The light
emission through optical transitions in atoms (or molecules) is given here not
in the detailed quantum mechanical way. Nevertheless we can use half-classical
estimation in a common approximation. The vibrating dipole sends electromagnetic
waves in the classical electrodynamics. The dipole is the dipole moment
when describing in mathematical way:
![]() ![]() |
(charge · distance [C · m]) |
where
is the vector coming from (+)charge to a vibrating (-)charge. According to the
"Conversion table" we can transform classical quantity into quantum mechanical
operator (
→µ)
obtaining average of distribution
<µ>
= òy*()
(− e
)
y(
)
dV
for any optical transition.
[Atom or molecule has a diameter of about few Angströms;
the light in visual wavelength range
is about 5000 Å. According to bigger scale of wave compare with atomic or
molecular sizes the electrical force of the field doesn't change in an atomic or
molecular spatial range.]
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However what are the wavefunctions y*, y
? If we consider our atom with both energy levels En
and Em and corresponding wavefunctions jn
and jm then y()
is the time overlapping of these both functions in a
time- dependent form:
y(,t)
= 1/Ö2[jn(
)
e−iEnt/
h
+ jm()
e−iEmt/
h]
(Since light is an electromagnetic radiation that vibrates in time we must time t in our future assumptions; 1/Ö2 is only for normalization).
Average of distribution calculation <µmn(,t)>:
![]() |
|
= −e/2{ ∫jn* ![]() ![]() |
+ òjn* ![]() ![]() |
|¾¾¾¾¾¾¾¯¾¾¾¾¾¾¾|
time-independent |
|¾¾¾¾¾¾¾¾¾¾¾¾¯¾¾¾¾¾¾¾¾¾¾¾¾¾¾|
time-dependent |
The time-independent part is vanished for even and odd wavefunction
j since |j|² is even (|j()|²
= |j(-
)|²)
and multiplication by
and finally integration over the whole volume gives finally 0.
Using the following short form w = (Em−
En)/h we see dipole
moment average of distribution vibrates as classical dipole and at the same time
produces electromagnetic field:
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As we will see later only the squared |<µmn>|² is relevant for all rearrangments in experiment so that complex conjugate part gives the same contribution as the first part of the formula. Now we would like to introduce the following abbreviation:
Rmn=
òjm*
jm dV
which will finally give:
<µmn> = -e Rmn eiwt |
If the above-mentioned integral Rmn vanishes transition is forbidden (the forbidden electrical dipole transition). The calculation of dipole matrix elements plays key role in optical transition calculations.
Atom or molecule having been in the excited energy state Em
moves into lower energy state En emitting a light simultaneously.
An ensemble of excited atoms (or molecules) emits a light independently from
each other. The radiation energy
Smn over time or volume for these spontaneous transitions is as
follows:
Smn = | Nm | hνmn | Amn |
Energy flux density / Jm-3s-1 | | | Transition probability for spontaneous emission / s-1 |
· | Energy difference Em− En ≡ Energy of Photons / J | ||
Particle density in an excited state / m-3 |
The transition probability Amn [s-1] making quantum mechanical calculations and transforming some relations from electrodynamics into quantum mechanics will be given below. We have already found that energy flux density S is as follows (in electrodynamics):
S = 4/3. Nm/c³. 1/4peo. <|d²µ/dt²|²>t
where brackets <>t are for time mean value of time-variable quantity (here one can see the second time derivative of dipole moment). The quantum mechanical dipole moment average of distribution is <µmn> = −eRmneiwt from which we can easily obtain the quantum mean value by using 2-fold derivation:
<|d²µ/dt²|²>t = |e Rmn w² eiwt|² = (2π)4νmn4 e² |Rmn|²
® Smn = 4/3. Nm/c³. 1/4peo· 16π4 νmn4 e² |Rmn|²
Smn = 16π³/3. e²/eoc³. νmn4|Rmn|² · Nm |
Amn = 8π²e²/3 |
There are induced processes which are caused by light influence on a particle
along with spontaneous emission that is characterized by Amn and is
independent of external field of radiation.
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The corresponding coefficients Bnm for absorption
and Bmn for induced emission have been already
given. The relations between them
are as follows:
Amn/Bmn = 8πhν³/c³ | and | Bnm/Bmn = gm/gn |
where gm and gn show the degeneracy of
corresponding energy levels. If degeneracies are equal we will have:
Bnm = Bmn |
The first equation represents important ν³-ratio
between spontaneous and induced emission; the second equation shows that
probabilities are the same for induced emission and (induced) absorption.
Reader can find more about lasers in
"Spektrum der Wissenschaften": Anwendungen des Lasers, ISBN: 3-922508-47
The relation with Rmn:
Bmn = e²/6eo |
The transition moment can be also represented by the so-called oscillator
force
f (this name came from the classical way of consideration when free
vibrating in three dimensions electron having mass me
will possess an oscillator force f = 1):
fmn = 4p
nmnme/3 |
The oscillator force f is dimensionless. The intensive transitions have
typical oscillator forces laying in the range between 0,1 and 1. The forbidden
transitions lay lower f ≈10-5.
The transition moment µmn , Rmn and fmn are theoretical quantities; but what do we measure in experiments and how do we come to these theoretical quantities? The answer is given here!