photons, and 15.1.A.2.i defines a photon as a massless, electrically neutral particle with energy proportional to the photon's frequency, with $E = hf$ and $\lambda = c/f$ as relevant equations. And it makes matter wavelike: 15.1.A.4 says particles can demonstrate wave properties, as shown by variations of Young's double-slit experiment, with the de Broglie wavelength in 15.1.A.4.i, which increases as the momentum of a particle decreases.

15.1.A.5 is the sentence that hands the next two topics their licence: values of energy and momentum have discrete, or quantized, values for bound systems described by quantum theory.

Suggested skills: 1.A, 2.A, 2.B, 3.B. No boundary statement.

Topics 15.2 and 15.3: quantization arrives at the atom

These two are a matched pair, and they are worth studying together. Topic 15.2 is the model; Topic 15.3 is what the model predicts you will see.

15.2, The Bohr Model of Atomic Structure, carries learning objective 15.2.A, describe the properties of an atom. It starts from structure rather than from quantum theory: atoms consist of a small, positively charged nucleus surrounded by one or more negatively charged electrons (15.2.A.1.i), the nucleus is made up of protons and neutrons (15.2.A.1.ii), the number of neutrons and protons can be represented using nuclear notation (15.2.A.1.iii), an ion is an atom with a nonzero net electric charge (15.2.A.1.iv), each atomic element has a unique number of protons (15.2.A.2), and the total number of neutrons and protons identifies the isotope of an element (15.2.A.2.ii).

The Bohr model proper is 15.2.A.3, and the CED is careful about its status: the Bohr model of the atom is based on classical physics and was the historical representation of the atom that led to the description of the hydrogen atom in terms of discrete energy states. Its relevant equations are Coulomb's law and the centripetal form of Newton's second law, $F_e = k \frac{q_1 q_2}{r^2}$ and $F_{net} = m \frac{v^2}{r}$, both of which you already own from Unit 10 and from AP Physics 1 circular motion. Then 15.2.A.3.ii supplies the quantum ingredient: the standing wave model of electrons accounts for the existence of specific allowed energy states of an electron in an atom, because the electron orbit's circumference must be an integer multiple of the electron's de Broglie wavelength.

Boundary statement, printed under Topic 15.2: the analysis and description of electron structure is limited to energy levels and will not include such advanced descriptions as orbitals, orbital shapes, or probability functions.

Suggested skills for 15.2: 1.A, 2.B, 2.C, 3.B.

15.3, Emission and Absorption Spectra, carries learning objective 15.3.A, describe the emission or absorption of photons by atoms. Its central rule is 15.3.A.2: energy can only be absorbed or emitted by an atom if the amount of energy being absorbed or emitted corresponds to the energy difference between two atomic energy states. 15.3.A.3 adds that transitions between two energy states of an atom correspond to the absorption or emission of a photon of a single frequency and, therefore, a single wavelength. 15.3.A.4 is the observational payoff: atoms of each element have a unique set of allowed energy levels and thereby a unique set of absorption and emission frequencies, and the unique set of frequencies determines the element's spectrum. 15.3.A.5 defines binding energy as the energy required to remove an electron from an atom, causing the atom to become ionized, and states that an atom in the lowest energy level (ground state) will require the greatest amount of energy to remove the electron from the atom.

Boundary statement, printed under Topic 15.3: in AP Physics 2, only energy level diagrams of single-electron atoms will be considered.

Suggested skills for 15.3: 1.A, 2.B, 2.C, 3.C.

Topics 15.4, 15.5 and 15.6: the three experiments that made light particulate

15.4, Blackbody Radiation (topic page) carries learning objective 15.4.A, describe the electromagnetic radiation emitted by an object due to its temperature. A blackbody is an idealized model of matter that absorbs all radiation that falls on the body, and if the body is in equilibrium at a constant temperature, then it must in turn emit energy (15.4.A.2). It emits a continuous spectrum that only depends on the body's temperature (15.4.A.3). Two printed equations belong here: Wien's law $\lambda_{max} = \frac{b}{T}$ in 15.4.A.3.ii, and the Stefan-Boltzmann law $P = A \sigma T^4$ in 15.4.A.3.iii. Suggested skills: 1.C, 2.C, 3.B, 3.C. No boundary statement. This is the only topic in Unit 15 that lists 1.C, create qualitative sketches of graphs, which tells you the intensity-against-wavelength curve is something you may be asked to draw.

