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1
Quantization of Energy
1.1 INTRODUCTION
In ancient times men had guessed that there were people, animals, land, beyond which there existed a
world of the ultra small before it was actually discovered. Thinkers had meditated on the way nature
had produced the world out of something quite formless. How was it, they queried, that it came to be
inhabited by its great diversity of things. Nature might have worked like a builder that makes a large
house out of small stones. Then what are these stones? Is there no limit to this division and subdivision
of matter? Are there particles so small that even nature is no longer able to break them up? The answer
was YES, so said the ancient philosophers. These particles were given the name ‘atom’. Their chief
property was that no further division is possible. The word ‘atom’ is Greek, means ‘non-divisible’.
What did an atom look like? In those times this question remained unanswered but as more
experimental results accumulated, the understanding of atomic structure became more and more clear.
An atom is made up of three basic particles, electrons, protons and neutrons. Their names and charges
are as follows:
Particle
Mass
Charge
Proton (mp)
1.672 × 10–27 kg
1.602 × 10–19 C
Neutron (mn)
1.674 × 10–27 kg
No charge
Electron (me)
9.109 × 10–31 kg
–1.602 × 10–19 C
= 1/1836 mp
1.2 BOHR MODEL OF HYDROGEN ATOM
On heating a body it emits electromagnetic radiation and when whole of this radiant energy is absorbed
by another body, the system will be in thermal equilibrium. A body that absorbs all the radiation is called
black body. Radiation emitted by a body in equilibrium with matter at a particular temperature is called
black body radiation. At low temperatures, radiation of long wavelength are emitted. At higher temperature
the amount of energy emitted is much greater and its principle component lies in the infrared region. At
still higher temperature, black body glows dull red, the white and afterward blue and the total amount
of energy radiated increase dramatically. Classical mechanics could not account for this energy distribution
in black body radiation at different temperatures. Max Planck derived the correct expression on the
basis of Quantum Theory of Radiation. According to this theory:
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Elements of Molecular Spectroscopy
1. Radiant energy is emitted or absorbed discontinuously in the form of tiny bundles of energy
known as quanta.
2. Each quanta is associated with a definite amount of energy E depending upon the frequency
of radiation, the two being related by the equation E = hn where E is energy in joules, n is
frequency of radiation in reciprocal seconds (s–1) and h is a fundamental constant known as
Planck’s constant with a numerical value of 6.626 × 10–34Js.
3. A body can emit or absorb energy only in whole number multiples of quantum i.e., 1 hn,
2 hn, 3 hn… . Energy in fractions of quantum cannot be lost or absorbed. This is known as
quantization of energy. Based on these postulates, Planck obtained the following expression
for energy density of black body radiation
E(n)dn =
8πhν 3
c
3
×
dν
exp(hν/kT ) − 1
This equation adequately accounts for black body radiation intensity at all wavelengths obtained
at different temperatures.
An atom has a minute but massive positively charged body called nucleus at its centre. All the
protons and neutrons are contained in the nucleus. Since the mass of an atom is entirely due to the
presence of protons and neutrons, it is concluded that almost the entire mass resides in the nucleus.
The positive charge of the nucleus is also due to the presence of protons.
The electrons of atoms are distributed around the nucleus in a way about which we shall talk later
on. Since the mass of the electron is 1/1836 the mass of proton, it does not contribute anything to the
mass of the atom. The electrons carry charge equal but opposite to that of protons. As the atom is
neutral, the number of electrons must be equal to number of protons in the nucleus. Most of the space
around the nucleus is empty except for the presence of extremely minute electrons. However, as
electron movement around nucleus is very fast, they cover almost all the space around the nucleus, and
thus atom appears as a sphere.
Bohr (1913) adopted these ideas for the hydrogen atom, a quite different system, and tried to give
it a structure. He postulated Atomic Theory as:
(i) An atom consists of a nucleus (containing protons and neutrons) surrounded by revolving
electrons. The electrons move around the nucleus in a circular paths called orbits. These
orbits were numbered 1, 2 …. which turned out to be the principal quantum number of an
orbit.
(ii) The columbic force of attraction between the electrons and nucleus hold the atom together.
(iii) Even though an electron in a particular orbit is constantly accelerated, yet each orbit in an
atom has a discrete energy. Each orbit is called a stationary state. As long as an electron is in
a stationary state, it does not radiate any energy.
