Download The Quantum Atom

1 The Quantum Atom
Vinay B Kamble
(Former Adviser, NCSTS & Former Director, Vigyan Prasar, New Delhi)
Email: [email protected]
The Plum Pudding
Most scientists of the nineteenth century accepted the idea that the chemical
elements consisted of atoms, but nothing was known about the atoms themselves. One
clue was the discovery that all atoms contained electrons. It implied that since electrons
carry negative charges, and atoms are neutral, positively charged matter of some kind
must be present in them. One suggestion made by the British physicist J. J. Thomson
(1856-1940), who had discovered the electron, in 1898 was that atoms are positively
charged lumps of matter with electrons embedded in them, that looked like raisins in a
fruit cake or plums embedded in a pudding (Figure 1).
Figure 1: Thomson Model of the atom
The electrons are attracted to the centre of the positive charge distribution and
repelled by one another according to the Coulomb’s law of electric interactions. The
normal state is attained when these two opposite sets of forces are in equilibrium. If an
atom is disturbed (excited) by a collision with another atom or a passing free electron,
its electrons would begin to vibrate about their equilibrium positions and emit a set of
characteristic frequencies, which should account for the observed atomic spectra. If
Thomson’s model is accepted, it should be possible to calculate by the methods of
classical mechanics the equilibrium distribution with a given number of electrons inside
the atom, and the sets of calculated characteristic vibration frequencies, which should
coincide with the observed line spectra of various elements. The spectra of two lightest
2 elements, hydrogen and helium, are shown in Figure 2. Thomson himself and his
students made such calculations. However, the results were quite disappointing.
Figure 2 : Simplicity of the hydrogen spectrum produced by the motion of one electron,
and complexity and seeming lack of order produced by two electrons in the spectrum of
helium. Only the visible lines in the spectra are shown.
It became evident that some revolutionary change in the Thomson’s classical
model of the atom was required. This was the situation when a young Danish physicist
Niels Bohr (1884-1962) appeared on the scene. He was born in Copenhagen, and
spent most of his life there. He worked on the theory of passage of charged particles
through matter; and received his doctorate from the University of Copenhagen in 1911.
He then sailed to England to broaden his scientific horizon; and arrived at the
Cavendish Laboratory of Cambridge University to join the group working under its
director, J. J. Thomson. He argued that if the electromagnetic energy of light is
“quantized”, it should be reasonable to assume that the mechanical energy of atomic
electrons is quantized too!
What we mean by quantization of light energy (or electromagnetic radiation) is
that the energy is restricted to definite amounts or packets of energy called the quanta
of light. As evidenced by the work of Max Planck (1858-1947) on quantum theory of
radiation, these quanta can assume only a discreet set of values with energies given by
hν, 2hν, 3hν, etc. Here, ’h’ is a constant called the Planck’s constant and ‘ν’ (the Greek
letter nu but pronounced as ’new’!) is the frequency of light. Intermediate values of
energy are forbidden. Bohr felt it odd that atomic systems built according to classical
mechanics of Newton (as Thomson’s model was!) should emit or absorb light in the
form of Planck’s quanta, as evidenced by their discreet spectra as in Figure 2! Certainly
this did not fit into the framework of classical physics!
Shooting with alpha bullets
It turned out that J. J. Thomson did not appreciate the revolutionary ideas of the
young Niels Bohr, and there followed a number of sharp arguments at times between
the two! Bohr therefore decided to spend the rest of his foreign fellowship at a place
3 where his still vague ideas would meet with less opposition. This is how he landed up at
Manchester were the chair of physics was held by Ernest Rutherford (1871-1937), a
New Zealander and former student of J. J. Thomson. Rutherford was a farmer’s son,
and as the story goes, he was on his family’s farm digging potatoes when he learned
that he had won a scholarship for graduate study at Cambridge. “This is the last potato I
will ever dig,” he said, throwing down the spade! Thirteen years later, in 1908,
Rutherford, although a physicist, received the Nobel Prize for his investigations in
chemistry of radioactive substances! In his speech, accepting the Nobel Prize, he noted
that he had observed many transformations in his work with radioactivity, but had never
seen one as rapid as his own, from physicist to chemist!
