1. Distribution of 3 particles with 6 units of energy.
a. The Maxwell-Boltzmann distribution.
| Energy
State | 0 1 2 3 4 5 6 | Microstates |
| 1 | 2 0 0 0 0 0 1 | 3 |
| 2 | 1 1 0 0 0 1 0 | 6 |
| 3 | 1 0 1 0 1 0 0 | 6 |
| 4 | 1 0 0 2 0 0 0 | 3 |
| 5 | 0 2 0 0 1 0 0 | 3 |
| 6 | 0 0 3 0 0 0 0 | 1 |
| 7 | 0 1 1 1 0 0 0 | 6 |
| Energy | Avg occupation M-B | Avg occupation E-B | Avg occupation F-D |
| 0 | 0.75 | 0.714285714 | 0.833333333 |
| 1 | 0.642857143 | 0.571428571 | 0.666666667 |
| 2 | 0.535714286 | 0.714285714 | 0.333333333 |
| 3 | 0.428571429 | 0.428571429 | 0.5 |
| 4 | 0.321428571 | 0.285714286 | 0.333333333 |
| 5 | 0.214285714 | 0.142857143 | 0.166666667 |
| 6 | 0.107142857 | 0.142857143 | 0.166666667 |
b. Einstein-Bose and Fermi-Dirac
There are 7 macrostates for the Einstein-Bose and 6 for Fermi-Dirac.
The average populations are shown above.
2. Helium-neon laser
a. Wavelength 632.8 nm, then energy = hc/wavelength = 1240 eV nm/632.8
nm = 1.95954488 eV
b. delta w/w = sqrt(2*k*T/Mc^2)= sqrt(2*8.67E-5*300/18.62E9)= 1.67146E-06
c. For cubic cavity of L=.1m or V= 1E-3 m^2 and w=632.8nm, the number
of modes is
Nm=8*pi*V*(delta w/w)/w^3= 8*pi*1E-3*1.67E-6/(632.8E-9)^3=
1.65637E+11 modes
3. Compare water vapor molecule separation of 3 nm and ice at 0.3 nm with DeBroglie wavelength of water molecule.
DeBroglie wavelength = h/p=hc/pc=hc/sqrt(2*KE*mc^2)=hc/sqrt(3*k*T*mc^2)=
DeBroglie wavelength = 1240 eV nm/sqrt(3*8.67E-5 eV/K*273 K*16.77E9
eV) = 0.035933842 nm
So for the gas there is no question, and for the solid the separation
is 10x the deBroglie wavelength, so even in ice
the Maxwell-Boltzmann statistics would appear to be valid.