(iv) Mathematical condition determines the size of an orbit. It states that the angular momentum
of an electron is an integral multiple of h/2p, where h is Planck’s constant
i.e.,
mvr = nh/2p
...(1.1)
where n = 1, 2, 3…. and is called the Principal Quantum Number. These numbers happen to
be the same numbers as stated in the first postulate.
Quantization of Energy
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(v) Emission or absorption of radiation energy takes place only if an electron jumps from one
stationary state of energy E1 to another stationary state of energy E2. The frequency of this
radiation is given by
n = (E2 – E1)/h.
Using these postulates let us try to give a structure to an atom.
(i) Radius of an Atomic Orbit
Consider a nucleus with a positive charge + Ze and an electron with a negative charge – e revolving
around nucleus with a velocity v, in a circular orbit of radius r. There are two forces: electrostatic force
of attraction and centrifugal force, acting on an electron.
Centrifugal force =
mv 2
r
Ze 2
Electrostatic force of attraction =
4πε0 r 2
. In order that an electron continues to move in a
particular stationary orbit, these two forces should balance one another.
2
i.e.,
Ze
mv 2
=
4πε0 r 2
r
1 Ze 2
1 2
mv =
2 4πε 0 r
2
...(1.2)
According to the fourth postulate the angular momentum of an electron is an integral multiple of
h/2p.
i.e.,
or
mvr =
v=
nh
2π
...(1.1)
nh
2πmr
...(1.3)
Substituting the value of v from equation 1.3 into 1.2 gives
r=
n 2 h2 ε0
Ze 2 mπ
...(1.4)
For a hydrogen atom in its ground state Z = 1, n =1 so that its radius is
r=
(6.626 × 10 −34 Js) 2 (8.85 × 10 −12 CNm −2 )
(1.672 × 10 −10 C) 2 (9.101 × 10 −11 kg )(3.141)
= 5.29 × 10–11 m = 52.9 pm
This value is in very good agreement with the experimental value and was the first achievement of
the Bohr model. From equation 1.4 the radius of the nth orbit can also be calculated as
...(1.5)
rn = n2 r1 = 52.9 n2 pm
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Elements of Molecular Spectroscopy
(ii) Velocity of Electron
Substituting the value of r from equation 1.4 in equation 1.2, we get
Z 2e4
v2
or
= 4n 2 h 2 ε 2
0
v=
Ze 2
2nh ε 0
...(1.6)
Velocity of electron in hydrogen atom in ground state comes to
v=
e2
2h ε 0
...(1.7)
On substituting the values of e, h, e0 this relation gives a value of v = 2.188 × 106 ms–1. This high
speed of the electron makes the atom appear spherical.
(iii) Energy of Stationery States
The total energy of the electron in a stationary state is equal to the sum of its kinetic and potential
energies.
1
Ze 2
mv 2 −
2
4πε 0 r
Substituting the value of kinetic energy from equation 1.2 into 1.8, so that
...(1.8)
1 Ze 2
Ze 2
1 Ze 2
−
=
−
E=
2 4πε 0 r 4πε 0 r
2 4 πε 0 r
Substituting the value of r from equation 1.4 in the above equation, we get
...(1.9)
E=
E= −
me 4 Z 2
8ε 02 n 2 h 2
...(1.10)
In this equation except n, all other parameters are constants; hence in a particular orbit electron
has a discrete energy that can be calculated. The negative sign in this relation implies that the electron
is bound to the nucleus by attractive force so that energy has to be supplied to the electron so as to get
separated from nucleus. As ‘n’ increases, the numerical value of the energy decreases, but on account
of the negative sign the actual energy will increase. This implies that outer orbits have greater energy
than inner ones. Further, the electron and nucleus are infinitely far apart when n = ∝ , and E = 0 so that
the atom get ionized. As they move close together, they are attracted and the energy of the system
becomes less than zero, i.e., negative. Thus, the energy of an electron in a stationary state is negative
as compared to the energy of a free electron.
(iv) Spectral Lines of Hydrogen Atom
The energy of the hydrogen atom when the electron is in the n1th orbit
E1 = −
me 4 Z 2 1
8ε 02 h 2 n12
and the energy of the atom when the electron is in the n2th orbit.
Quantization of Energy
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E2 = −
me 4 Z 2
1
8ε 02 h 2
n 22
Applying the Bohr’s fifth postulate radiation energy (hn) with frequency n is emitted when an
electron jumps from the n2th to the n1th orbit and thus is given by
hn = E2 – E1 =
1
me 4 Z 2  1
− 2
2 2  2
8ε 0 h  n1 n 2 
For the hydrogen atom Z = 1 and
n=
where
me 4
8ε 02 h 3
1
1
1
1
 2 − 2  = RH  2 − 2 
 n1 n2 
 n1 n2 
...(1.11)
me 4
is called the Rydberg constant, RH with a value of 3.289 × 1015 Hz.
8ε 02h 3
RH /c = 109724 cm–1.
The electron in an atom keeps on moving in an orbit but when it is heated it absorbs energy and
jumps from one energy state to another energy state. The excited electron returns from a higher energy
level to one of the lower energy levels when an emission spectrum is obtained. Relation 1.11 gives the
frequency of light so emitted. The five series of the hydrogen atom spectra are known by giving
different values to n1 and n2 in equation 1.11. These spectral lines can be generated as shown in Table
1.1.
or
Table 1.1: Generation of spectral lines from equation 1.11
Name of
spectra series
Spectral region
of series
Frequency and value of n1 and n2
Lyman Series
UV Region

