Unit 15 Modern Physics Comprehensive Study Guide

Quantum Theory and Wave-Particle Duality

• Quantization: This occurs when a physical quantity comes only in discrete units. When a quantity is restricted to certain values, it is said to be quantized.

  • Examples of Quantization:

    • Money: Currency is quantized; for instance, in US currency, there is no such thing as half a penny.

    • Microscopic Mass: At the microscopic level, mass is quantized because it is made of discrete particles like protons, neutrons, and electrons.

• Quantum Physics Definition: This is the branch of physics dedicated to the study of extremely small objects and the quantization of their physical properties on a very tiny scale.

• Quantized increments:

  • Mass: Quantized in increments of atomic mass units ($amu$).

  • Charge: Quantized in increments of elemental charge ($e$).

• Quantized Properties in AP Physics 2: The following properties will be treated as quantized throughout this unit:

  • Mass

  • Charge

  • Energy

  • Momentum

  • Angular Momentum

• Applications of Quantum Physics: Quantum mechanics is utilized to explain phenomena that classical physics cannot, including:

  • Blackbody radiation

  • Atomic spectra

  • The Photoelectric effect

  • Electron double-slit experiments

Blackbody Radiation and Atomic Spectra

• Blackbody Radiation:

  • Heated objects emit electromagnetic (EM) radiation, often causing them to glow.

  • Classical physics fails to explain the EM spectrum emitted by a heated object because the intensity at a given wavelength ($\lambda$) has discrete peaks rather than being continuous.

  • The peak wavelength ($\lambda$) emitted depends on the temperature of the blackbody.

  • Definition of a Blackbody: An idealized physical body that absorbs all incident EM radiation and emits radiation depending solely on its own temperature.

  • Energy in this context is quantized, meaning it can only be absorbed or emitted in discrete amounts.

• Atomic Spectra:

  • Electrons release light when they transition from higher energy levels to lower energy levels within an atom.

  • The emitted or absorbed light exists only in discrete lines, not a continuous smear of colors.

  • Electrons are restricted to transitions between specific energy levels, which are dependent upon the specific element.

  • Spectra Examples:

    • Sodium (Na): Unique discrete absorption and emission lines.

    • Nitrogen (N): Unique discrete signature.

    • Hydrogen (H): The simplest spectrum explained by the Bohr model.

    • Oxygen (O): Unique discrete signature.

The Nature of Light and Matter

• Photoelectric Effect:

  • When light of a sufficient frequency shines on certain surfaces, electrons are ejected.

  • The energy of these ejected electrons depends on the frequency of the light, not the intensity.

  • Quantum mechanics explains this by assuming light is composed of particles called photons.

• Electron Double-Slit Experiment:

  • When a beam of electrons passes through two narrow slits, they form an accumulation pattern on a screen consisting of fringes.

  • These fringes are similar to the bright and dark fringes created by interfering light waves.

  • This provides evidence that both electrons and light exhibit both particle-like and wave-like behavior.

• Wave-Particle Nature of Light Summary Chart:

  • Reflection: Exhibited by both Wave-like and Particle-like models.

  • Refraction: Exhibited by both Wave-like and Particle-like models.

  • Interference: Exhibited by the Wave-like model; NOT the Particle-like model.

  • Diffraction: Exhibited by the Wave-like model; NOT the Particle-like model.

  • Photoelectric Effect: Exhibited by the Particle-like model; NOT the Wave-like model.

• Photons:

  • Definition: A tiny packet of light energy.

  • A photon is a massless and electrically neutral particle that exists as a wave.

  • Its energy is proportional to its frequency.

  • Photons transfer quantized amounts of energy and momentum to atomic and subatomic particles.

• Mathematical Properties of Photons:

  • Energy formula: $E = hf$

  • Planck’s Constant ($h$): $h = 6.63 \times 10^{-34} \, Js$ or $h = 4.14 \times 10^{-15} \, eVs$

  • Speed in vacuum: A photon moves at $c = 3 \times 10^{8} \, m/s$ in empty space.

  • Speed in media: The speed changes based on the index of refraction: $n = \frac{c}{v}$. Therefore, $v = \frac{c}{n}$.

  • Newton's First Law: A photon moves in a straight line until it interacts with matter.

de Broglie Wavelength and Wave Nature of Matter

• Wave-like Behavior of Particles: Beams of electrons produce interference patterns just like EM waves when encountering double slits.