15.5, The Photoelectric Effect (topic page) carries learning objective 15.5.A, describe an interaction between photons and matter using the photoelectric effect. The photoelectric effect is the emission of electrons when electromagnetic radiation is incident upon a photoactive material (15.5.A.1), and emission requires a minimum frequency of incident light, called the threshold frequency (15.5.A.2). The evidence sentence is 15.5.A.2.ii: the energy of the emitted electrons is not dependent on the number of photons that are incident upon the material, which provides evidence that light is a collection of discrete, quantized energy packets called photons. The printed equation is $K_{max} = hf - \phi$. Boundary statement: where applicable, work functions for materials will be provided on the exam, and students are not expected to know values of work functions or variables of a material that influence the magnitude of its work function. Suggested skills: 1.B, 2.A, 2.D, 3.A, 3.C, five of them, matching Topic 15.7 for the longest list in the unit.

15.6, Compton Scattering (topic page) carries learning objective 15.6.A, describe the interaction between photons and matter using Compton scattering. A photon interacts with a free electron, and the photon that emerges from the interaction has a lower energy and longer wavelength than the incoming photon, with the magnitude of the change related to the direction of the photon after the collision (15.6.A.1). 15.6.A.2.i says Compton scattering can be explained by treating a photon as a particle and applying conservation of energy and conservation of momentum to the collision between the photon and electron, which makes this the one topic in Unit 15 that is literally a collision problem. The printed equation is $\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$. Boundary statement: AP Physics 2 includes full quantitative and qualitative treatments of conservation of momentum in two dimensions. That one expands the topic rather than restricting it. Suggested skills: 1.A, 2.B, 2.C, 3.C.

Read in order, the three build one argument. Blackbody radiation shows that a classical account of emitted light fails and a quantized one works. The photoelectric effect shows that the failure is not a bookkeeping trick, because a threshold frequency makes no sense for a wave whose energy you can raise by turning up the brightness. Compton scattering closes the case by making the photon obey conservation of momentum in a two-body collision, which is as particle-like as an object gets.

Topics 15.7 and 15.8: the nucleus

15.7, Fission, Fusion, and Nuclear Decay (topic page) is the only topic in the unit with two learning objectives.

15.7.A, describe the physical properties that constrain the behavior of interacting nuclei, subatomic particles, and nucleons, is a conservation topic. The strong force is exerted at nuclear scales and dominates the interactions of nucleons, protons or neutrons (15.7.A.1). Possible nuclear reactions are constrained by the law of conservation of nucleon number (15.7.A.2), and the behavior of the constituent particles of a nuclear reaction is constrained by laws of conservation of energy, energy-mass equivalence, and conservation of momentum (15.7.A.3). Fusion is the process by which two or more smaller nuclei combine to form a larger nucleus, as well as subatomic particles (15.7.A.6); fission is the process by which the nucleus of an atom splits into two or more smaller nuclei, as well as subatomic particles (15.7.A.7). The printed equation here is $E = mc^2$.

15.7.B, describe the radioactive decay of a given sample of material consisting of a finite number of nuclei, is the statistics half. 15.7.B.1.i is the honest sentence: the time at which an individual nucleus undergoes radioactive decay is indeterminable, but decay rates can be described using probability. Half-life and decay constant are linked by $\lambda = \frac{\ln 2}{t_{1/2}}$ (15.7.B.1.iii), and the population follows $N = N_0 e^{-\lambda t}$ (15.7.B.2), with $\ln\left(\frac{N}{N_0}\right) = -\lambda t$ listed as a derived equation. Suggested skills: 1.B, 2.B, 2.C, 3.A, 3.B. No boundary statement.

15.8, Types of Radioactive Decay (topic page) carries learning objective 15.8.A, describe the processes by which individual nuclei decay. It names the particles: an alpha particle, or helium nucleus, consists of two neutrons and two protons, and in Physics 2 only helium-4 nuclei will be considered (15.8.A.1.i); neutrinos and antineutrinos have no electrical charge and negligible mass (15.8.A.1.ii); positrons, or antielectrons, have an electric charge opposite that of an electron and the same mass as an electron (15.8.A.1.iv). Then the four processes in 15.8.A.2: alpha decay, beta-minus decay, beta-plus decay, and gamma decay. The conservation rule is 15.8.A.2.i: in all nuclear decays, nucleon number, lepton number, and charge are conserved.

The Topic 15.8 boundary statement is the longest in the unit and it removes real work, so carry it whole: AP Physics 2 does not expect students to memorize the processes by which specific isotopes decay or the half-lives of specific isotopes; neutron emission and electron capture are not included in the AP Physics 2 curriculum framework; additionally, types of neutrinos, the characteristics that distinguish neutrinos and antineutrinos, and an explanation or application of the weak force are not within the scope of this course. Suggested skills: 1.A, 2.C, 3.B, 3.C.

The Unit 15 equations, counted against the sheet

The AP Physics 2 equation sheet prints 129 equations in seven groups. The Modern Physics group holds exactly ten, counted line by line off the printed table, and every one of them belongs to this unit:

Two topics have no equation of their own in that group. Topic 15.2 borrows $F_e = k \frac{q_1 q_2}{r^2}$ from the Electricity group and $F_{net} = m \frac{v^2}{r}$ from Mechanics and Fluids, which is the CED's way of saying the Bohr model is assembled out of physics you already have. Topic 15.8 cites no equation at all.