1 
n = RH 1 − 2 
 n2 
n2 = 2, 3, ...
Balmer Series
Visible Region
1 1 
n = RH  4 − 2 
n2 

n2 = 3, 4, 5, ...
Paschen Series
Near IR Region
1 1 
n = RH  9 − 2 
n2 

n2 = 4, 5, ...
Brackett Series
IR Region
1
1
n = RH 16 − 2  n2 = 5, 6, ...
n2 

Pfund Series
Far IR Region
1
1
n = RH  25 − 2  n2 = 6, 7, ...
n2 

These transitions are shown diagrammatically in Fig. 1.1.
6
Elements of Molecular Spectroscopy
Fig. 1.1: Energy level diagram of the H atom and various transitions as depicted
by Bohr structure of hydrogen atom
Spectral lines so obtained are in very good agreement with experimental results.
Quantization of Energy
7
Limitations of Bohr’s Theory
1. It fails altogether to give a quantitative explanation for spectra of atoms having more than
one electron.
2. Bohr model picked up one hydrogen atom and tried to give it a structure. It does not deal
with a collection of hydrogen atom . Hence this treatment does not lay a base to calculate the
intensity of spectral lines.
3. It cannot explain the fine spectrum of hydrogen atom. In most cases, what were considered
to be single lines, proved to be clusters of closely spaced lines detected by more sophisticated
instruments.
(i) On placing hydrogen in a magnetic field and then recording spectra, the single line in
the earlier spectra split up into a number of closely spaced lines. This splitting is known
as the Zeeman effect.
(ii) It failed to explain the Stark effect. The effect of an electrostatic field on spectral
splitting is called the Stark effect.
4. It could not lay a basis for the periodic table of elements.
With further developments about the nature of subatomic particle, the idea of orbits of an electron
was jeopardized and hence contradiction regarding the problems of instantaneous jump of the electron
from one orbit to another was automatically eliminated.
1.3 IDEA OF WAVE-PARTICLE DUALITY
Classical physics acquaints us with two types of motions—corpuscular and wave motion. The first
type is characterized by localization of the object in space as shown by a trajectory motion. The second
type is characterized by delocalization in space. Localized objects do not corresponds to the wave
motion. In the world of macro-phenomena, the corpuscular and wave motions are clearly distinguished.
These usual concepts, however, cannot be transferred to quantum mechanics. The strict demarcation
between the two types of motions is considerably obliterated in micro-particles. The motion of microparticles is characterized simultaneously by wave as well as corpuscular properties. If macro-particles
and waves are considered as two extreme cases of motion of matter, micro-particles must occupy in
this scheme a place somewhere in between. They are not purely particles, they are not purely wave
like, but they are something qualitatively different. Moreover, how much it is particle in nature and how
much wave like depends on the conditions under which the micro-particle is considered. While in
classical physics a corpuscle and a wave are two mutually exclusive extremities, these extremities at
the level of micro-phenomena, combine dialectically within the framework of a single micro-particle.
This is known as wave-particle duality.
As early as 1917, Einstein suggested that quanta of radiation should be considered as particles
called photon possessing not only a definite energy but also a definite momentum i.e.,
DE = hn and p = hn/c. His suggestion was based on his interpretation of photoelectric effect. When
high frequency light falls on a metal surface electrons are emitted. This emission occurs only if the
frequency of incident light is greater than a threshold value characteristic of the metal. Intensity of
incident radiation plays no part in the emission. As the frequency is increased beyond the threshold
value, kinetic energy of the emitted electrons increase linearly to the frequency. These facts could be
explained as follows.
8
Elements of Molecular Spectroscopy
Light of frequency n can impart energy only in discrete amounts of magnitude hn. A light beam
of frequency possess a energy nhn, which could be regarded as containing n light corpuscles, called
photons. These photons of threshold value collide against the metal electrons and knock them off
leading to emission of electrons. In case photons have energy less than threshold value, no emission is
expected. However, if photons have energy higher than threshold value, the energy excess to threshold
value will be transferred to emitting electrons in the form of kinetic energy. In this interpretation
intensity of radiation plays no part.
In 1924, de Broglie suggested that duality should not only be extended to radiation but also to the
micro-particles and the idea was confirmed by de Broglie himself in 1927 with his discovery of electron
diffraction as shown by KCl crystal. For electrons the crystal lattice served as a diffraction grating,
while studying the passage of electrons through a thin foil, Davison and Germer observed characteristic
diffraction rings on the detector screen. Measurement of distances between diffraction rings for electrons
of a given energy confirmed the de Broglie theory that every electron of momentum p is associated a
wave of wavelength l given by
h
h
l = mv = p
...(1.12)
Such waves do not move with the velocity of light but with a velocity v and are called metallic
waves. Like electromagnetic waves, metallic waves are propagated in an absolute void; hence they are
not mechanical waves. Since these waves move with the velocity of particle in motion, they are not
electromagnetic waves. Let us take the examples of metallic waves.
PROBLEM 1.1 With the help of the Bohr model of the hydrogen atom one can calculate the velocity of
an electron in the ground state from equation 1.7, which comes to be 2.188 × 106 ms–1. Assuming the
mass of electron = 10–31 kg one can calculate the deBroglie wave associated with it.
SOLUTION
l=
6.6 × 10 −34 Js
(10
−31
−1
kg ) (2.188 × 10 ms )
6
= 3.01 × 10 −9 m
This is quite small and 10–9 corresponds approximately to the wavelength of X-ray, which can be
easily detected. Thus particle nature is characterized by the momentum and wave nature by the
wavelength. The two terms are inversely related to each other. For a macroscopic particle, which has
a large value of momentum, the wavelength as calculated from the deBroglie relation is too large to be
determined by experiment. For such a case the wave nature may be completely ignored and thus
particle has corpuscular nature governed by classical mechanics. On the other hand, all atomic particles
have dual character.
X-rays pass through the crystal almost unimpeded while electrons are totally absorbed even in a
millimetre thick crystal. Metal foils do show the electron diffraction pattern. Even when an electron
beam was directed at a small angle to the crystal, a diffraction pattern was observed. If it were