When Bohr arrived in Manchester, Rutherford was in the midst of his epochmaking studies of the internal structure of atoms. He was shooting at the atoms with
high energy alpha particles emitted by radioactive elements. In earlier studies, carried
out at the McGill University, in Canada, Rutherford had proved that alpha particles
emitted by radioactive elements were nothing but positively charged helium atoms.
Indeed, they are doubly ionised atoms of helium moving with high velocities. Emission
of alpha particles from unstable heavy atoms was often followed by emission of
electrons (Beta particles) and Gamma rays (which are high frequency electromagnetic
radiation like X-rays but much shorter wavelengths than X-rays, and more energetic
than X-rays). Back to fruit cake or plum pudding! Now, the most direct way to find out
what is inside a fruitcake or a plum pudding is to poke a finger into it! This is essentially
what Hans Geiger and Ernest Marsden, Rutherford’s students, did in 1911 at his
suggestion. They used as probes the fast alpha particles emitted by certain heavy
radioactive elements.
Figure 3: The Rutherford Scattering Experiment
4 Geiger and Marsden placed a sample of an alpha-emitting substance behind a
lead screen with a small hole in it (Figure 3), so that a narrow beam of alpha particles
was produced. This beam was directed at a thin gold foil about 10-6 m thick. A zinc
sulphide screen, which gives off visible flash of light when struck by an alpha particle,
was set on the other side of the foil with a microscope to see the flashes. It was
expected that the alpha particles would pass through the foil without hardly any
deflection, say typically less than 10. This is because according to Thomson’s model,
the electric charge inside an atom was assumed to be uniformly spread throughout its
volume. What they actually found was that although most of the alpha particles were
indeed not much deviated, only a few were scattered through very large angles. In fact,
some were even scattered in the backward direction! Rutherford remarked that “it was
incredible as if you fired a 15 inch shell at a piece of tissue paper and it came back and
hit you!”
Since alpha particles are relatively heavy, and those used in these
experiments had high speeds, and hence high kinetic energies, it was apparent that
powerful forces were required to cause such large deflections. This implied that an atom
is composed of a tiny nucleus in which a positive charge and nearly all its mass are
concentrated, with the electrons at some distance away. It was apparent that since most
alpha particles could go right through the gold foil, an atom must be largely an empty
space. When an alpha particle came near a nucleus, the intense electric field scattered
it through a large angle. The atomic electrons, being light and far away, do not affect the
alpha particles much. Comparing the relative scattering of alpha particles by different
elements provides a way to find nuclear charges of the atoms involved. All atoms of any
one element turned out to have the same nuclear charge, and it regularly varied from
element to element in the periodic table. Ordinary matter, thus, is mostly empty space. If
all the matter, electrons and nuclei, in our bodies could be packed closely together, we
would reduce to specks just visible with a microscope!
The Trouble with Classical Physics
We saw that the Rutherford model of the atom pictures a tiny, massive positively
charged nucleus surrounded by enough number of electrons at a relatively great
distance to render the atom electrically neutral as a whole. However, the electrons
cannot be stationary as a result of the electric force pulling them towards the nucleus. If
the electrons are in motion, dynamically stable orbits like those of the planets around
the Sun are possible, as shown for the hydrogen atom in Figure 4(A). This is a
straightforward application of Newton’s laws of motion and the Coulomb’s law of electric
force. This is in accord with the fact that atoms are stable. For sure, both the laws are
pillars of classical physics. However, it is not in accord with the electromagnetic theory also a pillar of classical physics. It predicts that an electric charge when accelerated
5 radiates energy in the form of electromagnetic waves, that is, electromagnetic radiation.
An electron pursuing a curved path is accelerated and hence should continuously lose
energy, spiralling into nucleus, just in a hundred-millionth of a second (Figure 4(B))!
