More than one distinct quantum state of a system can have a particular value of

Question

More than one distinct quantum state of a system can have a particular value of energy. Here we remember to count all the states in figuring out the probability that a small system has a particular energy value.

The molecule can be in one of six states. It exchanges energy with a thermal reservoir of other molecules at a temperature T. State 1 has EA, states 2 and 3 each have energy EB = 3EA, and states 4,5, and 6 each have energy EC=5EA. Now to simplify the calculations, let's set the temperature so that kBT=EA.

What is the probability that the molecule has energy EB?
P(EB) =

Explanation / Answer

average KE of the molecule = 3/2 kB T , T is the tempearature

we have 1 state with EA = kBT - degeneracy g1 = 1

2 - states with EB = 3kBT - degeneracy g2=2

3 stats with Ec = 5kBT - degeneracy g3 =3

The total number of particles occupying any state with energy Ei is

ni = gi A exp(-Ei /kB T)

na /nb = g1 /g2 exp( EB -EA)/kBT = e2 /2

na /nc = g1 /g3 exp( EC -EA)/kBT = e4 /5

Probabaility finding the molecule in EB = nb /(na + nb + nc )

= 2 e-2 /(1+ 2 e-2 + 5e-4 )

= 0.199

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