produced by a small number of diffracted electrons, at first glance it would appear that the electrons
impinged randomly on the plate. But there is one thing that attracts our attention. We measure the
aperture of the diaphragm from which the electrons emerged and project the outline onto the target. It
would seem that all the electrons should fit inside this outline, no matter how randomly they had fallen
on the photographic plate. However, some of the hits are far outside the boundary line.
Quantization of Energy
9
Every electron that hits the photographic plate, decompose Ag2S2O3 to Ag2S thereby leaving a
black spot. If the number of hits on the target is small there are closely bunched black places. If a line
is drawn through these places small rings appear. However, these rings are not well defined, but
improve as the number of electrons striking the plate increases. Thus graph of electron hits on
photographic plate is not a figment of imagination. It reflects the existence of a real wave of wavelength
h/mv associated with electron moving with velocity v.
1.4 HEISENBERG UNCERTAINTY PRINCIPLE
Consider a measurement of position of an atomic particle of mass m. If it has to be located within a
distance Dx then light with a wavelength of the size of a particle should be used to illuminate it. For the
particle to be seen, a photon must collide in some way with the particle, for otherwise the photon will
pass right through and the particle will appear transparent. The photon has a momentum, p = h/l and
during the collision some of the momentum will be transferred to the particle. The very act of locating
the particle leads to a change in its momentum. If a particle has to be located more accurately, light of
even greater momentum or smaller wavelength should be used. As some of the photon’s momentum is
transferred to the particle in the process of locating it, the momentum change of the particle becomes
greater. A careful analysis of this process was carried out by Warner Heisenberg who showed that it is
not possible to determine exactly how much momentum is transferred to electron. This means that if a
particle has to be located within a region Dx, then this causes an uncertainty in momentum of particle.
Heisenberg was able to show that if Dp is the uncertainty in the momentum then
Dx Dp ≥
h
4π
...(1.13)
The smaller the Dx, the greater the Dp and vice versa, but the product of the two is always given
by equation 1.13. This amounts to that it is impossible to measure simultaneously the position and
momentum of a particle with arbitrary precision. Such an interpretation may mean that the uncertainty
relation is responsible for limitations associated with the process of measurement.
One might be led to assume that a micro-particle possesses a definite co-ordinate as well as
definite momentum, but the uncertainty relation does not permit us to measure them simultaneously.
This may be an erroneous conclusion. Thus uncertainty relation is not that creates certain obstacles, to
the understanding of micro phenomena but it reflects certain peculiarities of the objective properties of
micro particles. The following two examples demonstrate the numerical consequence of the uncertainty
principle.
PROBLEM 1.2 Calculate the uncertainty of position of an automobile of mass 500 kg moving with a
speed of 50 + 0.001 km hr–1.
SOLUTION
The uncertainty of position is
1.05 × 10 −34 Js
D
=
= 3.779 × 10 − 2 m
−
−
2
4
1
4πm∆v 2(5 × 10 kg ) ( 2.77 × 10 ms )
This is a very small distance and wholly negligible as compared to the mass and velocity of the
automobile.
Dx =
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Elements of Molecular Spectroscopy
PROBLEM 1.3 What is the uncertainty of momentum of an electron in an atom so that Dx is 52.9 pm.
SOLUTION
1.05 × 10 −34 Js
D
=
= 9.9 × 10 −25 kg ms −1
Dp=
2∆x 2(52.9 × 10 −12 m)
Because of p = mv and mass of electron is 9.11 × 10–31 kg this value of Dv corresponds to:
Dv =
∆p 9.9 × 10 −25 kg ms −1
=
= 1.087 × 10 6 ms −1
m
9.11 × 10 −31 kg
Compare this uncertainty in velocity with velocity of electron in hydrogen atom, which is
2.188 × 106 ms–1. This is a very large uncertainty in speed and cannot be neglected. These two
examples show that although the Heisenberg Uncertainty principle is of no consequence for macroscopic
bodies it has very important consequence in dealing with the atoms and subatomic particles. This is
similar to the conclusion drawn for the application of deBroglie relation between wavelength and
momentum.
1.5 PROBABILITY
In classical mechanics the position and velocity of any particle for any instant of time can be predicted
with absolute certainty provided the forces acting upon it at that instant are known. However, it was
evident almost immediately that one could not apply classical mechanics directly to the motion of gas
molecules. The reason is quite simple. Even small volumes of gas, say 1 ml, contain 1019 molecules.
Now to give an accurate picture of their motion would require writing and solving 1019 equations of
motion. The gas molecules are never at rest, they are constantly colliding with other molecules bouncing
off some, running into others and these events occur millions of times every second. It is pre-posterous
even to imagine writing Newton’s equation for all molecules. Million of years would be spent in just
writing down the equations. More millions of millions of years in solving them. Meanwhile others
would have replaced all these motions.
In search for a reasonable way out, scientists saw that they should not be interested in the motion
of each individual molecule of gas colliding with other molecules with unbelievable rapidity. Rather,
should their interest lie in the state of the entire mass of gas, its temperature, density, pressure and other
characteristic properties?
In other words, there is no need to determine the velocities of the separate molecules. All
characteristics of the state of the gas should refer to the whole system of molecule as an assembly.
Now mainly the mean velocity of the gas molecules determines these characteristics. For example, the
higher the velocity higher the temperature. If in the process the gas does not change its volume then
there will be an increase in the pressure with rise in temperature. But to learn these relationships
accurately, one had to find some way to determine the mean velocity of the molecules. Here was where
the probability came in.
The behaviour of a large assemble of molecules can be described statistically where random
molecular motion has a definite form and hence every collision, every individual motion of a molecule
could be described by Newtonian law and that if one desired to solve millions of millions of equations,
he could express these motions with absolute precision and without any kind of mean values. We do