Figure 4(A)
Figure 4(B)
Figure 4(A): Force balance in the hydrogen atom
Figure 4(B): An atomic electron should, classically, spiral rapidly into the nucleus as it
radiates energy due to acceleration
Rutherford’s model somewhat resembled the Solar System. After all, Coulomb’s
law of electric attraction is mathematically identical with Newton’s law of Universal
Gravitation; and strictly speaking, similar losses of energy are expected also in the case
of planets of the Solar System. According to Einstein’s General Theory of Relativity,
gravitating masses also emit “gravitational waves” or gravitational radiation, which take
away energy. However, due to the small value of the gravitational constant, planetary
energy losses through gravitational emission are extremely small. Indeed, since their
formation some 4.7 billion years ago, the planets could not have lost more than a few
per cent of their original energy.
But, atoms do not collapse! There was something amiss with classical physics!
Something revolutionary was required to explain the stability of the atoms!
Atomic Spectra
Let us begin from the beginning. We shall need to understand the situation that
existed when Bohr entered the scene. The electromagnetic radiation emitted by free
atoms is concentrated at a number of discreet wavelengths giving rise to discreet
6 spectra. Each of th
hese wavele
ength comp
ponents is called a lin
ne because
e of the line
e that
v
the sp
pectrum, orr when we take a pho
otograph. These
T
liness are
is seen when we view
o the slit th
hrough whicch the light of different frequencies passes from
really the images of
atoms of
o a particula
ar element,, when theyy come dow
wn from a higher
h
energy state to their
normal energy sta
ate. Each kind
k
of ato
oms has itss own cha
aracteristic spectrum. This
st importan
nce becausse it makess spectrosccopy a verry useful to
ool in
feature is of utmos
al analysis.
chemica
Fig
gure 5: Hyd
drogen Spe
ectrum: The
e Balmer Se
eries.
The spectru
T
um of hydrogen is relatively siimple. Thiss is not su
urprising, since
s
hydroge
en is the sim
mplest atom
m with just one electro
on. Figure 5 shows the visible pa
art of
the hydrrogen spec
ctrum. The four
f
lines in
n the visible
e spectrum
m (designate
ed by α thro
ough
δ) were the first observed
o
b Balmer. Notice how
by
w the liness crowd to
ogether as they
approacch the ioniz
zation limitt in the near-ultraviole
et part of the spectrrum. Once
e the
electron has left the
t
atom, it is in an
n unbound state and
d its energy is no lo
onger
quantize
ed. When such
s
electro
ons return to the atom
m, they posssess random amoun
nts of
kinetic energies
e
ov
ver and abo
ove the binding energy. This revveals itself as
a the radia
ation
at the short-wave
s
length end
d of the sp
pectrum kn
nown as th
he continuu
um radiation
n. In
Figure 5,
5 the wavelength is denoted in
n nanomete
ers (1 nm = 10-9 m).. We often use
Angström
m units too
o (1 Å = 10-110 m), thus 700 nm = 7000
7
Å.
We may no
W
otice that th
he spacing in wavelen
ngths betw
ween adjace
ent lines of the
spectrum
m continuo
ously decre
eases with
h decreasin
ng wavelen
ngths (thatt is, increa
asing
frequenccy) of the liines, and th
he series of
o lines convverges to th
he so-called series lim
mit at
3645.6 Ǻ.
Ǻ The sho
ort wavelen
ngth lines are
a hard to
o observe experimenta
e
ally becausse of
their close spacing
g, and beca
ause they are
a in ultravviolet. The obvious
o
reg
gularity tem
mpted
many to
o look for an
n empirical formula th
hat would re
epresent th
he waveleng
gth of the lines.
Such a formula was
w
discove
ered by J. Balmer (1825-1898),, a school teacher and a
mathem
matician, in 1885.
1
He fo
ound that th
he simple re
elation
7 λ = 3646 n 2 / (n 2 – 1)
(in Ǻ units)
where n = 3 for Hα , n= 4 for Hβ , etc, was able to predict the wavelength of the first nine
lines of the series, which were all that were known at that time, to better than one part in
1000.
This discovery led to a search of similar empirical formulas that would apply to
series of lines which can sometimes be identified in the complicated distribution of lines
that constitute the spectra of other elements. In 1890, J. R. Rydberg (1854-1919) found
it convenient to deal with the reciprocal of the wavelength of the lines Κ instead of
wavelengths λ. In terms of the reciprocal of wavelength K, the Balmer formula can be
written as
K = 1/λ = RH (1/2 2 – 1/ n2) m -1
where RH is the so-called Rydberg constant for hydrogen. From recent spectroscopic
data, its value is known to be
RH = 10967757.6 ± 1.2 m -1 .
