CHE 208 Quantum Chemistry – Comprehensive Study Notes

Exam format and scoring
- Textbooks commonly used: Quantum Chemistry (McQuarrie, 2nd ed.) and Physical Chemistry (Atkins).
- Exam style: MCQs; 4 possible answers; 1 mark for a correct answer, -1 for a wrong answer, 0 for choosing “I don’t know.” Final exam is MCQ.
- Important: No penalty for the “I don’t know” option.

Why is this material important?

20th-century physics showed wave and particle pictures are two manifestations of the same phenomena at the atomic scale.
Wave–particle duality marks the first step in formulating quantum mechanics.
Quantum theory provides the best conceptual and mathematical framework for describing atoms and molecules.
Core concepts: quantum theory, quantum mechanics, quantum chemistry.

How scientific theory develops and why we need new theories

All scientific theories rely on the scientific method: observation → hypothesis → experiment → test/verification.
When classical physics fails (e.g., at atomic scales), new theories are needed to explain observations that cannot be reconciled with old models.

Classical physics failures that motivated quantum theory

Blackbody radiation (ultraviolet catastrophe when treated classically).
Photoelectric effect.
Discrete atomic spectra.
Double-slit experiment (wave-like interference and particle-like detection).

Blackbody radiation, the ultraviolet catastrophe, and Planck’s solution

Blackbody: an ideal object that absorbs and emits all frequencies completely.
Classical Rayleigh-Jeans law predicts energy radiated ↑ without bound as frequency → large (ultraviolet).
Planck’s resolution: energy exchange occurs in discrete quanta. For an oscillator of frequency v, allowed energies are multiples of $h
u$.
Planck distribution (spectral energy density) accounts for observed blackbody spectra and eliminates the ultraviolet catastrophe:
The expression you provided is part of the Planck distribution formula for blackbody radiation. It fully reads as:
$B(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/(\lambda k_B T)} - 1}$
This formula describes the spectral energy density (or spectral radiance) of electromagnetic radiation emitted by a blackbody at a given wavelength $\lambda$ and absolute temperature $T$ . It was derived by Max Planck and was crucial because it correctly predicts the observed blackbody spectra at all wavelengths, resolving the 'ultraviolet catastrophe' predicted by classical physics. It introduced the revolutionary idea that energy is quantized, meaning it can only be exchanged in discrete packets called quanta (or photons in the case of light).
Here's what the variables represent:
- $B(\lambda, T)$ : The spectral radiance of the blackbody at wavelength $\lambda$ and temperature $T$
- $h$ : Planck's constant ( $6.626 \times 10^{-34}\ \text{J s}$ )
- $c$ : The speed of light in a vacuum
- $\lambda$ : Wavelength of the emitted radiation
- $k_B$ : Boltzmann constant
- $T$ : Absolute temperature of the blackbody
- Planck constant: $h \,=\, 6.626 \times 10^{-34}\ \, ext{J s}$
- Reduced Planck constant: $\hbar = \frac{h}{2\pi}$
Wien’s displacement law relates the wavelength of peak emission to temperature: $λ_{\text{max}} T = b\quad (b\text{ is Wien’s constant})$
Stefan–Boltzmann law: energy density ∝ $T^4$; classical approach (Rayleigh–Jeans) fails at short wavelengths; Planck’s law provides the correct dependence.
Planck’s hypothesis established energy quantization and the quantum picture of matter–radiation.

The quantum theory of light and matter: key milestones

The photoelectric effect (1905) showed that light has particle-like quanta (photons) with energy $E_{\text{photon}} = h\nu$ .
- Absorption of a photon can overcome the work function φ; the leftover energy becomes the kinetic energy of the emitted electron:
  $\,\frac{1}{2}mv^2 = h\nu - φ$
Einstein’s photon model and the quantization of energy levels explain the observed threshold behavior and the dependence on frequency, not intensity, for photoelectrons.
De Broglie’s hypothesis (1924) extended wave-like properties to matter: each particle with momentum $p = mv$ has a wavelength
$\ λ = \frac{h}{p}$
Double-slit experiment with single particles demonstrates wave-like interference for matter and supports wave-particle duality.
The wave nature of matter leads to the concept of wave packets: a localized particle corresponds to a superposition of waves with a range of momenta/phases that interfere constructively in a localized region.