Quantization of Energy
11
not do that; of course in principle it could be done! We determine the motion of a gas by means of
probability laws, but underlying them are the exact laws of Newtonian Mechanics.
Due to the wave particle dual behaviour and uncertainty principle electrons, atoms and molecules
cannot follow the laws of Newtonian mechanics. This amounts to saying exact motion of electrons
cannot be traced even in principle. Hence quantum mechanics, also called wave mechanics, has been
developed in which basic probability laws are incorporated but modified by wave particle duality and
uncertainty principle. This will be explained by the following two examples:
1. Electron diffraction: The electron diffracted by crystal hit upon the photographic place to
give dark concentric rings but not all the electrons are unique here. There are certain greyish places
between the darkest and lightest sections. A mean number of electrons impinge on these portions. We
see this very clearly on the distribution curve of hits in our shooting game. An electron leaves its
source, passes through the diaphragm, is reflected from the crystal and is moving towards the
photographic plate. Where will it hit the plate?
Classical mechanics calculates the angle, distance and velocity with great accuracy and says,
“HERE’, which is usually not where it hits at all”.
Wave mechanics says, “I do not know exactly but the greatest probability is that it would have hit
the darkest rings, there is less probability that it would have hit the grey sections and it is hardly at all
that it may have impinged on the light rings”.
2. Structure of atom: Bohr’s treatment say that electrons are moving in circular rings called
orbits in which their exact position and velocity at any time can be located. But according to wave
mechanics say we are not interested in exact position and velocity and thus need not talk how an
electron is moving. Rather we say that there is certain probability in finding electron in a particular
orbital. This amounts to saying that every electron has some most probable distribution in an orbital and
thus behaviour of these orbitals give characteristic properties to constituting atom.
1.6 SCHRODINGER WAVE EQUATION
Classical mechanics deals with observable parameters such as position and momentum as function of
time or sometime of each other and Newton’s laws of motion enable these functions to be determined.
Quantum mechanics recognizes that all the information about the system is contained in its wave
function and that, in order to extract the information about the value of an observable parameter, some
mathematical operation must be done on the function. This is analogous to the necessity of doing an
act, an experiment on the system in order to make a measurement of its state. Quantum mechanics
really boils down to making correct selection of the operation appropriate to the observable parameter.
In the simple quantum mechanics that concerns us it turns out that the right way to determine the
momentum from a wave function is simply to differentiate it and then multiply the result of h/2pi,
where i = − 1 . Thus, gradient of wave function at a particular point determines the momentum. The
operator that extracts the position turns out to be simply multiplication by x, but this, as you can
imagine, is deceptively simple. Once we know what the operators are for the dynamical variables of
position and momentum, we can set up the operators for all observable parameters, because these can
be expressed as function of two basic variables. Thus, the kinetic energy in classical mechanics is a
function of the momentum namely p2/2m, therefore, the corresponding operators can be obtained by
∂ 2
replacing p2 by (h/2pi
) . This shows that curvature of the wave function determines the kinetic
∂x
energy.
12
Elements of Molecular Spectroscopy
With the discovery of uncertainty principle and wave particle dualism, orbit concept became
redundant and it became necessary to formulate a new mechanics to explain structure of atom. For this
Schrodinger formulated wave mechanics, which is used like any other law of nature. The solutions to
the wave equation are called wave functions, which give a complete description of the system and
quantized energy values. Our concern in this book will be to evaluate quantized energy levels only and
not to stationary state wave functions. This mechanics showed that fundamental laws of nature are not
dynamic but are statistical and thus probabilistic form of casualty, while the classical determinism is
just its limiting case. The first wave equation for a particle of mass m moving in space was given in the
form of a statement as
∂ 2ψ
∂x
or
2
+
∂ 2ψ
∂y
2
+
∂ 2ψ
+
2m
( E − V )ψ = 0
D
∇ 2ψ +
2m
( E − V )ψ = 0
D
∂z
2
The Schrodinger equation for any system can be set up in the following ways:
1. A system of particles can be represented in terms of generalized coordinates qi and generalized
∂q j
( with j = 1, 2, . . . . , N) . This representation is described by a function
velocities q j =
dt
y(q1, q2, …, qh), which determines all measurable quantities of the system. Any general law
so formulated will be independent of the coordinate system.
2. In wave mechanics we deal with linear operators. An operator is a symbol that tells you to
do something to whatever that follows. For example, consider dy/dx to be the d/dx, operator,
operating on the function y (x). An operator A is said to be linear if
A [c1 f1 (x) + c2 f2 (x)] = c1 A f1 (x) + c2 A f2(x)
...(1.13a)
Where c1 and c2 are possibly complex constants. Clearly the differential and integrate operators
are linear because these operators satisfy the 1.13a condition, i.e.,
d/dx [c1 f1 (x) + c2 f2 (x)] = c1 d/dx f1 (x) + c2 d/dx f2(x)
∫ [c1 f1 (x) + c2 f2 (x)]dx = c1 ∫ f1 (x)dx + c2 ∫ f2(x)dx
The square operator on the other hand is a non-linear because it does not satisfy the 1.13a
condition
SQR [c1 f1 (x) + c2 f2 (x)] = c12 f 12 (x) + c22 f 22 (x) + 2c1 c2 f1 (x) f2 (x)
≠ c12 f 12 (x) + c22 f 22 (x)
3. Hamiltonian operator Hop, appropriate to the problem, is written in terms of generalized
coordinates and momenta. The Hamiltonian may be defined as sum of kinetic energy and
potential energy as
p 2j
Hop =
∑ 2m + V(qj )
...(1.14)
j
Various Hamiltonians for the same problem differ only in the form of potential energy term.
Quantization of Energy
13
4. In quantum mechanics, momentum pj is defined as
pj =
∂
h ∂
= − iD
∂q j
2πi ∂q j
...(1.15)
where h is Planck’s constant, D = h/2p and i = − 1 . Replace each momentum pj in equation
1.14, wherever it occurs by such operators (Eq. 1.15) and we get quantum mechanical
Hamiltonian. Involvement of momenta and Planck’s constant represent the involvement of
de Broglie wave character of particle and Uncertainty principle. The quantum mechanical
Hamiltonian is as follows:

Hop =
D2 ∂2 
∑  − 2m ∂q 2j  + V(q j )
... (1.16)


5. Select an appropriate wave function, y and when Hamiltonian operates upon it Schrodinger
wave equation is obtained i.e.,
Hop y = E y
The differential equation so obtained is
j
 D2 ∂2 
−
 + V ( q )ψ = E y
... (1.17)
j
 2m ∂q 2j 
j 

This differential equation is called Schrodinger wave equation, for a so-called stationary
state of the system i.e., one whose energy does not vary with time. Equation 1.17 is the
characterization equation of the operator, Hop and E is the eigen value of Hop associated with
the eigen function y.
6. Eigen functions: Some operations or combination of operations and functions are such that
when operation is done the same function is regenerated, but perhaps multiplied by a number.
Thus differential of the function exp2x give 2exp2x which is the same function multiplied by
the number 2. When this occurs the function is said to be an eigen function of the operator
(in this case differential operator) and the numerical factor (2 in the example) is called the
eigen value of the operator.
Wave equation has the form Hy = Ey where Hamiltonian H is differential operator and y is
the wave function. This has the form of an eigen value equation with the energy E playing
the part of eigen value and wave function as eigen function. The wave function represents a
state of the system, and so y is often termed the eigen state.
7. The wave function is generally a complex quantity. The integral of probability density over N
particles and thus 3N coordinates is equated to one i.e.,
∑
∫ . . . . ∫ ψ ψdq1 dq2 . . . dq3N
*
or
∫ . . . . ∫ ψ ψdτ
*
=1
= 1 where dt = dq1, dq2, ... , dq3N
The condition is called the normalization condition.
...(1.18)
14
Elements of Molecular Spectroscopy
1.7 SIGNIFICANCE OF WAVE FUNCTION (O))
Wave equation so written is solved subject to the condition that wave function y be a continuous
function, which is not allowed to have singularities so that the normalization integral may diverge.
The wave function contains all the information about the dynamical properties of system. If the
wave function known, all the observable properties of the system in that state may be deduced by
performing appropriate mathematical operation.
The interpretation of y is based on a suggestion made by Born. The Born interpretation draws an
analogy with wave theory of optics in which square of the amplitude of an electromagnetic wave is
interpreted as intensity of radiation which amounts to number of photons present. The analogy for
particles is that wave function is an amplitude whose square indicates the probability of finding the
particle at each point of space. The Born interpretation of y is, therefore, that y*(x)y*(x)dx is
proportional to probability of finding the particle in an infinitesimal region between x and x + dx.
1.8 WAVE EQUATION FOR HYDROGEN ATOM
A system which consists of a positively charged nuclei and electron moving about it is found in the
hydrogen atom as well as in ions of He+, Li+2, Be+3 etc. According to Coulomb’s law, the force between
a pair of charged particles is operative with magnitude F = –eZe/4per2, where –e is the charge of
electron, Ze the nucleus charge and r the distance between the particles. The potential energy resulting
from this force is
r
∫
V = − Fdr = −
∞
Ze 2
4πε 0 r
...(1.19)
Now we are in a state to write down Hamiltonion for hydrogen like atoms.
The Hamiltonian for hydrogen like atoms is described as
H = KE of proton + KE of electron + PE of hydrogen atom
=
1
1
( p x2 + p 2y + p z2 ) electron + V
( p x2 + p 2y + p z2 ) proton +
2me
2m p
Since proton is 1836 times as heavy as electron, it may not be moving from its position, i.e., it
may be assumed to be stationary and hence kinetic energy of proton may be assumed to be zero. So
Hamiltonian reduces to:
H=
1
( p x2 + p 2y + p z2 ) + V
2me
...(1.20)
Replacement of momentum by quantum mechanical operator (given by equation 1.15) gives
quantum mechanical Hamiltonian.
H= −
D 2  ∂ 2
∂2
∂ 2  Ze 2
+
+
−
2µ  ∂x 2 ∂y 2 ∂z 2  4πε0 r
...(1.21)
When it operates upon a wave function y, we get Schrodinger wave equation for hydrogen atom.
Quantization of Energy
15
 ∂2
2µ 
Ze 2 
∂2
∂ 2 

+
+
ψ
ψ=0
+
+
E
 ∂x 2 ∂y 2 ∂z 2 
4πε 0 r 
D 2 


...(1.22)
This equation in cartesian coordinates cannot be solved. However, it can be solved if it is in polar
coordinates.
In spherical coordinates equation 1.22 becomes
1 ∂  r 2 ∂ψ 
1
1
Ze 2 
∂ 2ψ
∂ 
∂ψ  2µ 
+
+
+ 2 2
+
θ
ψ=0
sin
E