[If we multiply both the sides of the equation K = 1 / λ by c = 3 x 108 m / s, the
speed of light, RH would be replaced by R, where R = c x RH ≈ 3.290 x 1015 s-1. This
convention is also in vogue in spectroscopy since it gives frequency ν instead of inverse
of wavelength. Incidentally, 1/λ is called the wave number in spectroscopy and it gives
the number of waves in unit length.]
Formulas of this type were found for a number of series. We may write the
Balmer formula as
K = 1/λ = RH (1/m 2 – 1/ n2).
We now know the existence of five series of lines in the hydrogen spectrum as
described below.
Lyman:
in ultraviolet with m=1, n= 2, 3, 4,……….
Balmer: in near ultraviolet and visible with m=2 and n= 3, 4, 5,………..
Paschen: in infrared with m= 3, n= 4, 5, 6,…………..
Brackett: in infrared with m= 4, n= 5, 6, 7,………..
8 Pfund:
in infrared with m= 5, n= 6, 7, 8,………
The Rydberg constant played an important role in the development of quantum
theory of the atom by Niels Bohr.
Enter Niels Bohr
Although, theoretically Rutherford’s atoms cannot exist for more than a hundredmillionth of a second, in reality they exist for eternity! So, what to do about atoms built
according to Rutherford’s model? This was the question that confronted Bohr upon his
arrival in Manchester. Contradictions of this kind between theoretical expectations and
the observational facts on the other hand, sometimes even common sense, are the
main factors responsible for the development of science. The unrealistic behaviour of
Rayleigh-Jeans formula for spectral distribution of black body radiation at short
wavelengths, then famous as the “Ultraviolet Catastrophe”, led Max Planck to develop
the novel idea of light quanta in 1900. And this is how Michelson’s failure to detect the
motion of the Earth through ether led Einstein to the development of the Theory of
Relativity in 1905.
The situation was no different as regards the quantization of the atom. As stated
earlier, Bohr had a feeling that if electromagnetic energy is quantized, say, as in the
case of light quanta, mechanical energy must be quantized too, may be in a different
way. When an excited atom emits a light quantum with energy hν, its mechanical
energy must decrease by the same amount. The atomic spectra consist of a series of
discrete sharply defined lines. Hence, the energy differences between various possible
states of an atom must also have sharply defined values, and so must the absolute
energies of these states themselves. This is how Bohr approached the problem.
9 Figure 6: Bohr’s explanation of the Rydberg rule
An atom always has some internal energy. When the atom has fallen to the
lowest possible level, none of it is available for emission of a light quantum. Let us call
this level, E1, the normal or the ground state of the atom. An atom can exist in this state
forever. Let E2, E3, E4, etc, be the possible higher energy values (levels) of the different
states. Now consider that atom is excited to an excited energy state with energy Em.
This can be done by subjecting the gas to a very high temperature, as in the
atmosphere of the sun where the violent thermal collisions between the atoms bring
them into excited states, or by passing a high tension electric discharge through a glass
tube filled with a rarefied gas. In the latter case, the atoms are excited by the impact of
electrons rushing through the tube from cathode (negative terminal) to anode (positive
terminal). When the atom returns to some lower energy state En, as shown in the
diagram, it liberates the energy in the form of a light quantum, which we may write as,
E m,n = Em – En = hνm – hνn
or
νm,n = (Em – En) / h
Here, νm,n indicates that this particular frequency in the spectrum corresponds to
the transition from the mth quantum state to the nth quantum state. What is its
consequence? As shown in Figure 6, suppose we observe two lines in the spectrum of
some element a transition from 6th quantum state to the 4th; and from 4th to the 3rd.
Now, it is possible that the transition could also occur straight from the 6th quantum
state to the 3rd. We can then write,
10 ν 6,3 = ν 6,4 + ν 4,3
When we observe the frequencies ν
observe the frequency ν 3,2 and write
5,2
and ν
5,3
, it also follows that we may
ν 3,2 = ν 5,3 - ν 5,2
Based on these considerations, Bohr was led to believe that the mechanical
quantum states of the atom must look like as shown in Figure 7; and that the transitions
from different excited states to specific lower states should produce the various
spectroscopic series such as Lyman, Balmer, Paschen, Brackett and Pfund described
earlier. Thus, his ’original’ model of hydrogen atom was as shown in Figure 7.