Foundational postulates and mathematical framework (overview)

A quantum state $|\Psi\rangle$ can be expanded in the eigenbasis of any operator, where $ci = \langle ai|\Psi\rangle$ are probability amplitudes. The Born rule states that the probability of obtaining an eigenvalue $ai$ upon measurement is given by $|ci|^2$ .

Time evolution is governed by the Schrödinger equation (non-relativistic):
- Time-dependent: $i\hbar\frac{\partial}{\partial t}\,|\Psi(t)\rangle = \hat{H}|\Psi(t)\rangle,$ where $\hat{H} = \hat{T} + \hat{V}$ is the Hamiltonian.
- Time-independent form for stationary states: $\hat{H}|\Psi\rangle = E|\Psi\rangle.$
Observables are represented by Hermitian operators to ensure real eigenvalues; orthogonality and normalization of eigenfunctions follow from Hermiticity (Theorems 1 and 2).
Operators may not commute: non-commuting operators imply uncertainty relations and limits to simultaneous measurability (e.g., position and momentum): $[\hat{x}, \hat{p}] = i\hbar.$
The wavefunction $\psi(\mathbf{r},t)$ contains all information allowed by quantum mechanics; its modulus squared gives probability densities: $\rho(\mathbf{r},t) = |\psi(\mathbf{r},t)|^2.$ The wavefunction is generally complex and normalization ensures total probability equals 1.

Mathematical tools: wavefunctions, normalization, and eigenvalue theory

Normalization of a wavefunction in 1D: $\int_{-\infty}^{\infty} |\psi(x)|^2\,dx = 1.$ In 3D: $\int |\psi(\mathbf{r})|^2 \,d^3r = 1.$
Orthogonality of eigenfunctions: for a Hermitian operator \hat{A}, if \hat{A}|ai\rangle = ai|ai\rangle and \hat{A}|aj\rangle = aj|aj\rangle with $ai \neq aj$, then $\langle ai|aj\rangle = 0.$
Dirac notation basics (bra-ket): the matrix element of an operator \hat{A} between states is $\langle ai|\hat{A}|aj\rangle$.
Expectation value and probability: for a discrete spectrum, probability of $ai$ is $|\langle ai|\Psi\rangle|^2$; for a continuous variable, use probability density and integrals.

Operators and observables: linearity, Hermiticity, and commutators

Quantum-mechanical operators are linear: e.g., for any wavefunctions, the action of an operator on a sum is the sum of the actions.
Hermitian operators guarantee real eigenvalues (observable quantities) and orthogonal eigenfunctions for distinct eigenvalues.
Commutators determine simultaneous measurability: if two operators commute, their observables can be simultaneously known to arbitrary precision; otherwise, they obey uncertainty constraints.
Canonical commutation relation (prototype): $[\hat{x}, \hat{p}] = i\hbar.$
Consequences: momentum and kinetic energy operators have a specific commutation structure; position and momentum do not commute, explaining the Heisenberg uncertainty principle qualitatively.

The hydrogen atom and central potentials: angular momentum and spherical harmonics

In central potentials $V(r)$, use spherical coordinates (r, θ, φ).
The time-independent Schrödinger equation separates into radial and angular parts:
- Angular equation leads to angular momentum quantum numbers: $l = 0,1,2,…,n-1;\quad m_l = -l,-l+1,…,+l.$
- The angular solutions are spherical harmonics $Y_l^m(\theta,\phi).$
- The radial equation involves the radial function $R_{n,l}(r)$ and includes the angular momentum barrier with term $\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}$.
Orbital angular momentum: $\hat{L}^2|l,m\rangle = \hbar^2 l(l+1)|l,m\rangle,\quad \hat{L}_z|l,m\rangle = \hbar m|l,m\rangle.$ The eigenvalues of $\hat{L}^2$ are quantized, and the z-component has quantized values $m\hbar$.
Hydrogen energy levels (non-relativistic, single-electron, pointlike nucleus): $En = -\frac{m e^4}{2(4\pi\varepsilon0)^2\hbar^2}\frac{1}{n^2} \quad (n = 1,2,3,…)$
- Degeneracy for each $n$ is $n^2$ (taking into account all possible $l$ and $m_l$ values) unless spin is included/excluded.
Rydberg constant: $R = \frac{m e^4}{8\varepsilon_0^2 h^3 c}$ (commonly used in hydrogen spectroscopy).
Radial distribution function for hydrogen: $P(r)\,dr = r^2|R{n,l}(r)|^2\,dr.$ The radial part $R{n,l}(r)$ can be expressed in terms of associated Laguerre polynomials and an exponential decay; the Bohr radius $a_0$ appears as a natural length scale when the nucleus is treated as infinitely heavy.
Energy degeneracy of hydrogen: for a given $n$, energy depends only on $n$ (not on $l$ or $m_l$) in the non-relativistic Schrödinger description; spin and fine structure lift degeneracy in more complete treatments.

