Cosmology and Statistical Mechanics · BPhO 2012 #2

Physics · modern-physics

Problem

(a) In cosmology it is thought that the Universe is expanding and this results in an apparent velocity of recession

V_H

. This velocity is related to the distance from the observer R by the equation:

V_H = H_0 R

H_0

is Hubble's constant. A very distant galaxy at present (time

t_0

) is observed to be

R_0

away. How would you expect the distance of the galaxy to change with time? (b) It can be shown that in the case of a number of atoms with one electronic energy level

E_1

of value

\varepsilon

above the ground state, the number of electrons,

n_1

E_1

is related to the number in the ground state

n_0

by the equation:

\frac{n_1}{n_0} \approx e^{-\frac{\varepsilon}{kT}}

Where T is the absolute temperature and k is the Boltzman constant. Show that if the atom has two electronic levels

E_1

and

E_2

of energy

\varepsilon_1

and

\varepsilon_2

above the ground state then if

n_1 << n_0

and

n_2 << n_0

then:

\frac{n_2}{n_1} \approx e^{-\frac{\varepsilon_2 - \varepsilon_1}{kT}}

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