Figure 7: Bohr’s ‘original’ model of the hydrogen atom
Since the energy of the light quanta is equal to the energy difference between the
state from which it started and the state at which it ended, we can write the generalized
Balmer’s formula as
h ν m,n = Em – En = (- RH h / m 2) - (- RH h / n 2) .
We have slightly rearranged the two terms in the parenthesis, which represent
the energy levels Em and En. Why did we write them as negative quantities? The reason
is that one ascribes zero energy to the state of a system when all its parts are at infinite
distance from each other. In other words, if the energy is positive, the system becomes
unbound, that is, it cannot hold together. Planets moving around the Sun, or electrons
around a nucleus are a stable, and therefore a bound system; their total energy being
negative. To take it apart, an outside supply of energy would be required.
11 How did Bohr explain the energies of the different states of the hydrogen atom
given by the above formula?
The Quantum Atom
We are now all set to see how Niels Bohr developed a model which was in accurate
quantitative agreement with spectroscopic data for the hydrogen atom. It had a
particular attraction because the mathematics involved was very easy to understand. In
view of the circumstances we have outlined above, he was led to make the following
postulates:
Bohr’s Postulates
1. An electron in an atom moves in a circular orbit about the nucleus under the
influence of the Coulomb attraction between the electron and nucleus,
obeying the laws of classical mechanics.
2. Instead of the infinity of orbits which would be possible in classical
mechanics, it is only possible for an electron to move in an orbit for which its
orbital angular momentum L is an integral multiple of h / 2π, where h is
Planck’s constant.
3. Despite the fact that it is constantly accelerating, an electron moving in such
an allowed orbit does not radiate electromagnetic energy. Thus its total
energy remains constant.
4. Electromagnetic radiation is emitted if an electron, initially moving in an orbit
of total energy Em, discontinuously changes its motion so that it moves in an
orbit of total energy En. The frequency of the emitted radiation ν is equal to
the quantity (Em-En) divided by Planck’s constant h.
The first postulate bases Bohr’s model on the existence of the atomic nucleus.
The second postulate introduces quantization of the angular momentum, which is
nothing but the product of mass, velocity, and radius of the particle. In the case of
electron it would be given by L = mVr. The third postulate removes the problem of
stability of an electron moving in a circular orbit, due to the emission of the
electromagnetic radiation required of the electron in classical theory. In other words, this
particular feature of classical theory is not valid in the case of the atomic electron. For
sure, this postulate was based on the fact that atoms are observed by experiment to be
stable. The fourth postulate,
ν m,n = (Em – En) / h
12 is really just Einstein’s postulate that the frequency of a photon of electromagnetic
radiation is equal to the energy carried by the photon divided by Planck’s constant.
Mechanical Stability of the atom
For stability, the electric force must be balanced by the centrifugal force. Thus,
e 2 / r 2 = (mV 2 / r)
From where we get, V = e / √ (m r).
Quantization of the Angular Momentum
Now, if the force acting on a particle is in a radial direction, as in the present
case, the angular momentum L= mVr remains constant. According to the second
postulate, which is really the quantization condition, we have,
L = mVr = nh / 2π
n = 1, 2, 3......
If we solve for V and substitute in the above equation, we obtain for the radius r
the following relation.
r = [ (h 2 / (4π 2 e 2 m)] n2
n = 1, 2, 3,.........
V = nh / 2π m r =(e2 / r)
n = 1, 2, 3, ..........
and
So, what have we achieved? The application of angular momentum quantization
condition has restricted the possible circular orbits to those of radii given as above, and
are proportional to the square of the quantum number n. With known values of h, m and
e, the smallest orbit (n=1) turns out to be ≈ 0.5 Å, which is in good agreement
experimentally. We may interpret it as the radius of the hydrogen atom in the normal
state. Next, if we calculate the orbital velocity of the electron in the smallest orbit from
above, we find V = 2.2 x 106 m/sec, which is less than 1 per cent of the velocity of light,
and hence use of classical mechanics instead of relativistic mechanics is justified for
Bohr model.