• Wave-Particle Duality of Matter: Particles can demonstrate wave-like behavior, quantified by the de Broglie wavelength.

• de Broglie Wavelength Formula: $\lambda = \frac{h}{p}$

  • $p$ is the momentum of the particle.

  • Because $h$ is so small ($6.63 \times 10^{-34} \, Js$), the momentum must be extremely small for the wavelength to be observable.

  • If a system's size is comparable to its de Broglie wavelength, quantum theory is required to describe the system (i.e., energy and momentum must be treated as quantized).

  • Derivation for Energy: Given $p = mv$ and $K = \frac{1}{2} mv^2$, then $v = \sqrt{\frac{2E}{m}}$ and $p = m \sqrt{\frac{2E}{m}} = \sqrt{2mE}$. Thus: $\lambda = \frac{h}{\sqrt{2mE}}$.

• Everyday Matter: Objects like a baseball have such large masses that their momentum is enormous compared to $h$. Consequently, their de Broglie wavelength is far beyond the range of human perception and cannot be observed.

The Bohr Model of Atomic Structure

• Atomic Basics:

  • Atoms are the smallest building blocks of an element.

  • Composed of subatomic particles: Protons, Neutrons (nucleons), and Electrons.

• Isotopes and Ions:

  • Isotopes: Atoms of the same element (same number of protons) with varying numbers of neutrons, resulting in different masses and nuclear stabilities.

  • Ions: Non-electrically neutral atoms. Positive ions have fewer electrons; negative ions have more electrons.

• Atomic Numbers and Masses:

  • Atomic Number ($Z$): Number of protons; defines the element.

  • Atomic Mass: Average mass of isotopes. Dominated by nucleons ($m_p \approx m_n \approx 1.67 \times 10^{-27} \, kg$).

  • Electron Mass ($m_e$): $9.11 \times 10^{-31} \, kg$. Four orders of magnitude smaller than nucleons; negligible for atomic mass calculations.

• Bohr Model Specifics:

  • Niels Bohr proposed electrons move in stable, circular orbits around the nucleus.

  • Energy Transitions: Electrons must gain energy to move to higher (larger) orbits and lose energy to move to lower orbits.

  • Quantization: Transitions must occur exactly between discrete levels; electrons cannot exist between orbits.

  • Orbital Requirement: The orbital circumference must be an integer multiple of the electron's de Broglie wavelength to form standing waves.

  • Spectral Lines: Explains hydrogen's discrete spectral lines. Each $\lambda$ corresponds to specific energy differences ($\Delta E$). The model only works perfectly for hydrogen.

Emission and Absorption Spectra Detail

• Relationship Between Energy and Wavelength:

  • $E = hf$

  • $v = f \lambda \rightarrow f = \frac{v}{\lambda}$

  • $E = \frac{hc}{\lambda}$

  • Constants: $hc = 1.99 \times 10^{-25} \, Jm = 1240 \, eVnm$.

  • The greater the wavelength ($\lambda$), the lower the energy ($E$).

• Mechanisms:

  • Absorption: An electron absorbs a photon to move to a higher energy level.

  • Emission: An electron releases a photon to move to a lower energy level.

• Visualizing Spectra:

  • Emission Lines: Bright colored lines against a black background.

  • Absorption Lines: Black lines against a continuous rainbow spectrum (ROYGBV).

  • This is evidence of particle nature: one electron transition involves exactly one photon.

Detailed Blackbody Laws

• Basic Properties:

  • All objects emit radiation based on temperature.

  • Blackbodies absorb all incident radiation; the emitted light is a continuous spectrum independent of what was absorbed.

  • Heat relationship: The hotter the object, the shorter the peak wavelength ($\lambda_{max}$).

• Wien's Law:

  • $\lambda_{max} = \frac{b}{T}$

  • $b = 2.90 \times 10^{-3} \, mK$ (Wien's constant).

  • As temperature ($T$) increases, $\lambda_{max}$ decreases.

• Stefan-Boltzmann Law (Power of radiation):

  • $P = \sigma A T^4$

  • $A$: Surface area.

  • $\sigma$: Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, W / (m^{2} K^{4})$).

