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Exercises for Chapter 1 21

orbital frequencyωgiven by eqn 1.4. Verify

that this equation follows from eqn 1.3.

(c) In the limit of large quantum numbers, the

quantum mechanical and classical expressions

give the same frequencyω. Show that equat-

ing the expressions in the previous parts yields

∆r=2(a 0 r)^1 /^2.

(d) The difference in the radii between two ad-

jacent orbits can be expressed as a difference

equation.^44 In this case ∆n=1and

∆r

∆n

∝r^1 /^2. (1.45)

This equation can be solved by assuming that

the radius varies as some powerxof the quan-

tum numbern, e.g. if one orbit is labelled

by an integernand the next byn+1, then

r = anx andr′ = a(n+1)x. Show that

∆r=axnx−^1 ∝nx/^2. Determine the powerx

and the constanta.

Comment. We have found eqn 1.8 from the cor-

respondence principle without considering angular

momentum. The allowed energy levels are easily

found from this equation as in Section 1.3. The re-

markable feature is that, although the form of the

equation was derived for high values of the prin-

cipal quantum number, the result works down to

n=1.

(1.13)Rydberg atoms

(a) Show that the energy of the transitions be-

tween two shells with principal quantum num-

bersnandn′=n+ 1 is proportional to 1/n^3

for largen.

(b) Calculate the frequency of the transition be-

tween then′ =51andn= 50 shells of a

neutral atom.

(c) What is the size of an atom in theseRydberg

states? Express your answer both in atomic

units and in metres.

Web site:

http://www.physics.ox.ac.uk/users/foot

This site has answers to some of the exercises, corrections and other supplementary information.

(^44) A difference equation is akin to a differential equation but without letting the differences become infinitesimal.