Basic and fundamental of quantum mechanics (Theory)

Basic and Fundamental of
Quantum Mechanics (Only
theory Part)
Mr.Halavath Ramesh
Department of Chemistry
Loyola College-Chennai
University of Chemistry
E-mail: halavathramesh39@gmail.com

Introduction
Quantum Mechanics is one of the greatest intellectual
developments of the 20th century.Today,quantum mechanics has
merged and assisted almost all disciplines of science such as
chemistry ,physics,biology,medicine,computing and soon.
Quantum Mechanics is one of the most remarkable discoveries of
contemporary sciences over the last century. Quantum mechanics
arose from Max Plank’s solution in 1900 to the black-body
radiation and Albert Einstein’s 1905 paper on photoelectric effect.
The term “ Quantum Mechanics” was first used in a paper by
Max Born in 1924.However,even after 116 years of the birth of
concept, it is still considered the weirdest of all sciences. The
learner is not able to digest the quantum concepts and is unable to
appreciate the importance of quantum mechanic in science.

A chemist’s intention to learn quantum mechanics is to be
able to understand the structure ,Bonding and Reactivity of
/in/between chemical entities be it an atom or a molecules,
which is defined by the behaviour of electrons(microscopic
matter).To be able to appreciate the role of quantum mechanics
in chemistry, one must have a sound background in quantum
mechanics. So, here is an attempt to make you understand the
beauty of the subject.
What is Quantum Mechanics?
The term “ Quantum Mechanics” is made up of two word:
QUANTUM + MECHANICS. The term “Mechanics” refers to

The science of the motion of the body. The other word is
“Quantum” which is Latin for “ amount” and in modern
conventions is used to represent the smallest possible discrete unit
of any physical property.
Quantum Mechanics replaces Classical Mechanics at the
atomic or subatomic levels( electron and nuclei in atoms and
molecules).It gives the laws of motions of microscopic objects (the
way classical mechanics gives for macroscopic objects).So, we can
say that Quantum Mechanics is the theoretical sciences of
microscopic matter.
But then what happens to matter at the microscopic
level that classical mechanics? After all , matter is matter i.e.,
anything that has mass and occupies space.

Classical mechanics failed to explain certain
experimental phenomena related to microscopic matter
correctly which led to the origin of quantum mechanics. In fact
, this difference puzzles people as there is a complete disconnect
between the understanding of the universe based on classical
mechanics and quantum mechanics. This is because we don’t
build our understanding of quantum mechanics. Although we
use identical terminology such as particles ,wave , position,
momentum,etc.as in classical mechanics, in quantum mechanics
they attain a very different nature.

Origin of Quantum Mechanics
Till the end of the 19th century, Classical Mechanics
was considered to be the only right and undisputed theoretical
science. But soon some new experimental phenomena were
observed which could not be explained by classical mechanics.
This included Black Body Radiation, Photo-electric effect,
Compton effect, Atomic Spectrum , Heat capacity of solid and so
on. In all ,the correct interpretations of the above mentioned
experimental phenomena gave two important conclusions:
1.Energy is quantized or one can say that it can be transferred
only in discrete packets called quanta.
2. Light ( or radiation ) exhibits particle –like behaviour.

These two new concepts were the basis for the origin
of the field of quantum mechanics. Classical Mechanics failed to
explain the above mentioned experimental phenomena as
classical mechanics considered an electromagnetic radiation
solely a wave phenomenon.
Now we find that there is some particles
character associated with radiation ( along with the wave
phenomenon ) as well. It seems that radiation shows a dual
nature. In some cases, it behaves like a wave ( reflection,
refraction , diffraction,etc.) and some times it manifests itself as
a particle , the photon ( a photon is a single quantum of
electromagnetic energy or one can say , photons are quanta of

Electromagnetic energy which means that the energy is quantized
and can only be transferred in discrete units ( or packets) of size
hv (v is the frequency) , Compton effect,etc.). Neither picture is
wrong.
Energy is Quantized Black body radiation Atomic spectrum
Heat capacity of solids
Light is composed of particles Photoelectric effect Compton effect

In 1924, Louis De Broglie suggested some
logic to this situation. Broglie said that nature manifests itself in
two forms- matter and radiation .And if radiation has dual
behaviour, then by virtue of symmetry matter should also have
dual behaviour. Broglie suggested that particles have wave like
properties characterised by a wavelength as λ = h/p , where λ is
the wavelength (wave nature) and p is the momentum of the
particles [ p=mv( v is the velocity ) particles nature]. The wave
nature of matter was first confirmed by Davisson and Germer
experiment. This established the wave particles Duality in
matter.

