Physics 3313 - Lecture 18

4/5/2010 13313 Andrew Brandt

Monday April 5, 2010Dr. Andrew Brandt

1. Take Home quiz on CH6 due2. HW7 due Weds. 4/7 3. HW8 (TBA) due Weds. 4/144. Test Monday 4/19 on Ch 7-9

Adding Angular Momenta

4/5/2010 3313 Andrew Brandt 2

| |l s j l s

• What is the total angularmomentumwhen I add thespin angular momentumof an electron to itsorbital angular momentum?•l=1 +s=1/2?•3/2?

(general for any l or s)

Total Angular Momentum• If j and mj 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

4/5/2010 33313 Andrew Brandt

Total Angular Momentum (J)

4/5/2010 3313 Andrew Brandt 4

Total Angular MomentumNo external magnetic field:• Only Jz can be known because the uncertainty principle forbids Jx or Jy

from being known at the same time as Jz.

4/5/2010 53313 Andrew Brandt

Total Angular MomentumWith an internal magnetic field:• will precess about .

4/5/2010 63313 Andrew Brandt

Total Angular Momentum• Now the selection rules for a single-electron atom become

– Δn = anything Δℓ = ±1– Δmj = 0, ±1 Δj = 0, ±1

• Hydrogen energy-level diagram for n = 2 and n = 3 with the spin-orbit splitting.

4/5/2010 73313 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.

4/5/2010 83313 Andrew Brandt

LS Coupling• The notation for a single-electron atom becomes

n2S+1 LJ

• 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.

4/5/2010 93313 Andrew Brandt

LS Coupling

4/5/2010 103313 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 .

4/5/2010 113313 Andrew Brandt

Examples

4/5/2010 3313 Andrew Brandt 12

• 9.1 Historical Overview• 9.5 Classical and Quantum Statistics• 9.2 Maxwell Velocity Distribution• 9.3 Equipartition Theorem• 9.4 Maxwell Speed Distribution• 9.6 Fermi-Dirac Statistics• 9.7 Bose-Einstein Statistics

CHAPTER 9Statistical PhysicsStatistical 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)4/5/2010 133313 Andrew Brandt

Historical OverviewBenjamin 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.

4/5/2010 143313 Andrew Brandt

Historical OverviewEinstein• Published a theory of Brownian motion, a theory that supported the

view that atoms are real.

Bohr• Developed atomic and quantum theory.

4/5/2010 153313 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.

4/5/2010 163313 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.

• FMB tells the relative probability that an energy state is occupied at a given temperature.

1( )kT

4/5/2010 173313 Andrew Brandt

9.2: Maxwell Velocity Distribution• There are six parameters—the position (x, y, z) and the velocity

(vx, vy, vz)—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

4/5/2010 183313 Andrew Brandt

Classical Distributions• Rewrite Maxwell speed distribution in terms of energy.

• For a monatomic gas the energy is all translational kinetic energy.

where

4/5/2010 193313 Andrew Brandt