Quantum Orbitals and Quantum Numbers

Exam logistics and preparation

Formula sheet: posted by end of today on Canvas in the same module as the practice exam; it is the last page of the exam and helps decide what to put on the periodic table. You do not need to bring the sheet to the exam.
Bring a calculator and the periodic table; if you forget the periodic table, the formulas and a periodic table are provided on the exam.
Wednesday class is optional.
Monday: SI review session for the exam; sign up early if you have academic accommodations (spots are limited).
Exam timing and format: the exam is on Wednesday at 08:15 PM. The exam uses a bubble sheet for answers; the worksheet involves showing work but does not count toward the exam score.
After the exam (Thursday): a worksheet based on the five questions that went most poorly the prior night; you must rewrite them and show work; this is for reflection and potential extra credit (not affecting exam score).
Sunday: homework deadline remains.
Monday start: Chapter 3 (orbitals and elements); those Monday topics won’t appear on Exam 1 but will appear on Exam 2.
Quick reminder: if you have questions about accommodations or the exam, ask ahead of time.

Light and matter have a dual nature; electrons behave as waves (wave-particle duality).
Four key calculational equations to be comfortable with:
- \lambda = \frac{h}{mv}
  Momentum and wavelength (De Broglie relation).
- E = hf
  Energy of a photon; relates to frequency.
- f = \frac{c}{\lambda}
  or equivalently
  E = \frac{hc}{\lambda}
  connecting energy, frequency, and wavelength.
- c = \lambda f
  Speed of light relation.
Momentum relation: p = mv
Heisenberg uncertainty principle: \Delta x\,\Delta p \ge \frac{h}{4\pi}
Wave function and probability:
- The wave function is (\psi(x,y,z)); the probability density is |\psi|^2.
- The probability is related to brightness of a wave; brighter corresponds to higher probability density.
From the wave function to observable quantities:
- The square of the wave function gives the probability density of finding the electron in a region.
- The radial probability distribution is
  P(r) = |\psi(r,\theta,\phi)|^2 \; 4\pi r^2
  (the factor (4\pi r^2) accounts for spherical shell volume).
Orbital concepts:
- Delocalization: electrons are spread out over a region (an orbital), not a fixed path like a planet.
- Orbital vs. orbit: an orbital is a volume/region; an orbit is a classical path.
- Nodes: places where the wave function changes sign and the probability density is zero; important for understanding orbital shapes.

The Schrödinger equation in stationary form involves:
- Kinetic energy term (second derivative)
- Potential energy term (V, e.g., electron–nucleus attraction in hydrogen; many-body interactions complicate this for larger atoms)
- Total energy E; wave function (\psi) yields information about energy and electron distribution.
For hydrogen-like systems, orbitals can be described by quantum numbers and are derived from solving the equation; for multi-electron atoms, exact solutions are not analytical and require approximations.
The wave function solutions give probability densities; the square of the wave function provides electron density information.

1s orbital: spherical, no nodes; probability highest near the nucleus and falls off with distance.
2s orbital: spherical, has 1 node (where probability is zero); larger spatial extent than 1s.
3s orbital: spherical, has 2 nodes; larger still.
2p orbitals: three orientations (p ax, py, pz); each has a nodal plane (for example, the pz orbital has a node in the xy plane).
3d orbitals: five shapes (dxy, dxz, dyz, dx^2−y^2, d_z^2); typically visualized as clover shapes or with z^2 lobes; nodes can be planes or conical surfaces depending on the orbital.
Radial nodes and angular nodes: total number of nodes roughly correlates with the principal quantum number n; the angular part (l) introduces angular nodes (planes) in certain orbitals.

Three core quantum numbers (the fourth, spin, is not covered yet):
- Principal quantum number: (n) (n = 1, 2, 3, …); defines energy level or shell.
- Azimuthal (orbital angular momentum) quantum number: (l) with values (0 \le l \le n-1).
- Magnetic quantum number: (ml) with values (-l \le ml \le l).
Orbital naming by (l):
- (l = 0) → s orbitals (spherical, 1 orbital per (n)).
- (l = 1) → p orbitals (3 orbitals per (n): px, py, p_z).
- (l = 2) → d orbitals (5 orbitals per (n): d{xy}, d{xz}, d{yz}, d{x^2−y^2}, d_{z^2}).
- (l = 3) → f orbitals (7 orbitals per (n)).
Number of orbitals in a subshell: N_{orbitals} = 2l + 1
For a given (n), allowed (l) values are 0 to (n-1); e.g.,
- (n=1): only (1s) ((l=0)).
- (n=2): (2s) and (2p) ((l=0,1)).
- (n=3): (3s), (3p), (3d) ((l=0,1,2)).
Naming conventions for orientation: p orbitals labeled by axis ((px, py, pz)); d orbitals labeled by pair of axes (e.g., (d{xy}, d{xz}, d{yz}, d{x^2−y^2}, d{z^2})).

Be able to identify orbital types and counts from quantum numbers; know the number of orbitals in a subshell and how many nodes a given (n, l) combination has.
Remember the physical meaning: probability density, radial distribution, and nodes influence where electrons are likely to be found.
The particle-in-a-box and boundary-condition concepts illustrate why mathematical solutions require certain continuity and boundary constraints.
For exam formatting: bubble sheet counts; show work on worksheets when required; worksheets are practice and may help with later exams.

Key relationships: p = mv, \quad \lambda = \frac{h}{p}, \quad E = hf, \quad c = \lambda f = \frac{c}{\lambda} h (check consistency of units)
Uncertainty: \Delta x\,\Delta p \ge \frac{h}{4\pi}
Proton–electron potential energy in hydrogen is simple; in multi-electron atoms, electron–electron repulsion complicates exact solutions.
Orbital shapes and counts: s (1 per (n)); p (3); d (5); f (7). Nodes count roughly = (n-1) for the orbital; p orbitals have angular nodes (planes).