Physics Lecture 18 4/5/ Andrew Brandt Monday April 5, 2010 Dr. Andrew Brandt 1.Take Home quiz on CH6 due 2.HW7 due Weds. 4/7 3.HW8 (TBA) due Weds. 4/14 4.Test Monday 4/19 on Ch 7-9

Adding Angular Momenta 4/5/ Andrew Brandt2 What is the total angular momentum when I add the spin angular momentum of an electron to its orbital angular momentum? l=1 +s=1/2? 3/2? (general for any l or s)

Total Angular Momentum If j and m j are quantum numbers for the single electron (hydrogen atom). Quantization of the magnitudes. The total angular momentum quantum number for a single electron can only have the values 4/5/ Andrew Brandt

Total Angular Momentum (J) 4/5/ Andrew Brandt4

Total Angular Momentum No external magnetic field: Only J z can be known because the uncertainty principle forbids J x or J y from being known at the same time as J z. 4/5/ Andrew Brandt

Total Angular Momentum With an internal magnetic field: will precess about. 4/5/ Andrew Brandt

Total Angular Momentum Now the selection rules for a single-electron atom become – Δn = anythingΔℓ = ±1 – Δmj = 0, ±1Δj = 0, ±1 Hydrogen energy-level diagram for n = 2 and n = 3 with the spin-orbit splitting. 4/5/ Andrew Brandt

LS Coupling This is used for most lighter atoms in general and for heavier atoms as well when the magnetic field is weak. If two electrons are in a single subshell, S = 0 or 1 depending on whether the spins are antiparallel or parallel. For given L, there are 2S + 1 values of J. For L > S, J goes from L − S to L + S. For L < S, there 2L + 1 possible J values. The value of 2S + 1 is the multiplicity of the state. 4/5/ Andrew Brandt

LS Coupling The notation for a single-electron atom becomes n 2S+1 L J The letters and numbers are called spectroscopic symbols. There are singlet states (S = 0) and triplet states (S = 1) for two electrons. There are separated energy levels according to whether they are S = 0 or 1. Allowed transitions have ΔS = 0. Non-allowed (forbidden) transitions are possible between singlet and triplet states with much lower probability. 4/5/ Andrew Brandt

LS Coupling 4/5/ Andrew Brandt

jj Coupling For the heavier elements the nuclear charge causes the spin-orbit interactions to be as strong as the force between the individual and. 4/5/ Andrew Brandt

Examples 4/5/ Andrew Brandt12

9.1Historical Overview 9.5Classical and Quantum Statistics 9.2Maxwell Velocity Distribution 9.3Equipartition Theorem 9.4Maxwell Speed Distribution 9.6Fermi-Dirac Statistics 9.7Bose-Einstein Statistics Statistical Physics CHAPTER 9 Statistical Physics Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906 by his own hand. Paul Ehrenfest, carrying on his work, died similarly in Now it is our turn to study statistical mechanics. Perhaps it will be wise to approach the subject cautiously. - David L. Goldstein (States of Matter, Mineola, New York: Dover, 1985) 4/5/ Andrew Brandt

Historical Overview Benjamin Thompson (Count Rumford) Put forward the idea of heat as merely the motion of individual particles in a substance. James Prescott Joule Demonstrated the mechanical equivalent of heat. James Clark Maxwell Brought the mathematical theories of probability and statistics to bear on the physical thermodynamics problems. Showed that distributions of an ideal gas can be used to derive the observed macroscopic phenomena. His electromagnetic theory succeeded to the statistical view of thermodynamics. 4/5/ Andrew Brandt

Historical Overview Einstein Published a theory of Brownian motion, a theory that supported the view that atoms are real. Bohr Developed atomic and quantum theory. 4/5/ Andrew Brandt

9.5: Classical and Quantum Statistics If molecules, atoms, or subatomic particles are in the liquid or solid state, the Pauli exclusion principle prevents two particles with identical wave functions from sharing the same space. There is no restriction on particle energies in classical physics. There are only certain energy values allowed in quantum systems. 4/5/ Andrew Brandt

Classical Distributions Boltzmann showed that the statistical factor exp(−βE) is a characteristic of any classical system where with k = Boltzmann constant and T is temperature in Kelvin Maxwell-Boltzmann factor for classical system: The energy distribution for classical system: n(E) dE = the number of particles with energies between E + dE. g(E)= the density of states, is the number of states available per unit energy range. F MB tells the relative probability that an energy state is occupied at a given temperature. 4/5/ Andrew Brandt

9.2: Maxwell Velocity Distribution There are six parameters—the position (x, y, z) and the velocity (v x, v y, v z )—per molecule to know the position and instantaneous velocity of an ideal gas. These parameters six-dimensional phase space The velocity components of the molecules are more important than positions, because the energy of a gas should depend only on the velocities. Define a velocity distribution function. = the probability of finding a particle with velocity between. where 4/5/ Andrew Brandt

Classical Distributions Rewrite Maxwell speed distribution in terms of energy. For a monatomic gas the energy is all translational kinetic energy. where 4/5/ Andrew Brandt