Quantization of total energy of the atom
13 Next, let us calculate the total energy E of an atomic electron moving in one of
the allowed orbits. Total energy is the sum of the kinetic energy KE and the potential
energy PE of the electron. We have,
E = KE + PE = ½ mV2 -
e2 / r.
First term is the kinetic energy and the second term is the potential energy of the
electron moving in the orbit of radius r. The potential energy is negative because the
Coulomb force is attractive. Using the relation V = e / √ (m r) we calculated above, and
substituting it in the equation for E, we calculated above, we can write E as,
E= - ½ e2 / r
We are now ready for the final leap. Substitute the value of r obtained above, we
get,
E= - (2 π 2 m e4 / h 2) / n 2
n = 1, 2, 3,............
Thus, we find that the quantization of the angular momentum of the electron
leads to the quantization of its total energy. Energy of each level as evaluated from this
equation is shown in Figure 8. Note that the lowest, or most negative, allowed value of
total energy occurs for the smallest quantum number n=1. As n increases, the total
energy of the quantum state becomes less negative, with E approaching zero as n
approaches infinity. Since the state of the lowest energy is the most stable state for the
electron, we see that the normal state of the electron in a one-electron atom is the state
for which n=1.
Figure 8: Energy level diagram for the hydrogen atom.
14 Rydberg Constant
Let us now come back to Rydberg’s constant. When we calculate the frequency
radiation emitted when the electron makes a transition from moving in an orbit with
quantum number m discontinuously to an orbit with quantum number n, using Bohr’s
fourth postulate, we find that,
νm,n = (Em - En) / h = + (2 π 2 m e4 / h 3) [(1/ n 2) – (1/ m 2)]
In terms of wave number (reciprocal of wavelength) K = 1/ λ = ν /c, we get,
K = (2 π 2 m e4 / h 3) [(1/ n 2) – (1/ m 2)].
This is Balmer’s formula. We can at once recognise that the Rydberg constant R
= 2 π m e4 / h 3 ! When Bohr substituted into this expression the numerical values for
e, m e, and h, he obtained R = 3.289 x 1015 sec -1, in close agreement with the
experimentally obtained empirical value 3.290 x 1015 sec -1 . We have, for the Lyman
series, n =1; for the Balmer n =2; for the Paschen n = 3; for the Brackett n = 4; and for
the Pfund n = 5. These series are illustrated in terms of the energy level diagram in
Figure 9.
2
Figure 9: Five known series of the hydrogen spectrum
15 Elliptical orbits
Bohr’s paper on the quantization of the hydrogen atom was followed by that of a
German physicist, Arnold Sommerfeld. He tried to extend the Bohr’s ideas to elliptical
orbits. As in the case of the gravitational field of the Sun in which the planets move in
elliptical orbits, there is no reason why the electrons should move only in circular orbits,
and not in the elliptical orbits, as proposed by Bohr. In an elliptical motion, the motion of
a particle in a central field is characterized by two polar coordinates, its distance ‘r’ from
the centre of attraction, and its position angle ‘θ’, measured with respect to the major
axis of the ellipse. The value of r is maximum when θ is zero, decreases to its minimum
value at θ = π, and increases again to its maximum value at θ= 2π. In the case of
Bohr’s circular orbits, r remains constant while θ changes. Thus, the motion along
Sommerfeld’s elliptical orbits is characterized by two independent coordinates, r and θ.
The quantized elliptical orbit, therefore, ought to have two quantum numbers associated
with r and θ, which are denoted by nr and nθ , called the radial quantum number and the
azimuthal quantum number. According to Bohr’s quantum conditions for radial and
azimuthal components of mechanical action, both must be integral of h with nr and nθ.
