Module 1 Notes: History, Quantum Theory, Hydrogen Atom, Schrödinger Equation

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In-person Workshops and Course News

• In-person workshops start this week! Workshop problems are posted to Blackboard in the Module 1 folder. You do not need to turn in these workshops.
• HW 2 due Sunday at 5:00 pm in Aktiv. Will cover Lectures 4 and 5.
• New Information posted to “Student Resources” Folder in Blackboard:
  • Procedures and schedule for virtual make-up workshops on Thursday and Friday – these are offered in case you miss your regular workshop due to illness or other conflicts outside of your control
  • WSL Office hour schedule
  • UCS tutoring hours
  • Office hours this week
  • Prof. Knowles in Hutchison 457: Thursday 3-4 pm, Friday 11 am-12 pm
  • Prof. Matson in Hutchison B45: Thursday 2-3 pm, Friday, 12-1 pm

Module 1: Topic List

• History of Atomic Theory and discovery of subatomic particles
• Structure of nucleus
• Why we need quantum theory to understand Chemistry
• Key experiments that illustrate problems with classical physics
• Rules of quantum mechanics: Schrödinger equation, Uncertainty Principle
• Hydrogen atoms explained with quantum theory: wavefunctions, orbitals and energies
• Spin and the Pauli Principle
• Development of the Periodic Table
• Periodic Trends in properties of the elements

Learning Objectives

• Cite experimental evidence for the existence of atoms and nuclei
• Identify an element based on its atomic number and mass
• Determine when it is necessary to use quantum mechanics
• Cite experimental evidence for quantum theory
• Explain some of the mysteries of quantum mechanics to others
• Calculate spectra and ionization energies of hydrogen
• Recognize spatial patterns of orbitals
• Rationalize the relative energies of orbitals in polyelectronic atoms
• Determine electron configurations of atoms
• Rationalize the structure of the Periodic Table
• Understand periodic trends in ionization energy, electron affinity and electronegativity

Summary of Last Lecture: Quantum Implications

• Observations that “broke” classical physics and their quantum implications:
  • Blackbody radiation
  • Photoelectric effect
  • Line Spectra
  • Energy is quantized; light behaves as particles; atomic energy levels are quantized
• Key equations:
  • $E = n h \nu, \quad h = 6.626 \times 10^{-34} \ \text{J} \cdot \text{s}, \quad E_{\text{photon}} = h \nu = \frac{h c}{\lambda},$
  • $E_n = -\frac{2.18 \times 10^{-18} \ \text{J} \ Z^{2}}{n^{2}}.$
  • Note: The last expression is for hydrogen (Z = 1); for general Z: $E<em>n = -\frac{m e^{4} Z^{2}}{8 \varepsilon</em>{0}^{2} h^{2} n^{2}} = -2.18 \times 10^{-18} \ \text{J} \ \frac{Z^{2}}{n^{2}}.$

Origins of Bohr’s Model of the Atom

• Bohr posited that electrons orbit the nucleus at discrete, well-defined distances.
• Classical balance of forces:
  • $\frac{m v^{2}}{R} = \frac{Z e^{2}}{4 \pi \varepsilon_{0} R^{2}}$
• Total Energy = kinetic energy + potential energy:
  • $E = \frac{1}{2} m v^{2} - \frac{Z e^{2}}{4 \pi \varepsilon_{0} R}$
• Angular momentum quantization (Bohr’s postulate):
  • $m v R = n \frac{h}{2 \pi} = n \hbar$
• For Hydrogen, Z = 1
• Final expression for the quantized energy levels of a hydrogen-like atom:
  • $E<em>n = - \frac{m e^{4} Z^{2}}{8 \varepsilon</em>{0}^{2} h^{2}} \frac{1}{n^{2}}$
  • $= -2.18 \times 10^{-18} \ \text{J} \ \frac{Z^{2}}{n^{2}}$
• Orbit radius relation (Bohr radius concept):
  • $R<em>{n} = \frac{4 \pi \varepsilon</em>{0} \hbar^{2}}{m e^{2}} \frac{n^{2}}{Z}$
• Connection to de Broglie waves:
  • De Broglie wavelength: $\lambda = \frac{h}{m v}$
  • Standing-wave condition: $2 \pi R = n \lambda, \quad n = 1,2,3,\dots$
  • Equivalent expression for angular momentum: $m v R = n \frac{h}{2 \pi}$

De Broglie Wavelength and Bohr’s Model

• De Broglie relation:
  • $\lambda = \frac{h}{m v}$
• Circular standing-wave condition:
  • $2 \pi R = n \lambda, \quad n = 1,2,3,\dots$
• Equivalence to angular-momentum quantization:
  • $m v R = n \frac{h}{2 \pi}$
• The de Broglie picture provides a wave explanation for Bohr’s energy quantization