A direct consequence of wave particles duality of
matter as well as radiation led to the Heisenberg Uncertainty
principle which states that the position and momentum of a
particles cannot be simultaneously measured with arbitrarily
high precision Δx. Δpx>- h/4 π. Thus the more precisely we
determine a particles' position ,the more we disturb its motion
(momentum).
With the Heisenberg uncertainty principle
came the concept of “ Orbital” Bohr’s atomic model was one
model that was widely appreciated but was later replaced by the
quantum theory of atom. Bohr postulated that the electrons in
an atom revolve round the nucleus in fixed circular paths called

orbits. This concept of orbit is not valid as per the Heisenberg
uncertainty principle because the trajectory of a particles can
only be defined if its position and momentum are known
simultaneously with precision. An important consequence of the
Heisenberg uncertainty principle is that one cannot determine
the path of a moving microscopic particle. Bohr’s concept of orbit
failed and was replaced by orbital.
Here came in the concept of probability. In terms of
uncertainty principle, one can only predict the probability or
relative chances of locating an electron in a particular region of
space around the nucleus i.e, ..one can only predict where an ele
ctron is most likely to be found.

Basic and fundamental of quantum mechanics (Theory)

Quantum Mechanics replaces Classical Mechanics at the atomic or subatomic
levels ( electrons and nuclei in atoms and molecules). It gives the laws of
motion of microscopic objects ( the way classical mechanics gives for
macroscopic objects).So, we can say that Quantum Mechanics is the
theoretical science of microscopic matter.

Fundamental of Quantum Mechanics
The dual behaviour of matter and uncertainty principle
gave birth to quantum mechanics. These ideas inspired
Schrodinger and Heisenberg and they independently formulated
quantum mechanics in 1925 ( Schrödinger –wave mechanics and
Heisenberg –Matrix mechanics) ,to study the behaviour of
microscopic matter.
At first sight, the two approaches
appeared different but later Dirac and Newman showed that in
essence the two formulations are mathematically equivalent.
Here ,we will be highlighting the basis of the popular Schrodinger
quantum theory only.

Schrodinger Quantum Mechanics:
Schrodinger proposed quantum theory to
explain the behaviour of microscopic particles taking into account
the wave nature of particles as suggested by de broglie.This
approach revolves around a partial differential equation now
popularly known as the Schrödinger equation, which describes
the behaviour of microscopic particles by means of a function
called the wave –function
There are two forms of Schrödinger equation :
1. Time dependent ( used for non-conservative systems
where energy changes with time)
2. Time independent ( deals with conservative systems
where energy of the system remains constant with respect to
time).

1.The wave –function is a function of particle’s position and
time ,
(x,y,z,t) in the time dependent Schrödinger equation , whereas it is a function of
position only ,
(x,y,z) in the time independent Schrödinger equation.
Over here , we will be restricting our discussion to Schrödinger's
time independent quantum mechanics.

2.The time independent Schrodinger equation is of the form,
Where the Hamiltonian operation H^ ( an operation is a
mathematical command that tells you what to do and what
follows; for every measurable property or observable in classical
mechanics, there is a corresponding operator in quantum
mechanics)acts on the wave function si and the result is
proportional to the same wave function si (stationary state) and
the proportionality constant, E which is the energy of the state
si.H^ is taken as sum of kinetic energy operator (T^) and
potential energy operation (v^);
H^=(T^) +(v^)

Wave-Function: wave function si (also called state function or
Eigen function) is the store house of information and is the
heart of Schrödinger equation as it contains all the information
about the system it describes.
Interestingly, wave function in
itself does not have any explicit meaning. Max Born gave the
correct statistical interpretation of the wave function for which
he was awarded the Nobel Prize in 1954.According to born,si
has no physical significance. It is merely a mathematical
function of the coordinates of the system. He called si as
probability amplitude and si square or si.si quare.star,the
probability density of the system is the measure of probability
density at that point ( probability of finding a particles in a
given space).

In the context of atoms, an atomic orbital(si) is a three
dimensional region around the nucleus within which the
probability of finding an electron with a certain energy is
maximum . There is no limit to the number of solutions of the
Schrödinger equation. However, a number of conditions are
required for a physical realistic solution of wave function. So, an
acceptable well behaved wave- function is the one which is single
valued, continuous and doubly differentiable, finite, satisfy
boundary conditions and normalized.
Operation of Quantum Mech-anics
Having studied the basics of Schrödinger quantum
mechanics, the problem is how to solve a system quantum
mechanically using the Schrödinger equation the following four
steps are followed:
1. Writing the Schrödinger equation for the system
2. Defining boundary conditions
3. Solution of Schrödinger equation
4. Extracting information out of wave function.

