1. Physics 3313 - Lecture 18 4/5/20101 3313 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

2. Adding Angular Momenta 4/5/20103313 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)

3. 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 4/5/201033313 Andrew Brandt

4. Total Angular Momentum (J) 4/5/20103313 Andrew Brandt4

5. 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. 4/5/201053313 Andrew Brandt

6. Total Angular Momentum With an internal magnetic field: will precess about. 4/5/201063313 Andrew Brandt

7. 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. 4/5/201073313 Andrew Brandt

8. 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. 4/5/201083313 Andrew Brandt

9. 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. 4/5/201093313 Andrew Brandt

10. LS Coupling 4/5/2010103313 Andrew Brandt

11. 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. 4/5/2010113313 Andrew Brandt

12. Examples 4/5/20103313 Andrew Brandt12

13. 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 1933. 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) 4/5/2010133313 Andrew Brandt

14. 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. 4/5/2010143313 Andrew Brandt

15. 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. 4/5/2010153313 Andrew Brandt

16. 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. 4/5/2010163313 Andrew Brandt

17. 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. 4/5/2010173313 Andrew Brandt

18. 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 4/5/2010183313 Andrew Brandt

19. Classical Distributions Rewrite Maxwell speed distribution in terms of energy. For a monatomic gas the energy is all translational kinetic energy. where 4/5/2010193313 Andrew Brandt