Practice: de Broglie wavelength problem

• Question: What is the de Broglie wavelength of an electron moving at velocity $v = 0.01 c$?
• How does this compare to the distance between the electron and nucleus in the ground-state of the hydrogen atom?
• Note: Ground-state radius given: $R_{n=1} = 5.3 \times 10^{-11} \ \, \text{m}$

Summary of Last Lecture (continued): Wave-Particle Duality

• Consequence: Electron Diffraction
• Davisson and Germer at Bell Labs observed an interference pattern consistent with Bragg’s formula
• Bragg formula:
  • $n = \text{order of diffraction}, \quad d = \text{spacing of atoms}, \quad \theta = \text{scattering angle}$
  • $n \lambda = 2 d \sin \theta$
• Classical expectations for double-slit: particles vs waves (Feynman reference)

Double-Slit in Electron Microscopy and Matter-Wave Wavelengths

• Electron microscope setup (one electron at a time)
• Question: What determines the wavelength of a particle?
• De Broglie wavelength connection to Bohr’s angular momentum quantization (see above)

De Broglie Wavelength of Matter and Bohr (Detailed)

• Reiteration of: $\lambda = \frac{h}{m v}$ and $2 \pi R = n \lambda, \quad m v R = n \frac{h}{2 \pi}$
• The wave nature of matter underpins atomic structure and bonding

Practice: de Broglie wavelength and hydrogen comparison (summary)

• Provided in the previous practice problem; links to how mass, velocity, and confinement determine wavelength and allowed orbits

Summary Observations: Wave-Particle Duality

• Four key observations:
  1) Blackbody radiation
  2) Photoelectric effect
  3) Line spectra
  4) Electron diffraction
• Core ideas:
  • Energy is quantized; light behaves as particles (photons)
  • Atomic energy levels are quantized
  • Matter behaves as waves
• Key relations (repeated):
  • $E = n h \nu, \quad h = 6.626 \times 10^{-34} \ \text{J} \cdot \text{s}$
  • $E_{\text{photon}} = h \nu = \frac{h c}{\lambda}$
  • $E_n = -2.18 \times 10^{-18} \ \text{J} \frac{Z^{2}}{n^{2}}$

Electron Diffraction (Davisson–Germer) and Bragg’s Formula

• Davisson and Germer demonstrated electron diffraction consistent with wave-like behavior of electrons
• Bragg’s formula: $n \lambda = 2 d \sin \theta$
• The observation that particles can display interference patterns supports wave-particle duality

Schrödinger Equation

• Schrödinger proposed an equation to describe electron waves (and generalized to other systems):
  • $\hat{H} \psi = E \psi$
• Hamiltonian operator is a set of mathematical operations defining total energy
  • $\hat{H} = \text{“Kinetic energy operator”} + \text{“Potential energy operator”}$
• Wavefunction is the object whose squared magnitude gives probabilities
• In this course, focus on the position of the electron: where are we most likely to find it within an atom?
• Key definitions:
  • $\hat{H} \psi = E \psi, \quad \hat{H} = -\frac{\hbar^{2}}{2 m} \nabla^{2} + V(\mathbf{r}), \quad \psi = \text{wavefunction}$

What does the wavefunction mean?

• Wavepacket: a cross between a wave and a particle; wavefunction for a free electron (V(x) = 0)
• For a 1D coordinate x:
  • $\Delta x \; \psi^{2}(x) \Delta x$ is the probability of finding the particle in [x, x+Δx]
• If a position measurement is made, the particle will “choose” a specific location
• Repeated measurements yield a histogram that maps out $\psi^{2}(x)$ over x
• In atoms or molecules, $\psi^{2}$ is proportional to the electron density; in 3D this corresponds to the electron density per unit volume

The Schrödinger Equation as a Wave Equation (1D Box Analogy)

• To understand standing waves, consider a particle in a box with walls separated by distance a
• For a standing wave to exist, the wave must have zero amplitude at the walls (boundary conditions)
• Nodes occur where the wave crosses zero between walls
• The allowed wavelengths satisfy
  • $\lambda = \frac{2 a}{n}, \quad n = 1,2,3,\dots$

Hydrogen Atom: Schrödinger Equation in 3D

• The hydrogen problem leads to standing waves on a sphere (spherical waves)
• These standing waves are characterized by multiple quantum numbers
• Note: The slide images are schematic cartoons; they illustrate the concept of standing waves on a circle or a sphere, not literal wavefunctions for hydrogen

Schrödinger Equation: 3D Formalism

• General form in 3D coordinates:
  • $\hat{H} \psi = E \psi, \quad \hat{H} = -\frac{\hbar^{2}}{2 m} \nabla^{2} + V(x,y,z)$
• In Cartesian coordinates, the kinetic energy operator expands as:
  • $-\frac{\hbar^{2}}{2 m} \left( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}} \right)$
• Potential energy operator simply multiplies by V(x,y,z): $\hat{V} \psi = V(x,y,z) \psi$