With this four step mechanism one can obtain the
entire information about the system in terms of wave function
and associated energy for a given state. In fact for solving any
problem quantum mechanically ,this four step process is
required to be followed. However, the most difficult part of
this four step process is the solution of Schrödinger equation
as there is no universally accepted unique method to solve this
equation. The entire quantum science revolves around the
solution of this equation for a given system.
Quantum Mechanics & Atomic Structure:
Several attempts were made to explain the structure of an
atom but the correct interpretation came in the 19th century
with quantum mechanics.
The simplest chemical system-the
hydrogen atom-consists of one electron and one nucleus. The
most rewarding outcome of the solution of Schrödinger

Equation for the hydrogen atom is the occurrence of a set of integers called quantum
Number, which are n( Principal quantum number; n=1,2,3….) l( Azimuthal quantum
number=0,….n-1) and m(Magnetic quantum number;m—1,…,0…,+1).The quantum
numbers so defined help to designate the electron present in an orbital. The
distribution of electrons of an atom in its various orbital's gives the electronic
configuration. So, the quantum mechanical solution of the hydrogen atom lays the
foundation of the Atomic structure.
Quantum Mechanics & Spectroscopy :
Quantum mechanics provides the theoretical basis of spectroscopy, which is the
study of the interaction of electromagnetic radiation with matter. Spectrum is observed
during transition in a state of a system and this transition from one energy level to the
other is selective (selection rules),i.e., not all transitions are allowed which is a
consequence of quantization (or discreteness) of energy as given by quantum mechanics
( time-dependent Schrödinger equation) .

Beside quantum mechanic has not only assisted chemistry but
almost all other disciplines of science such
as,physics,biology,medicine,computing and so on over the year
evolving a better understanding of nature and has also
precipitated a new block of super-smart real-time applications…..

Various structural properties are obtained using quantum mechanical
interpretations of spectra, which help in structure elucidation. For the
hydrogen atom, the quantum mechanical results so obtained successfully
predict all aspect of the hydrogen atom spectrum.
Quantum Mechanics & Chemical Bonding:
A Chemical bond may be defined as the force that holds
the atoms together in a molecules. However, the Schrodinger
equation can not be solved exactly for a multi-electron system
due to the presences of electron-electron repulsion terms in the
Hamiltonian. The fundamental difficulty arises due to the fact
that each electron repels every other electron so that the motion
of each electron is dependent on the motion of all the other.
However, solution of reasonable accuracy can be obtained using
approximate methods. And henceforth, the chemical bonding in
molecules is explained quantum mechanically via two popular
theories Viz., Valence Bond Theory(VBT) and Molecular
Orbital Theory(MOT).

Both these approaches assume a guess/approximate wave –
function but the physical interpretation is different. Valence Bond Theory
considers bond formation by overlapping of valence electrons in atomic
orbital's and gives the concept of hybridization. Whereas, Molecular Orbital
Theory describes bonding in terms of the combination and arrangement of
atomic orbital's to form molecular orbital's that are associated with the
molecule as a whole.
This way the quantum mechanical treatment has been extended via
more appropriate approximations to conjugated molecules (a popular
approach to study the structure of conjugated molecules is via Hückel
Molecular Orbital Theory), complexes and even polymers. The results/trends
obtained from quantum mechanical approximations of various multi-electron
atoms and molecules are in good agreement with the experimental results and
it is for this reason that quantum mechanical wave-function interpretations
have found acceptability.
One needs to remember that the entire quantum
mechanical interpretation lies in the solution of the Schrödinger equation
which gives energy(particles character ) and associated with the microscopic
matter. In fact the solution of the Schrödinger equation for wave function is
the most difficult part as there is no unique method to solve it.

If the Schrödinger equation is solvable exactly , we get an
exact wave-function for a system. But ,for multi-electron atoms or even for
molecules, the exact solution of Schrodinger equation is not possible. In such
cases , approximations are used where we build/guess an appropriate wave-
function based on certain reasonable parameters and then try to solve the
equation to obtain energy values.
Once the wave-function is known , one can
calculate any property of the system using appropriate quantum mechanical
operators via methods such as eigenvalue equation or mean value theorem.
The ultimate goal of quantum chemistry is to obtain wave-function si for a
given system. Conclusively if one understands the basis of quantum
mechanics, one can apply and solve any problem following the four-step
quantum mechanical operation.
Because of paucity of space and time,
detailed mathematical formulation and more factual discussions about the
subject are out of scope of this article. But one can say that quantum
mechanics is essential for understanding every aspect of chemistry.
Besides, quantum mechanics has not only
assisted chemistry but almost all other disciplines of science such as,
physics,biology,medicine,computing and so on over the years evolving a better

Understanding of nature and has also precipitated a new block of
super-smart real time applications which includes ultra-precise clocks,un-
crakable codes, super-powerful computers, improved microscopes, biological
compasses ,GPS,lasers,telecommunications, smart phones,MRI scanners-the
list goes on.

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Basic and fundamental of quantum mechanics (Theory)

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