4πε 0 r 
r ∂r  ∂r  r sin θ ∂φ 2 r 2 sin θ ∂θ 
∂θ  D 2 
...(1.23)
With centre of mass of the electron and nucleus as the origin of coordinates, m is the reduced
mass of the atom, E is its total energy. Since wave equation (1.23) involves the use of momentum
operator, hence it includes de Broglie dual character of atomic particles. Further, since it has Planck’s
constant, it incorporates Heisenberg uncertainty principle.
1. In the equation 1.23, y function can be written as a product of functions Rnl (r) Q (q) F (f)
written more briefly as Rnl Q F which gives three differential equations. Solution of these
differential equations, give three solutions corresponding to three coordinates r, q and f.
Relation 1.24 between principal quantum number, n, and energy, E is given by:
E=–
µZ 2e 4
8ε 20 n 2 h 2
n = 1, 2, 3,......
...(1.24)
Since principal quantum number n appears only in radial wave function Rnl (r), the energy of
atoms is related, to the distance of the electron from the nucleus and not to the angular
momentum. However, the angular momentum has its importance in selection rules, which
states that n may change by any number but l must change by ±1 and m may change by 0,
±1 in a transition. These two factors were not there in Bohr theory. The value of energy
given by equation 1.24 is same as was given by Bohr theory in equation 1.10 but the operating
mechanics is different.
2. The orbital angular momentum comes from function Q part and solution of corresponding
differential equation gives the solution as:
L=
l (l + 1) D
...(1.25)
where l = 0, 1, 2, …… (n – 1). The values calculated from equation 1.25 are the only exact
values of angular momentum L.
3. Though the magnitude of orbital angular momentum are known, yet the orientations of
orbital momentum with respect to an external reference comes from function F part and
solution of corresponding differential equation is L z = mD .
where m = – l, – (l – 1), …..0, …….. (l – 1), l
...(1.26)
As long as there is no external magnetic field, all orientations of the angular momentum L
possess the same energy. For one value of l there can be 2l+1 values of m, as for example,
when l = 2. The parameter m can have five values 2, 1, 0, –1, –2. This means that for a
16
Elements of Molecular Spectroscopy
magnitude of angular momentum L = 6 D , in a magnetic field only five orientations are
allowed. Similar solutions are obtainable for any one-electron atom problem.
The principal quantum number n and azimuthal quantum number decides radial distributions
of the electron and thus the values of r. The permitted values of these numbers are:
Principal quantum number, n
Azimuthal quantum number, l
1
0
2
0, 1
3
0, 1, 2
4
0, 1, 2, 3
The function Q depends only on angle q, therefore, describes the electron distribution as a
function of angle q. These functions again depend upon two quantum numbers l and m.
Though the permitted values of m are 0, ±1, ±2… ±l, the Q functions depend only on the
magnitude of l. The orientation of orbitals is decided by the value of quantum number m.
Thus, the total wave function y which constitutes what is known as orbital dependence on
the quantum numbers n, l and m, i.e., different y functions for different orbital have different
values of n, l and m and hence different behaviour of the different electrons in an atom. It is
customary to designate the values of l by letters as given below:
Values of l
Designation of atomic orbital
0
s
1
p
2
d
3
f
4
g
5
h
The magnetic quantum number describes the Z-component of the angular momentum of the
electron through equation 1.26.
The energy of the electron depends only on the value of n and not at all on l and m. Thus, all
y functions with same value of n but different values of l and m are degenerate as follows:
n=1
l=0
m=0
n=2
l=0
m=0
l=1
m = +1
E1 =
µe 4
8ε 02 h 2
E2 =
1
E1
4
E3 =
1
E1
9
m=0
m = –1
n =3
l=0
l=1
l=2
m=0
m = +1
m=0
m = –1
m = +2
m = +1
m=0
m = –1
m = –2
Quantization of Energy
17
1.9 SPECTRA OF HYDROGEN ATOMS
Atomic spectra are obtained when transition of electron from one wave function (or orbital) to another
wave function takes place. A more rigorous quantum mechanical study of transition between quantum
states indicates that certain restrictions in the change in the values of l and m must be satisfied. The
transitions, which do not follow these restrictions, are forbidden. These restrictions are referred to as
selection rules.
Selection Rules
(i) n may change by any integer i.e., Dn = any value
(ii) l must change by ±l value i.e., Dl = ±1
(iii) m may change by ±1 or not at all i.e., Dm = 0, ±1
For example, if an electron changes its principle quantum number from n = 2 to n = 1, it must go
from a state of l = 1 to l = 0, i.e., the transition 1s ¬ 2p is allowed. The transition 1s ¬ 2s where
Dl = 0 is forbidden. Similarly, 2s ¬ 2p, 2p ¬ 3s, 2p ¬ 3d are allowed transition but
2s ¬ 3s, 2p ¬ 3p, 2s ¬ 3d are not allowed. Some of the hydrogen atom transitions are given in
Table 1.2.
Table 1.2
Series name
Allowed transition
Lyman series
Balmer series
n1 = 1
n1 = 2
n2 = 2, 3, 4…
n2 = 3, 4, 5…
Paschen series
n1 = 3
n2 = 4, 5, 6…
1s ¬ n2 p
2s ¬ n2 p
2p ¬ n2s
2p ¬ n2d
3s ¬ n2 p
3p ¬ n2s
3d ¬ n2Tp
3p ¬ n2d
3d ¬ n2 f
1.10 USE OF ATOMIC TERM SYMBOLS TO DESCRIBE ATOMIC SPECTRA
Atomic term symbols are sometimes called spectroscopic term symbols because atomic spectral lines
can be assigned to transitions between states that are described by atomic term symbols. For example,
consider atomic hydrogen. Exact solution of Schrodinger wave equation for hydrogen can be obtained
from equation 1.24 as
En =
µe 4
8ε 20 n 2 h 2
...(1.24)
It is peculiar to the simple 1/r Columbic potential of a hydrogen atom that the energy depends
only on the principal quantum number. An electron in a 3s, 3p or 3d orbital, for example has the same
energy E3 in equation 1.24. As n increases coupling between spin angular momentum and orbital
angular momentum takes place. This introduces a new quantum number J. States with different values
of J will have different energies and thus have different term symbols.
18
Elements of Molecular Spectroscopy
Term symbol that takes into account the spin orbit coupling is written as follows:
1. First take the l’s of each electron outside the closed shell and from them calculate orbital
angular momentum vector L. The magnitude of L can equal is (l 1+l 2 ), (l 1+l 2 –1)…
(l1 – l2). Each L can have 2L + 1 various values. A notation grew up in which different L
values are described according to symbol.
L =
0
1
2
3
4
5
6
Symbols
S
P
D
F
G
H
I
2. The spins of each electron outside the closed shell are coupled to give resultant spin, Ŝ.
3. Each L and Ŝ vector is coupled to get resultant J vector.
The particular state is said to be a multiple or can have a multiplicity equal to the number of J
values. The energy state with quantum number L, Ŝ , J is indicated by a code symbol, which is called
term symbol as:
2 Ŝ +lL
Term Symbol
Ĵ
For example, 4D7/2 means L = 2 and thus d state; Ŝ = 3/2 or 2 Ŝ + 1 = 4 and J = 2 + 3/2 = 7/2.
The electronic configurations and corresponding term symbols for various states of atomic
hydrogen are given in Table 1.3.
Table 1.3: The first few electronic states of atomic hydrogen
Electronic configuration
Term symbols
Energy / cm–1
1s
1s 2S1/2
000
2p
2p 2P1/2
82258.917
2s
2s 2S1/2
82258.942
2p
2p 2P3/2
82259.272
3p
3p 2P1/2
97492.198
3s
3s 2S1/2
97492.208
3p, 3d
3p 2P3/2, 3d 2D3/2
97492.306
3d
3d 2D5/2
97492.342
4p
4p 2P1/2
102823.835
4s
4s 4p 2S1/2
102823.839
4p, 4d
4p 2P5/2, 4d 2D3/2
102823.881
4d, 4f
4d 2D5/2, 4f 2F3/2
102823.896
4f
4f 2F7/2
102823.904
Quantization of Energy
19
Let us use Table 1.3 to take a closer look at the hydrogen atom spectrum. In particular, let us look
at Lyman series which is the series of transitions from n = 1 state to states of higher n. Rydberg
formulae Table 1.1 can be used to calculate the frequencies of the lines in the Lyman series. The
frequencies of lines in the Lyman series are given by
1 

