Chapter 4. Quantum Mechanics Part 1 **
Recommended Article : 【Chemistry】 Chemistry Table of Contents
2. Introduction of Matter Waves
1.History of Light
⑴ Overview
① Wave Theory : The theory that light is a wave
② Corpuscular Theory : The theory that light is composed of particles
③ The history of light research can be described as the history of determining whether light is a wave or a particle
⑵ History up to Modern Times
① Aristotle (BC. 384-322) : When we see the world, something emitted from our eyes is reflected from objects and seen
② Hasan Ibn al Haytham (965-1040) : Claimed the similarity between the anatomical structure of the eye and the pinhole camera
③ Francesco Maria Grimaldi (Bologna) : Explained diffraction of light as particles in 1660
④ Huygens : Advocated the wave theory of light. Effectively explained reflection and refraction
○ Treatise on light (1690)
○ Thought light is a wave that travels through a medium called “aether”
⑤ Newton : Argued that light consists of particles called “corpuscles”
○ Opticks (1704)
⑥ Diffraction Experiment
⑦ Scattering Experiment
⑧ Thomas Young (1773-1829)
○ Double-slit interference experiment (1801-1803)
○ Measurement of light’s wavelength
⑨ Augustine Fresnel (1788-1827)
○ Advocated the wave nature of light
○ Developed methodology for diffraction based on Huygens’ principle (1818)
⑩ James Clerk Maxwell (1831-1879)
○ Formulated mathematical theories for electricity and magnetism into 4 Maxwell’s equations
○ Presented theory on electromagnetic wave propagation (1873)
○ Calculated the speed of electromagnetic waves and found it to be equal to the known speed of light
⑪ Heinrich Hertz (1857-1894) : Created and detected electromagnetic waves predicted by Maxwell (1887)
⑶ Photoelectric Effect : Adopted as evidence for the particle nature of light with Einstein’s interpretation
① Overview
○ Definition : Phenomenon where incident light collides with a metal plate, emitting photoelectrons
○ Work Function : Minimum energy required for the photoelectric effect to occur, essentially ionization energy
○ Threshold Frequency : Minimum frequency of light for the photoelectric effect to occur, h × threshold frequency = work function
○ Maximum Kinetic Energy of Photons : Incident light energy - work function
Figure. 1. Relationship between Voltage, Photoelectric Current, and Light Intensity]
② Interaction Between Matter and Light
Figure. 2. Energy Levels and Photoelectric Effect]
○ X = a. Y = b. Z = c
○ Photoelectrons aren’t emitted when illuminating X and Z separately. Emission occurs when illuminating Y
○ Energy of photon Y equals the sum of energies of photons X and Z
○ No emission of photoelectrons from P when both X and Z are illuminated
○ Reason : A photon interacting with an atom can at most interact with one
③ Applications
○ Charge-Coupled Device(CCD)
○ X-ray Photoelectron Spectroscopy (XPS)
⑷ Blackbody Radiation
① Definition : Phenomenon where all objects with energy emit light
② Blackbody : Object that absorbs all incident energy and emits all absorbed energy completely
③ Number of Wave Modes (number of modes)
○ Based on Standing Waves in the present
○ 1D Standing Waves : Depending on the length L of the string, there exist various standing waves with quantum numbers n (where n is a natural number)
○ 3D Standing Waves : Depending on the vector of quantum numbers (l, m, n), various waves (light in this case) exist
○ Mapping quantum numbers to orthogonal coordinates : Considering an octant since l, m, n are positive integers
○ Number of Wave Modes : If we consider a grid of points within a 1/8 sphere with radius p and center at the origin as N*(p),
○ Relationship between Number of Modes (N*) and Frequency (ν)
○ This equation doesn’t consider the possibility of two waves with opposite phases having the same quantum numbers
○ Conclusion : Given the volume V = L3, the number of modes per unit volume N = N* / V,
④ Rayleigh-Jeans Law
○ Overview : Analyzing blackbody radiation as waves should lead to an observed UV catastrophe
○ Average vibrational energy of a system : Allocates a degree of freedom of 2 for vibrations, unlike translational and rotational motion
○ Average emitted energy per unit volume at frequency ν
○ UV Catastrophe : Blackbody emits infinite energy as the wavelength approaches 0
○ Actually, near-zero wavelengths have vanishing intensity
⑤ Planck’s Law
○ Max Planck introduced particle nature and assumed E = hν for successful explanation in 1900
○ Energy of a single photon
○ Probability of having n photons with frequency ν : Probability of particles having a specific energy follows the exponential function of the [Maxwell-Boltzmann distribution](https://jb243.github.io/pages/1338#:~:text=%E2%91%B6-,%EB%A7%A5%EC%8A%A4%EC%9B%B0%2D%EB%B3%BC%EC%B8%A0%EB%A7%8C%20%EC%86%8D%EB%8F%84%20%EB%B6%84%ED%8F%AC,-(Maxwell%2DBoltzmann%20speed)
○ Average energy of the system
○ Average emitted energy per unit volume at frequency ν
○ Planck’s Curve : Distribution of emitted radiant energy from a blackbody based on wavelength. The distribution only depends on temperature
Figure. 3. Planck’s Curve
○ Total energy per unit volume of the system
○ Photon flux
⑥ Stefan-Boltzmann Law : The energy emitted by a blackbody per unit area per unit time is proportional to the fourth power of the blackbody’s absolute temperature T(K)
○ In real objects, the equation is sometimes multiplied by the reflectance ε
○ σ : Stefan-Boltzmann constant, 8.22 × 10-11
⑦ Wien’s Displacement Law : The wavelength λmax (μm) at which the maximum radiated energy occurs is inversely proportional to the absolute temperature T(K) of the blackbody
○ α : Wien’s constant, 2.89 × 103
⑧ Pauli Exclusion Principle
○ Definition : No more than one electron with identical quantum numbers can exist in the same orbital
○ Why the Planck Curve appears as a continuous graph
○ When many atoms gather, energy levels slightly shift, causing energy levels to appear continuously
Figure. 4. Splitting of Energy Levels due to Orbital Overlap
Figure. 5. Formation of Energy Bands According to Orbital Overlap
⑸ Compton Scattering
① Phenomenon where a stationary electron and a photon undergo elastic collision when light is incident on the electron.