Hydrogen Atom Schrödinger Equation: Key Concepts

• Chemistry is largely governed by the Coulomb potential between nuclei and electrons
• For hydrogen (one electron, one proton):
  • Coulomb potential is spherically symmetric:
  • $V<em>r = -\frac{e^{2}}{4 \pi \varepsilon</em>{0} r}$
• The goal is to obtain wavefunctions (orbitals) and energies that satisfy the equation
• These orbitals provide the basis for all chemical bonds

Quantum Numbers and Orbitals

• Each orbital is defined by a set of integers known as quantum numbers (Table 6.1 in the textbook)
• Generally, there is one quantum number per spatial dimension (in the full treatment there are multiple quantum numbers for the hydrogen problem: n, l, ml, ms in the full solution)
• The hydrogen orbitals are the eigenfunctions of the Schrödinger equation with the Coulomb potential; in many chemistries courses these orbitals form the basis for bonding and electron configurations

Important Notes and Connections

• Experimental foundations link classical physics failures to quantum concepts: blackbody radiation, photoelectric effect, line spectra, electron diffraction; these motivate quantization and wave-particle duality
• De Broglie hypothesis connects particle and wave pictures: matter waves have wavelength $\lambda = \frac{h}{m v}$ which explains the Bohr quantization and leads to standing-wave interpretations of atomic orbitals
• The Schrödinger equation provides a quantitative framework to calculate allowed energies and spatial distributions of electrons in atoms, beyond Bohr’s simple model
• The Coulomb potential is central to most chemistry because it captures how charged particles (nuclei and electrons) influence each other’s energy; the spherical symmetry greatly simplifies the hydrogen problem and provides intuition for multi-electron atoms

Quick Reference: Key Equations

• Quantization of energy for photons and atoms:
  • $E = n h \nu, \quad h = 6.626 \times 10^{-34} \ \text{J} \cdot \text{s}$
  • $E_{\text{photon}} = h \nu = \frac{h c}{\lambda}$
  • Hydrogen-like energy levels: $E<em>n = - \frac{m e^{4} Z^{2}}{8 \varepsilon</em>{0}^{2} h^{2}} \frac{1}{n^{2}} = -2.18 \times 10^{-18} \ \text{J} \frac{Z^{2}}{n^{2}}$
• Bohr model relationships:
  • Centripetal-Coulomb balance: $\frac{m v^{2}}{R} = \frac{Z e^{2}}{4 \pi \varepsilon_{0} R^{2}}$
  • Kinetic + Potential energy: $E = \frac{1}{2} m v^{2} - \frac{Z e^{2}}{4 \pi \varepsilon_{0} R}$
  • Angular momentum quantization: $m v R = n \frac{h}{2 \pi} = n \hbar$
  • Energy levels (explicit): $E<em>n = - \frac{m e^{4} Z^{2}}{8 \varepsilon</em>{0}^{2} h^{2}} \frac{1}{n^{2}}$
• Bohr radius and radii:
  • $R<em>n = \frac{4 \pi \varepsilon</em>{0} \hbar^{2}}{m e^{2}} \frac{n^{2}}{Z}$
• De Broglie wavelength and standing waves:
  • $\lambda = \frac{h}{m v}$
  • $2 \pi R = n \lambda, \quad n = 1,2,3,\dots$
• Bohr–de Broglie connection:
  • $m v R = n \frac{h}{2 \pi}$
• Standing-wave in a 1D box:
  • $\lambda_n = \frac{2 a}{n}, \quad n = 1,2,3,\dots$
• Schrödinger equation in 1D/3D:
  • $\hat{H} \psi = E \psi, \quad \hat{H} = -\frac{\hbar^{2}}{2 m} \nabla^{2} + V(\mathbf{r})$
• Probability interpretation of the wavefunction:
  • $\Delta x \ \psi^{2}(x) \Delta x \approx \text{Probability of finding particle in } [x, x+\Delta x]$
• Hydrogenic Coulomb potential:
  • $V<em>r = -\frac{e^{2}}{4 \pi \varepsilon</em>{0} r}$

Practice and Review Prompts

• Describe an experiment where we need to think of electrons as waves and how that helps to rationalize the results.
• Describe an experiment where we need to think of light as particles and how that helps to rationalize the results.
• Define the de Broglie wavelength in both English and quantitatively. What is its significance? How does it make clear that we need quantum theory to understand chemistry?
• What was Bohr’s basic idea to explain the energies the electron is allowed to have? How does his theory explain why hydrogen atoms can only emit a few colors of light? How do we determine which colors?
• State the Uncertainty Principle. What are the practical implications?

Office Hours and Course Logistics (Bonus Reference)

• Prof. Knowles: Hutchison 457 — Thu 3–4 pm, Fri 11 am–12 pm
• Prof. Matson: Hutchison B45 — Thu 2–3 pm, Fri 12–1 pm
• WSL Office hour schedule and UCS tutoring hours (as posted in Blackboard)
• Quiz and practice problem reminders (e.g., Ionization Energy of hydrogen; links to notes and problem sets)