University:
Boston University
Course:
CAS PY 521 | Electromagnetic Theory I
Academic year:

2022
Views:

55

Pages:

3
Author:

Tara Guerrero

CH101 - Quantum Numbers Handout Wavefunctions, Quantum Numbers, and the nature of electrons Electrons in atoms behave as waves (delocalized). Attempts to “measure” e–’s cause them to localize (particle). The picture in the book is the result of many electrons measured! 1 electron (b) = 1 dot in (a) The Wavefunction (𝜓) of an electron is the MATHEMATICAL function describing the electron wave. Just like sin(𝑥) describes a 1D wave, 𝜓 is a function of 𝑛, 𝑙, 𝑚! , 𝑚" : that describes the 3D wave that IS the electron. 𝑛, 𝑙, 𝑚! , 𝑚" are the quantum numbers that describe the wave! 1 Quantum numbers (n, l) n = total # of loops (or # nodes–1) (principal quantum number) ⟶ Energy of the wave: En = –(2.18 aJ)Z2/n2 ⟶ Size of the wave: radius = 52.9 pm n2/Z l = # of nodal planes (0, …, n–1) (azimuthal quantum number) ⟶ Wave of the shape (sphere, dumbbell, etc) 2 1 CH101 - Quantum Numbers Handout Quantum numbers (𝑚4 , 𝑚5 ) ml magnetic quantum number 0, ±1, … ±l (ml is about the orientation) (Check out falstad.com/qmatom to visualize them!) pz dz2 ml = 0: ml = ±1: px, py dxz dyz ml = ±2: dx2-y2 dxy ms spin quantum number -½, +½ Nothing is spinning!! This is an intrinsic property of the electron (no reason) – magnets have two orientations, so too here. 3 Electrons have a quantized magnetic moment § Stern and Gerlach using silver atoms (which have a single valence electron) § Silver atoms travelling through an uneven magnetic field are deflected up or down: (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result. § Electrons have a magnetic moment that can be aligned in only two opposite directions. § The magnetic moment was thought to be due to the electron spinning on its axis. We now know this is not the case (it would have to spin unphysically fast). Rather, it is now understood instead to be an intrinsic property of the electron. Nonetheless, the magnetic moment is referred to as the “spin” of the electron. 4 2 Common parlance and notation Electron: unique set of 4 QN’s (n, l, m l, m s) that describe a unique wave § Two electrons cannot have the same set of QNs (Pauli exclusion principle) § The electron IS the WAVE (e.g., 𝜓!" , 𝑚 " =+1/2) Orbital: the unique set of first three QNs for an electron in an atom (n, l, and m l) § There can be exactly two electrons with the same (n, l, and m l) § The electron isn’t IN in the orbital; it IS the orbital (e.g., 𝜓 #$!" ) Subshell: subset of the orbitals with the same energy and shape (n and l) § e.g., 2p subshell is 2px + 2py + 2pz; 4d subshell is 4dxy, 4dzy, 4dxz, 4dx2-y2, 4dz2 Shell: set of (degenerate, in H) orbitals (same energy, same n) 5 3