Particle-in-a-box: 1D, 2D, and 3D quantization

1D box (infinite potential well): particle of mass $m$ confined to $0 \le x \le L$ with $V(x)=0$ inside and $V=\infty$ at the walls.
- General solution inside: waves $e^{ikx}$ and $e^{-ikx}$; boundary conditions give standing waves.
- Normalized eigenfunctions: $\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right),\quad n=1,2,3,…$
- Energies: $E_n = \frac{\hbar^2 k^2}{2m} =\frac{\hbar^2}{2m}\left(\frac{n\pi}{L}\right)^2 = \frac{n^2 h^2}{8mL^2}.$
- Quantum number $n$ labels the discrete energy levels; number of nodes $n-1$.
2D box: separable into two 1D problems; energies add: $E{nx,ny} = \frac{\hbar^2\pi^2}{2m} \left(\frac{nx^2}{Lx^2} + \frac{ny^2}{Ly^2}\right).$ Degeneracy arises when different pairs $(nx,n_y)$ yield the same sum.
3D box: similarly separable; energies depend on three quantum numbers $(nx,ny,n_z)$. Degeneracy generalizes with multiple states sharing the same energy.
Boundary conditions enforce that only certain wavefunctions are acceptable, which is the origin of quantization in a finite region.

Free motion, translational states, and tunneling

Free particle in 1D: potential $V(x)=0$ everywhere; general solution is a superposition of plane waves $e^{ikx}$ and $e^{-ikx}$ with continuous energy spectrum $E = rac{\hbar^2 k^2}{2m}$; not quantized because the box is infinite.
Wavefunction normalization for continuous spectra uses delta-function normalization; physical interpretation relies on wave packets and finite regions.
Tunneling: a particle can penetrate and traverse a classically forbidden region due to nonzero wavefunction inside the barrier, a purely quantum effect described by the Schrödinger equation in multiple dimensions.

Time evolution, superposition, and measurement

Time evolution preserves linearity: a linear combination of solutions is also a solution (superposition principle).
Stationary states have time dependence only through a phase factor $e^{-iEt/\,\hbar}$; probability densities are time-independent for stationary states.
Superposition of stationary states yields time-dependent probability densities and expectation values.
Measurement and collapse: measurement of an observable yields eigenvalues with probabilities determined by projection onto the eigenbasis; after measurement, the system collapses to the measured eigenstate.
Expectation value and uncertainty:
- Expectation value: $\langle Q\rangle = \int \psi^*(\mathbf{r},t)\hat{Q}\psi(\mathbf{r},t)\,d^3r.$ If the state is a superposition, this is the probability-weighted average.
- Uncertainty (standard deviation) arises when the state is not an eigenstate of the observable.

Wavefunctions, normalization, and the role of the wave packet

The wavefunction contains all information about the system compatible with quantum mechanics for a given state; its modulus squared yields probability densities.
Normalization ensures total probability is 1:
$\int |\psi|^2 \,d^3r = 1.$
The wavefunction can be expressed as a superposition of eigenfunctions of any complete set of commuting observables (e.g., energy eigenfunctions, angular momentum eigenfunctions).
For a 1D particle in a box, the wavefunctions are sine functions, and normalization constant is $N=\sqrt{2/L}$; energies are discrete due to boundary conditions.