The symbol key printed beside the Modern Physics group is short and contains one real trap: $\lambda$ means wavelength or decay constant, on the same ten-line table. In $\lambda = h/p$ it is a length in metres; in $N = N_0 e^{-\lambda t}$ it is a rate in inverse seconds. The rest of the key reads $A$ area, $E$ energy, $f$ frequency, $K$ kinetic energy, $m$ mass, $N$ number of particles, $p$ momentum, $P$ power, $t$ time, $T$ absolute temperature, $\theta$ angle, and $\phi$ work function.

The constants are all in the Table of Information at the front of the sheet: $h = 6.63 \times 10^{-34}$ J s $= 4.14 \times 10^{-15}$ eV s, $hc = 1.99 \times 10^{-25}$ J m $= 1240$ eV nm, $c = 3.00 \times 10^8$ m/s, $1$ eV $= 1.60 \times 10^{-19}$ J, Wien's constant $b = 2.90 \times 10^{-3}$ m K, the Stefan-Boltzmann constant $\sigma = 5.67 \times 10^{-8}$ W/(m$^2$ K$^4$), the electron mass $m_e = 9.11 \times 10^{-31}$ kg, and $1$ u $= 1.66 \times 10^{-27}$ kg $= 931$ MeV/$c^2$.

What is not printed is worth the same attention. Neither the equation sheet nor the required course content of Unit 15 contains a Rydberg formula, a Rydberg constant, a hydrogen energy-level formula, a Bohr radius, a table of work functions, or the half-life of any isotope. If a question needs one of those, the question supplies it.

Boundary statements, and where the unit stops

Unit 15 prints boundary statements under exactly five of its eight topics. Do not assume the pattern: 15.1, 15.4 and 15.7 have none at all.

Four of the five take work off your revision list. The 15.6 statement is the odd one out: it is an expansion, telling you that two-dimensional momentum conservation is fully in scope for Compton scattering, both quantitatively and qualitatively.

Taken with the 15.2 and 15.3 statements, the shape of the atom on this exam is fixed: single-electron energy level diagrams, no orbitals, no multi-electron spectra. That is a smaller object than the atom a chemistry course builds, and the diagrams you are handed on the exam reflect it.

How Unit 15 is tested, and a study order

The AP Physics 2 exam is 3 hours long. Section I is 42 multiple-choice questions in 85 minutes for 50 percent of the score. Section II is 4 free-response questions in 95 minutes for the other 50 percent: Question 1 Mathematical Routines, Question 2 Translation Between Representations, Question 3 Experimental Design and Analysis, Question 4 Qualitative/Quantitative Translation. A four-function, scientific, or graphing calculator is allowed on both sections. The 12 to 15 percent weighting applies to the multiple-choice section.

The CED's own sample exam questions show the unit turning up in both sections. Sample multiple-choice question 4 is aligned to learning objective 15.7.B and skill 2.A. Sample multiple-choice question 11 is aligned to 15.4.A, essential knowledge 15.4.A.3, and skill 2.D, and asks how the emitted power changes when a blackbody's peak wavelength is halved. Sample free-response question 4, the Qualitative/Quantitative Translation question, is worth 8 points and is aligned to two learning objectives at once, 15.1.A and 15.5.A, with skills 2.A, 2.D, 3.B and 3.C.

That pairing is worth noticing. The exam does not treat duality and the photoelectric effect as separate facts. It changes one variable, the wavelength of the incident light, then asks you to predict the direction of change in another, justify it conceptually, derive the symbolic relationship, and check that the derivation agrees with the justification.

A workable order:

1. Start with 15.1. Get $E = hf$, $\lambda = c/f$ and $\lambda = h/p$ fluent in both joules and electron volts before anything else. Four of the eight topics are unusable until that arithmetic is automatic.
2. Do 15.4, 15.5 and 15.6 next, as one block. They are three answers to the same question, and each has exactly one printed equation.
3. Then 15.2 and 15.3 together. The Bohr model on its own is a piece of history; paired with a spectrum it becomes a prediction you can check.
4. Finish with 15.7 and 15.8. Neither depends on the photon arithmetic the earlier topics run on, and 15.8 in particular is bookkeeping with conserved quantities.
5. Practise the comparison shape, not just the calculation. Skills 2.C and 2.D are what the unit opener flags, and both ask what happens to a quantity when something else changes. "Double the temperature, what happens to the emitted power" is closer to the exam's habit than "compute the emitted power".

Unit 15 closes out the AP Physics 2 course. It leans on Unit 14 for the wave language it inverts, and on Units 9 to 11 for energy, temperature and the electric force, so it is a reasonable unit to leave until last.