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4 Early atomic physics

charges of magnitudeeis characterised by the combination of constants

(^5) Older systems of units give more suc- e (^2) / 4 π 0. (^5) This leads to the following relation between the angular fre-
cinct equations without 4π 0 ;someof
this neatness can be retained by keep-
inge^2 / 4 π 0 grouped together.
quencyω=v/rand the radius:
ω^2 =
e^2 / 4 π 0
mer^3

. (1.4)

This is equivalent to Kepler’s laws for planetary orbits relating the square

of the period 2π/ωto the cube of the radius (as expected since all steps

have been purely classical mechanics). The total energy of an electron

in such an orbit is the sum of its kinetic and potential energies:

E=

1

2

mev^2 −

e^2 / 4 π 0

r

. (1.5)

Using eqn 1.3 we find that the kinetic energy has a magnitude equal

to half the potential energy (an example of the virial theorem). Taking

into account the opposite signs of kinetic and potential energy, we find

E=−

e^2 / 4 π 0

2 r

. (1.6)

This total energy is negative because the electron is bound to the proton

and energy must be supplied to remove it. To go further Bohr made the

following assumption.

Assumption I There are certain allowed orbits for which the electron

has a fixed energy. The electron loses energy only when it jumps between

the allowed orbits and the atom emits this energy as light of a given

wavelength.

That electrons in the allowed orbits do not radiate energy is contrary

to classical electrodynamics—a charged particle in circular motion un-

dergoes acceleration and hence radiates electromagnetic waves. Bohr’s

model does not explain why the electron does not radiate but simply

takes this as an assumption that turns out to agree with the experi-

mental data. We now need to determine which out of all the possible

classical orbits are the allowed ones. There are various ways of doing this

and we follow the standard method, used in many elementary texts, that

assumes quantisation of the angular momentum in integral multiples of


(Planck’s constant over 2π):

mevr=n
, (1.7)

wherenis an integer. Combining this with eqn 1.3 gives the radii of the

allowed orbits as

r=a 0 n^2 , (1.8)

where the Bohr radiusa 0 is given by

a 0 =

^2

(e^2 / 4 π 0 )me

. (1.9)

This is the natural unit of length in atomic physics. Equations 1.6 and

1.8 combine to give the famous Bohr formula:

E=−

e^2 / 4 π 0

2 a 0

1

n^2

. (1.10)