The hydrogen atom in detail (central potential theory)

The hydrogen problem uses spherical coordinates; the total wavefunction factors as
$\Psi{n\ell m}(r,\theta,\phi) = R{n\ell}(r)Y_{\,\ell}^{m}(\theta,\phi).$
Radial equation determines $R{n\ell}(r)$; angular equation yields spherical harmonics $Y\,\ell^{m}(\theta,\phi)$.
Principal quantum number $n = 1,2,3,…$; orbital angular momentum $\ell = 0,1,2,…,n-1$; magnetic quantum number $m = -\ell, -\ell+1, …, \ell$.
Energy spectrum depends primarily on $n$ (non-relativistic Schrödinger equation for a Coulomb potential):
$En = -\frac{m e^4}{2(4\pi\varepsilon0)^2 \hbar^2}\frac{1}{n^2}.$
Degeneracy for each $n$ is $n^2$ (including spin would increase degeneracy depending on coupling).
The angular part $Y\ ^m(\theta,\phi)$ are the eigenfunctions of $\hat{L}^2$ and $\hat{L}z$ with eigenvalues $\hbar^2 l(l+1)$ and $\hbar m$ respectively; the spherical harmonics are the angular eigenfunctions.
The radial distribution function $P(r)dr = r^2|R_{n\ell}(r)|^2dr$ shows where the electron is likely to be found; for hydrogen, radial functions exhibit nodes and exponential decay with increasing $r$.
Bohr radius $a_0$ often enters as a natural length scale for atomic orbitals.
Hydrogen spectral series (Balmer, Lyman, etc.) can be understood from the Rydberg formula.

The Planck distribution, Wien’s law, and the Planck constant in depth

Planck’s hypothesis quantizes energy, leading to correct blackbody radiation curves and the concept of energy quanta.
Planck distribution (spectral power per unit wavelength):
$B(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/(\lambda k_B T)} - 1}.$
The Planck distribution reduces to the Rayleigh–Jeans law in the long-wavelength limit (low energy per mode) and deviates at short wavelengths, preventing the ultraviolet catastrophe.
Planck’s constant numerical value: $h = 6.626\times 10^{-34}\ \text{J s};\quad \hbar =\frac{h}{2\pi} \approx 1.055\times 10^{-34}\ \text{J s}.$
Planck also connected to Boltzmann statistics in his derivation of energy quantization for oscillators.

Quantum concepts of energy and heat capacity (Einstein/quantization perspective)

Classical heat capacity fails at low temperatures: the classical prediction tends to a constant value (e.g., $CV \to 3R$ for certain models), whereas experiments show $CV \to 0$ as $T\to 0$.
Einstein model: each atom (oscillator) has quantized vibrational energy $En = n h\nu$; excitation probabilities follow Boltzmann distribution; at low $T$, few oscillators are excited, so $CV\to 0$.
High-temperature limit recovers classical Dulong–Petrov-like behavior ($C_V \approx 3R$ for monatomic solids), while at low temperatures quantum effects suppress vibrational contributions.
This quantum treatment of lattice vibrations was a key success of quantum theory beyond atomic spectra.

Wave–particle duality in experiments and the role of photons

Photoelectric effect demonstrates particle nature of light: photon energy $E_{ph} = h\nu$ drives electron emission; kinetic energy of emitted electrons depends on frequency, not intensity:
$\frac{1}{2}mv^2 = h\nu - φ,$ where φ is the work function of the material.
Wave picture predicts intensity-dependent kinetic energy, which is incorrect; photon picture resolves the discrepancy.
De Broglie relation connects particle momentum and wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$ for nonrelativistic particles.
The wave-packet view explains localization: a localized particle corresponds to a superposition of waves with different momenta, forming a packet that travels with a group velocity.

The Schrödinger equation: foundations and separability

Time-dependent Schrödinger equation (TDSE):
$i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t) = \hat{H}\psi(\mathbf{r},t).$
Time-independent Schrödinger equation (TISE) for stationary states:
$\hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r}).$
Postulate 4: General state can be expanded in eigenfunctions of any operator of interest; expansion coefficients are the Fourier-like amplitudes (Fourier coefficients) that give the probability distribution over eigenstates.
Postulate 5 (Born rule): probability of measuring eigenvalue $ai$ is $|\langle ai|\Psi\rangle|^2$; the probability density for continuous spectra is given by $|\psi(a)|^2$.
For separable problems, use the method of separation of variables to reduce partial differential equations to ordinary differential equations when the potential is separable.
The Hamiltonian for many-particle systems includes kinetic energy and interaction terms; in multi-particle QM, the eigenvalue problem becomes more complex but retains the same fundamental principles.

