Three short counting exercises.

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch Microstates, multiplicity and the statistical entropy

Problem

Three short counting exercises. ((a)) A row of

7

distinguishable two-state units, each fixed in place and showing up or down, all arrangements having the same energy. How many microstates has the macrostate with exactly

3

units up? ((b)) An Einstein solid of just

2

identical oscillators at fixed sites shares

q = 7

indistinguishable units of energy

h\nu

, either oscillator being free to hold any whole number of units. How many microstates has this macrostate? ((c)) A particle is confined to a line of length

a = 1.0\,\mathrm{\mu m}

with rigid ends, so its de Broglie standing waves fit whole half-wavelengths and the allowed momenta are

p_n = nh/2a

with

n = 1, 2, 3, \ldots

. How many of these states have momentum less than

1.0\times10^{-26}\,\mathrm{kg\,m\,s^{-1}}

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