  • Power is directly proportional to area and proportional to temperature to the fourth power. If temperature doubles, power increases by a factor of $2^4 = 16$.

• Astronomical Application: Star temperatures are inferred from color (Red stars are cooler, blue stars are hotter).

The Photoelectric Effect and Work Function

• Ejection of Electrons:

  • Incident photons must have enough energy to overcome the material's work function ($\phi$).

  • Formula: $K_{max} = hf - \phi$

  • $K_{max}$ is the maximum kinetic energy of the ejected electron.

  • $\phi$ is a unique physical property of the specific metal.

• Experimental Determination:

  • Stopping Potential ($\Delta V$): The voltage required to stop the current of ejected electrons. $eV_{stop} = K_{max}$.

  • Graphing: In a plot of $K_{max}$ (or $V_{stop}$) vs. Frequency ($f$):

    • Slope: Equal to Planck's constant ($h$).

    • x-intercept: Threshold frequency ($f_0$), where $hf_0 = \phi$.

    • y-intercept: Negative of the work function ($-\phi$).

Compton Scattering

• The Process: A photon collides with a free electron. The collision is elastic, meaning both momentum and kinetic energy are conserved.

• The Effect: The incoming photon transfers some energy and momentum to the electron. The emerging (scattered) photon has lower energy, lower frequency, and a longer wavelength ($\lambda$).

• Calculations:

  • Conservation of Momentum: $p_{initial} = p_{final}$. Since momentum is a vector, x and y components must be solved separately.

  • Change in Wavelength: $\Delta \lambda = \frac{h}{m_e c} (1 - \cos(\theta))$

  • $\theta$ is the scattering angle of the photon.

Nuclear Physics: Fission, Fusion, and Decay

• Nuclear Forces: The nucleus is held together by the Strong Force, which overcomes the electrical repulsion between protons. It is extremely powerful but acts only over very short distances.

• Conservation Laws:

  • Energy ($E = mc^2$)

  • Mass-Energy Equivalence

  • Momentum

  • Charge

  • Nucleon number (Protons + Neutrons)

• Fission: A nucleus splits into smaller nuclei and subatomic particles (e.g., $U-235$ bombarded by a neutron).

• Fusion: Two smaller nuclei combine to form a larger nucleus (e.g., hydrogen fusion in stars).

• Mass Defect: In these reactions, "missing mass" ($\Delta m$) is converted into energy ($E$).

• Radioactive Decay Rates:

  • Half-life ($t_{1/2}$): Time for half of a sample to decay.

  • Decay Constant ($\lambda$): $\lambda = \frac{\ln(2)}{t_{1/2}}$.

  • Remaining Nuclei: $N = N_0 e^{-\lambda t}$.

Types of Radioactive Decay

• Alpha Decay (\alpha): The nucleus ejects an alpha particle (${}^4_2 He$), consisting of 2 protons and 2 neutrons. The atomic number decreases by 2, and the mass number decreases by 4.

• Beta-Minus Decay (\beta^-): A neutron changes into a proton. It emits an electron ($e^-$) and an antineutrino ($\bar{\nu}$). The atomic number increases by 1; the mass stays the same.

• Beta-Plus Decay (\beta^+): A proton changes into a neutron. It emits a positron ($e^+$) and a neutrino ($\nu$). The atomic number decreases by 1; the mass stays the same.

• Gamma Decay (\gamma): The nucleus emits a high-energy photon ($\gamma$) to reach a lower energy state. No change in atomic number or mass.

Questions & Discussion

• Q: Explain how an orbiting electron around a hydrogen nucleus supports the wave description.

  • A: Electrons can only exist in discrete energy levels because the orbit must accommodate an integer number of de Broglie standing waves.

• Q: Why do incandescent bulbs feel hot compared to LEDs?

  • A: Incandescent bulbs emit much of their energy in the infrared range (heat), whereas LEDs emit mostly in the visible range.

• Q: What will happen if a $14 \, eV$ photon hits an atom with an ionization energy of $12 \, eV$?

  • A: The electron will be ejected because the photon energy exceeds the ionization energy.

• Q: In a photoelectric experiment, what happens to current if we adjust voltage until it stops?

  • A: The current becomes zero at the stopping potential ($V_s$). This value allows for the calculation of $K_{max}$: $K_{max} = qV_s$.