Separation of variables in spherical coordinates and angular momentum (recap)

For central potentials $V(r)$, use spherical coordinates; wavefunction factorizes as
$\Psi(r,\theta,\phi) = R(r) Y_{\ell}^{m}(\theta,\phi).$
Angular equation yields quantization of angular momentum, producing spherical harmonics $Y{\ell}^{m}(\theta,\phi)$ with $\hat{L}^2 Y{\ell}^{m} = \hbar^2 \ell(\ell+1)Y{\ell}^{m},\quad \hat{L}z Y{\ell}^{m} = \hbar m Y{\ell}^{m}.$
The separation constant is conventionally written as $\ell(\ell+1)$.
The full angular part contributes the familiar $2\ell+1$ degeneracy for a given $\ell$ due to $m = -\ell, …, \ell$.

Normalization, orthogonality, and Hermitian operators (Dirac/Born framework)

Hermitian operators guarantee real eigenvalues and orthogonal eigenfunctions for distinct eigenvalues.
Orthogonality/normalization in Dirac notation: $\langle\,\ell m|\ell' m'\rangle = \delta{\ell\ell'}\delta{mm'}.$
The spectral decomposition of an operator allows expansion of a state in its eigenbasis:
$|\Psi\rangle = \sumi ci|ai\rangle,\quad ci = \langle a_i|\Psi\rangle.$

Postulates and interpretation (condensed list)

Postulate 1: State vector $|\Psi\rangle$ or wavefunction $\psi(\mathbf{r},t)$; normalization to unity.
Postulate 2: Observables correspond to linear, Hermitian operators.
Postulate 3: Measurements yield eigenvalues; eigenfunctions form a basis; measurement collapses state to the eigenstate.
Postulate 4: States can be expanded in eigenfunctions of any observable; linear combination principle.
Postulate 5: Probabilities are given by squared magnitudes (Born rule); expectation values derived from the state.
Postulate on interpretation: The total energy operator (Hamiltonian) $\,\hat{H}$ governs dynamics; eigenvalues/eigenfunctions determine quantized observables.
The concept of stationary states: eigenfunctions of $\,\hat{H}$ with definite energy $E_n$; time evolution yields a simple phase factor.

Additional concepts essential for quantum chemistry

The Hamiltonian and kinetic energy operator:
$\hat{T} = -\frac{\hbar^2}{2m}\nabla^2,\quad \hat{H} = \hat{T} + \hat{V}.$
The Schrödinger equation in 3D is extended to many-electron systems; the Laplacian in spherical coordinates and the volume element $d^3r = r^2\sin\theta\,dr\,d\theta\,d\phi$ appear in integrals.
Boundary conditions and normalization ensure physically meaningful, finite wavefunctions.
Spin and the Stern–Gerlach experiment illustrate intrinsic angular momentum and spin quantization; spin-1/2 particles have two eigenstates (up/down) along a chosen axis.
Pauli exclusion principle applies to fermions: no two electrons can occupy the same quantum state simultaneously.
Angular momentum coupling and Zeeman effect: splitting of spectral lines in a magnetic field arises from coupling of spin and orbital angular momentum to the external field.

Practical examples and worked-idea prompts (from the transcript)

Example: Estimating the de Broglie wavelength
- Given an electron accelerated through a potential $V = 40\ \text{kV}$, the momentum is $p = \sqrt{2 m e V}$ and the wavelength is
  $\lambda = \frac{h}{p} = \frac{h}{\sqrt{2 m e V}}.$
- Substituting $m$ (electron mass) and constants yields around $\lambda \approx 6.1 \text{ pm}$ for 40 kV.
Exercise: Wavelengths in different systems (neutron at temperature, tennis ball) illustrate how quantum effects are tiny for macroscopic objects and large for tiny particles.
Planck distribution plots and finite-temperature spectra illustrate how quantum statistics diverge from classical predictions at short wavelengths or low temperatures.