Sommerfeld obtained for energy of the quantized elliptical motion the formula,
E nr, nθ = - Rh / (nr + nθ) 2
Figure 10: Some elliptical Bohr-Sommerfeld orbits. The nucleus is located at the
common focus of the ellipses indicated by the dot
For nr = 0, we get the Bohr’s circular orbit as a special case. If nr ≠ 0, we get
elliptical orbits with different degrees of ellipticity (Figure 10). However, what is
interesting is the fact that the energies of all the orbits corresponding to the same value
of nr + nθ is exactly the same, in spite of their different shapes. nr + nθ is usually denoted
by n, called the principal quantum number. However, when we treat the hydrogen atom
relativistically, we obtain somewhat different results, because the mass of the particle
16 increases with velocity according to theory of relativity. In elliptical motion, the velocity of
the electron varies for different points of the trajectory, and hence its mass also varies at
every point. In this case, the energies of different orbits corresponding to the same
principal quantum number n (but for different nr and nθ ) splits into several closely
located components. Hence, a single spectral line, resulting from the transitions
between two quantum levels characterized by two principal quantum numbers m and n,
splits into a number of components spaced closely. We need a spectral analyser with
very high dispersive power to observe this splitting, and this is why it came to be known
as the “fine structure” of the spectral lines. The frequency differences between the fine
structure components depend on the so called “fine structure constant” α given by α =
e2 / hc = 1 / 137. It is dimensionless, and its smallness accounts for the closeness of the
fine structure components.
A Seminal Achievement
In this article, we have tried to highlight the points in the development of what is
now referred to as the old quantum theory. In many respects, this theory was very
successful. It had some undesirable aspects and hence its own share of criticism. To
begin with, it tells us how to deal with periodic systems only; however, there are many
systems of physical interest that are non-periodic. It does tell us how to calculate the
energies of the allowed states of a system, and the frequency of the quanta emitted or
absorbed when a system makes transition between allowed states, however, it does not
tell us how to calculate the rate at which such transitions take place. Further, the theory
is applicable only to a one-electron atom. The alkali elements (Li, Na, K, Rb, Cs) can be
treated approximately since they are in many respects similar to a one-electron atoms.
The theory fails badly when applied to neutral He atom, which contains two electrons.
Finally, the entire theory somehow seems to lack coherence, appears ad hoc; and
intellectually unsatisfying.
True, some of the objections stated above are of very fundamental nature, and
much effort was directed towards developing a quantum theory that was free of such
objections. One must, however, admit that the old quantum theory was one of those
seminal achievements that transformed the scientific thought. However, a more general
approach to atomic phenomena was required. Such an approach was developed in
1925 and 1926 by Erwin Schrödinger, Werner Heisenberg, max Born, Paul Dirac, and
others under the apt name of quantum mechanics. The picture of atomic structure
provided by quantum mechanics is the antithesis of the picture, used in old quantum
theory, of electrons moving in well defined orbits. Nevertheless, the old quantum theory
is still often employed as a first approximation to the more accurate description of
quantum phenomena provided by quantum mechanics.
17 By the early 1930s, the application of quantum mechanics to problems involving
nuclei, atoms, molecules, and matter in solid state made it possible to understand a vast
body of data – “ large part of physics and the whole of chemistry”, according to Dirac.
Quantum mechanics has survived every experimental test thus far.
References
1. Inward Bound - Of matter and forces in the physical world by Abraham Pais,
Oxford university Press 1986
2. Introduction to Modern Physics by Richtmyer, Kennard and Lauritsen , Mcgrow
Hill 1955
3. Thirty Years that Shook Physics by George Gamow, Dover Publications 1966
4. Quamtum Physics by Robert Eisberg and Robert Resnick, Seconfd Edition, John
Wiley 1985
5. Concepts of Modern Physics by Arthur Beiser, Sixth Edition, Tata Mc Grow Hill
2005
6. The path to the quantum atom, John Heilbron, Nature 2013(498) 27 published on
06 June 2013
Note: This article is a tribute to Niels Bohr and the hundred years of his trilogy of
publications in The Philosophical Magazine on the quantum theory of the hydrogen
atom. I have benefited from several excellent sources while writing it; however, I would
like to mention two sources in particular - Inward Bound by Abraham Pais; and The
Thirty Years that Shook Physics by George Gamow, both brilliant physicists; and active
contributors to the quantum revolution. Most of the description and the figures
appearing in this article are taken from these two sources.
Author