② Evidence of the particle nature of light.
⑹ Wave Nature of Electrons
① Davisson-Germer Experiment : Diffraction observed when electron beam is incident on a nickel crystal.
② Thomson’s Electron Scattering Experiment (1925)
○ Obtained diffraction pattern of electrons similar to X-ray diffraction when electron beam is incident on a metal foil.
Figure. 6. Thomson’s Electron Scattering Experiment
Left shows X-ray diffraction pattern, right shows electron diffraction pattern.
○ Conclusion : Experimentally proved that electrons, previously thought of as particles, can undergo diffraction.
○ Inference : If electrons have wave properties, their precise trajectories cannot be determined.
2. Introduction of Matter Waves (1925)
⑴ Assumptions
① Proposed by de Broglie.
② All objects with momentum possess wave-like properties.
⑵ Similarity with Photon Equation
① Relativity Theory and Photon Equation
② Quantum Mechanics and Photon Equation
③ Final Equation
⑶ de Broglie Matter Wave Equation
3. Bohr’s Atomic Model
⑴ Principle 1. de Broglie Matter Waves, Stationary Wave Conditions
① Principle 1-1. Coulomb’s Law for Electrons
② Principle 1-2. Electrons satisfy de Broglie’s Matter Wave Equation
③ Principle 1-3. Stationary Wave Conditions : Electrons move around the nucleus (incorrect assumption), n-th energy orbit is a multiple of the wavelength.
○ Fundamental cause of discontinuity (quantization)
○ Example
Figure. 7. Stationary Wave Conditions for n = 2 (a) and n = 3 (b)
④ Premises
○ Z : Nuclear charge. e : Charge of electron. k : Coulomb’s constant.
○ Hydrogen-like atom : Atom with only one electron. The number of protons may not be 1.
⑤ Velocity
⑥ Radius : Proportional to n squared.
⑦ Momentum
⑧ Energy Level
⑨ Rydberg’s Constant (R∞)
⑩ (Footnote) Energy level equation reveals useful relationships that can ease memorization.
⑪ Significance : Clarification of previously known quantized atomic spectra (values closely match)
⑫ Limitations
○ Not well-suited for multi-electron atoms other than hydrogen.
○ Theoretical prediction of atomic collapse as electrons lose energy.
⑵ Principle 2. Vibrational Conditions : Energy is absorbed or emitted when electrons transition between energy levels.
Figure. 8. Spectrum of Hydrogen Gas
① Rydberg Formula
② Lyman Series
○ Emission Lines : Emitting ultraviolet radiation when transitioning from n > 1 energy level to n = 1.
○ Absorption Lines : Absorbing ultraviolet radiation when transitioning from n = 1 to n > 1.
○ These emitted or absorbed lines are called the Lyman series.
③ Balmer Series
○ Emission Lines : Emitting visible light when transitioning from n > 2 energy level to n = 2.
○ Absorption Lines : Absorbing visible light when transitioning from n = 2 to n > 2.
○ These emitted or absorbed lines are called the Balmer series.
④ Paschen Series : Also known as the Bohr series.
○ Emission Lines : Emitting infrared radiation when transitioning from n > 3 energy level to n = 3.
○ Absorption Lines : Absorbing infrared radiation when transitioning from n = 3 to n > 3.
○ These emitted or absorbed lines are called the Paschen series.
⑤ Brackett Series
○ Emission Lines : Emitting when transitioning from n > 4 energy level to n = 4.
○ Absorption Lines : Absorbing when transitioning from n = 4 to n > 4.
○ These emitted or absorbed lines are called the Brackett series.
⑦ Pfund Series
○ Emission Lines : Emitting when transitioning from n > 5 energy level to n = 5.
○ Absorption Lines : Absorbing when transitioning from n = 5 to n > 5.
○ These emitted or absorbed lines are called the Pfund series.
⑧ Humphrey Series
○ Emission Lines : Emitting when transitioning from n > 6 energy level to n = 6.
○ Absorption Lines : Absorbing when transitioning from n = 6 to n > 6.
○ These emitted or absorbed lines are called the Humphrey series.
⑶ Principle 3. Limitations of Electron Transitions
① Law of Spin Conservation : Spin of an electron remains unchanged during transition.
② Selection Rules : Change in angular quantum number Δℓ = ±1.
③ Example : 1s → 2p is allowed, but 1s → 2s is not allowed.
⑷ Application : Flame Test
① Overview
○ Established through alchemical studies in the Middle Ages.
○ Indirectly implies quantized energy levels of electrons.
② Flame test used as an elemental detection method.
○ Chemical species emitting visible light under electron transition limitations are restricted.
○ Not prominently distinctive due to the energy difference between ns and np orbitals.
③ Examples
Table. 1. Examples of Flame Tests
Figure. 9. Sodium Flame Test
⑸ Application : Laser
Figure. 10. Principles of Laser
Input: 2018.12.28 22:40
Modification: 2022.09.12 19:25