Connections to foundational principles and real-world relevance

Quantum theory replaces deterministic Newtonian predictions with probabilistic outcomes governed by wavefunctions and operators.
Quantization explains many real-world phenomena: electronic structure of atoms, chemical bonding, spectral lines, molecular vibrations, and reaction dynamics.
The probabilistic interpretation (Born rule) underpins measurement outcomes in spectroscopy, imaging, and chemical reactivity.
The formalism (operators, eigenvalues, commutators) provides the mathematical toolkit used in computational chemistry, spectroscopy, and materials science.

Quick references to key formulas to memorize

Planck distribution (spectral energy density per wavelength):
$B(\lambda, T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{hc/(\lambda k_B T)} - 1}.$
Planck constant and reduced form:
$h = 6.626\times10^{-34}\ \text{J s},\quad \hbar = \frac{h}{2\pi}.$
de Broglie wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}.$
Time-dependent Schrödinger equation: $i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r},t) = \hat{H}\psi(\mathbf{r},t).$
Time-independent Schrödinger equation: $\hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r}).$
Hamiltonian for a particle in a potential: $\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(r).$
Angular momentum operators: $\hat{L}^2|\ell m\rangle = \hbar^2 \ell(\ell+1)|\ell m\rangle,\quad \hat{L}_z|\ell m\rangle = \hbar m|\ell m\rangle.$
Hydrogen energy levels: $En = -\frac{m e^4}{2 (4\pi\varepsilon0)^2 \hbar^2}\frac{1}{n^2}.$
Radial distribution: $P(r)dr = r^2|R_{n\ell}(r)|^2dr.$
Particle in a 1D box: $\psin(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right),\quad En = \frac{n^2h^2}{8mL^2}.$
Normalization: $\int |\psi|^2 d^3r = 1.$
Orthogonality: $\langle ai|aj\rangle = \delta_{ij}.$
Commutator and uncertainty (canonical): $[\hat{x},\hat{p}] = i\hbar.$
Expectation value: $\langle Q\rangle = \langle\Psi|\hat{Q}|\Psi\rangle.$
Spherical coordinates volume element: $d^3r = r^2\sin\theta\,dr\,d\theta\,d\phi.$
Spherical harmonics normalization and degeneracy: for each $\ell$, there are $(2\ell+1)$ states with $m=-\ell,…,\ell$.

Study tips and course guidance

Read actively and repeatedly; the material builds conceptually (the slides emphasize “READ” multiple times).
Work through exercises like estimating de Broglie wavelengths and analyzing Planck/blackbody problems to connect formulas with physical intuition.
Practice deriving the angular-part separation constant and the angular equation to reinforce the role of $\ell$ and $m$.
Familiarize yourself with the 1D, 2D, and 3D particle-in-a-box problems as foundational examples of quantization due to boundary conditions.
Remember the three big experimental pillars that motivated quantum theory: blackbody radiation, photoelectric effect, and discrete spectra.

Summary takeaway

Quantum chemistry rests on a small set of robust postulates: states, observables as Hermitian operators, measurement outcomes as eigenvalues with probabilities given by the state’s projections, and evolution via the Schrödinger equation.
Quantization arises from boundary conditions and intrinsic wave properties; degeneracy and selection rules follow from angular momentum algebra and symmetry.
The hydrogen atom serves as the canonical model for applying these principles, illustrating discrete energy levels, angular momentum quantization, and the role of spherical harmonics.
The wavefunction is a fundamental object whose squared magnitude yields probabilities; the formalism provides precise, testable predictions across spectroscopy, chemical bonding, and molecular dynamics.

References (conceptual alignment with lecture material)

Textbooks: McQuarrie’s Quantum Chemistry and Atkins’ Physical Chemistry (9+ edition references).
Key topics covered in the lecture: blackbody radiation, Planck’s law, photoelectric effect, de Broglie hypothesis, Schrödinger equation, Born interpretation, Hermitian operators, commutators, angular momentum, spherical harmonics, hydrogen atom, particle in a box, tunneling, spin, Pauli principle, and the foundational postulates of quantum mechanics.