n = 109677.8 1 − 2  cm–1
 n 
n = 2, 3 ......
which gives following results:
Table 1.4: Calculated Lyman series of lines of hydrogen atom
Transition
Frequencies / cm–3
1®2
82258.20
1®3
97491.18
1®4
102822.73
1®5
105290.48
Table 1.3 shows three states for n = 2 and so we do know which state to use to calculate the
transition frequency into the ground state 1s 2S1/2. The changed selection rules for allowed transitions
incorporating spin orbit coupling are:
DL = ±1
and
DJ = 0, ± 1.
In the DL = 0 case, the transition from a state with J = 0 to another state with J = 0 is forbidden.
Thus Lyman series of atomic hydrogen for the allowed transitions are:
np
2P
1/2
®
1s 2S1/2
np
2P
2/2
®
1s 2S1/2
No other transitions into the 1s 2S1/2 ground state are allowed. The frequencies associated with
the 2 ® 1 transitions can be computed from Table 1.3 and their values are:
n = (82258.917 – 0.000) cm–1 = 82258.917 cm–1
n = (82259.272 – 0.000) cm–1 = 82259.272 cm–1 respectively.
and
Thus, n = 2 to n =1 transition which occurs at a frequency n = 82258.20 cm–1 if we ignore spin
orbit coupling consist of two closely spaced lines. These closely spaced pair of lines are called doublet,
and so do we see that under high resolution, the first line of Lyman series is a doublet. Table 1.3 shows
that all the lines of Lyman series are doublets and that the separation of the doublet lines decreases with
increasing n. The increased spectral complexity caused by spin orbit coupling is called fine structure.
Similarly, lines in the 3d 2D to 2p 2P transition can be calculated. There are two 2p states in
atomic hydrogen 2p 2P3/2. The transition from
3d 2D states into 2p 2P1/2 are:
3d 2D3/2 ® 2p 2P1/2
n = (97492.306 – 82258.917) = 15233.389 cm–1
3d 2D3/2 ® 2p 2P3/2
n = (97492.306 – 82259.272) = 15233.034 cm–1
20
Elements of Molecular Spectroscopy
3d 2D5/2 ® 2p 2P3/2
n = (97492.342 – 82259.272) = 15230.070 cm–1
3d 2D5/2 ® 2p 2P1/2
n = (97492.342 – 82258.917) = 15233.425 cm–1
Note that the 3d 2D5/2 ® 2p 2P1/2 transition is not allowed because DJ = 2.
1.11 WAVE EQUATIONS FOR SOME SYSTEMS
1. Particle in a One Dimensional Box
A particle of mass m is constrained to remain strictly in a one dimensional box of length, l with no
seeping into or through the walls of the container. It is so called because the confinement can be
achieved by arranging a zero potential energy within the box but to rise perpendicularly to infinity outside it. Thus, particle will have only kinetic energy with in the box. So, Hamiltonian for such a system
will have only kinetic energy term i.e.,
1  h ∂ 
p2
=


H = kinetic energy =
2m 2m  2πi ∂x 
=
Wave equation
−h
2
∂2
...(1.27)
8π 2 m ∂x 2
Hy = Ey
or
–
h2
∂ 2ψ
8π 2 m ∂x 2
= Ey
...(1.28)
Solution to this wave equation
E=
h2n2
8ml 2
where n = 1, 2, ......
...(1.29)
2. For a Rigid Rotator
For a rigid rotator, again, the molecule is continuously rotating and hence potential energy may be taken
equal to zero and moment of inertia, I = µr2. This relation reduces two-body problem to one body
problem. Thus entire energy is the kinetic energy of body. The Hamiltonian may be defined as
h2  ∂ 2
∂2
∂2 
Hop = − 2  2 + 2 + 2 
µ  ∂x
∂y
∂z 
...(1.30)
The Schrodinger wave equation for rigid rotator is given
−
D 2  ∂ 2 ψ ∂ 2ψ ∂ 2 ψ 
+
+

 = Ey
2µ  ∂x 2 ∂y 2 ∂z 2 
...(1.31)
It is convenient to transform the above equation into spherical coordinates r, q and f. This
transformation is lengthy, hence only transformed expression is given in equation (1.32).
−
D2
2µ
 1 ∂  2 ∂ψ 
1
1
∂ 
∂ψ 
∂ 2ψ 
r
 sin θ
+ 2
+ 2
 2

∂θ  r sin 2 θ ∂φ 2  = Ey
 r ∂r  ∂r  r sin θ ∂θ 
...(1.32)
Quantization of Energy
21
Taking 1/r2 as a common term so that µr2 = I = moment of inertia of molecule. Since in this
∂ψ
=0
treatment, it is assumed that molecule behaves as rigid rotator hence the first term involving
∂r
and equation 1.32 reduces to
−
1 ∂ 2ψ 
D2  1 ∂ 
∂ψ 
 sin θ
+


2I  sin θ ∂θ 
∂θ  sin 2 θ ∂φ 2  = Ey
...(1.33)
Solution of equation 1.33 gives
h
E
=
J( J + 1) = BJ (J + 1) cm–1
hc 8π 2 Ic
where B =
h
...(1.34)
cm −1 and is called rotational constant. J is called rotational quantum number which
8π Ic
can have integer values i.e., J = 1, 2, 3…...
2
3. Linear Harmonic Oscillator
Harmonic oscillators occur in classical mechanics when restoring force on a body is proportional to
displacement. A force = – kq implies the existence of a potential energy U = 1/2 kq2 where k is called
force constant and is characteristic of a bond, and q is the displacements. Since it is linear oscillator, its
kinetic energy is confined to one dimension. Hence Hamiltonian may be defined as
D2  ∂ 2  1 2
H = 2  2  − 2 kq
µ  ∂q 
...(1.35)
When this Hamiltonian operates upon wave function y it gives wave equation for harmonic
oscillator.
∂ 2ψ
1 2
8π 2 µ .
...(1.36)
E − 2 kq  y = 0
2
2
∂q


h
The solution to this differential equation gives quantized oscillator energy levels given by
E=
+
h
2π
k
1
V + 
µ
2
where V is vibrational quantum number confined to the integer values V = 0, 1, 2…. This implies the
h k
, when the oscillator is in its lowest energy state,
4π µ
with V = 0 all the energy cannot be removed from an oscillator.
existence of a zero point energy for V = 0 of