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Biju Patnaik University of Technology BPUT
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LESSON NOTES ON Applied Chemistry (1st SEM)
Learning Objectives: At the end of Class the student will be able
•
•
To understand the basics of molecular interactions.
To understand the basics of various heterogeneous equilibrium systems
MODULE-I
Basic concepts and postulates of quantum mechanics. :
Quantum mechanics is the science of the very small. Quantum mechanics explains the behavior of matter
and its interactions with energy on the scale of atoms and subatomic particles.
Postulates of Quantum Mechanics
There are six postulates of quantum mechanics as follows:
Postulate 1.
The state of a quantum mechanical system is completely specified by a function Ψ(x, y, z, t) that depends
on the coordinates of the particle(s) and on time. This function, called the wave function or state function,
has the important property Ψ ٭Ψ dτ that is the probability that the particle lies in the volume
element dτ located at r at time t.
The wave function must satisfy certain mathematical conditions because of this probabilistic
interpretation. For the case of a single particle, the probability of finding it somewhere is 1, so that we
have the normalization condition
It is customary to also normalize many-particle wave function to 1. The wave function must also be
single-valued, continuous, and finite.
Postulate 2.
To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum
mechanics. This postulate comes about because of the considerations raised: if we require that the
expectation value of an operator
is real, and then
must be a Hermitian operator. Some common
operator occurring in quantum mechanics are collected in Table:
Observable
Observable
Operator
Name
Symbol
Symbol
Position
Momentum
Kinetic energy
Potential energy
Total energy
Angular
momentum
=x/y/z

Postulate 3.
In any measurement of the observable associated with operator
, the only values that will ever
be observed are the eigenvalues , which satisfy the eigenvalue equation
This postulate captures the central point of quantum mechanics--the values of dynamical variables can be
quantized (although it is still possible to have a continuum of eigenvalues in the case of unbound states).
If the system is in an eigenstate of
with eigenvalue
, then any measurement of the quantity
yield .
Postulate 4.
If a system is in a state described by a normalized wave function
observable corresponding to
will
, then the average value of the
is given by
Postulate 5.
The wavefunction or state function of a system evolves in time according to the time-dependent
Schrödinger equation
The central equation of quantum mechanics must be accepted as a postulate
Postulate 6.
The total wave function must be anti symmetric with respect to the interchange of all coordinates of one
fermions’ with those of another. Electronic spin must be included in this set of coordinates.
The Pauli Exclusion Principle is a direct result of this anti symmetry principle.
de-Broglie's concept and uncertainty principle
de-Broglie’s concept
In 1924, Lewis de-Broglie proposed that matter has dual characteristic just like radiation. His concept
about the dual nature of matter was based on the following observations:(a) The whole universe is composed of matter and electromagnetic radiations. Since both are forms of
energy so can be transformed into each other.
(b) The matter loves symmetry. As the radiation has dual nature, matter should also possess dual
character.
According to the de Broglie concept of matter waves, the matter has dual nature. It means when the
matter is moving it shows the wave properties (like interference, diffraction etc.) are associated with it
and when it is in the state of rest then it shows particle properties. Thus the matter has dual nature. The
waves associated with moving particles are matter waves or de-Broglie waves.
Wavelength of de-Broglie waves: Consider a photon whose energy is given by
E= hυ= hc/λ
– – (1)

If a photon possesses mass (rest mass is zero), then according to the theory of relatively , its energy is
given by E=mc2 – – (2)
From (1) and (2),we have
λ = h/p = h/mc ------------- (3)
If instead of a photon, we consider a material particle of mass m moving with velocity v, then the
momentum of the particle, p=mv. Therefore, the wavelength of the wave associated with this moving
particle is given by: h/mv
λ = h/p (But here p = mv) --------(4)
This wavelength is called DE-Broglie wavelength.
Special Cases:
1. de-Broglie wavelength for material particle:
If E is the kinetic energy of the material particle of mass m moving with velocity v, then E=1/2 mv2=1/2
m2v2= p2/2m
Or
p=√2mE
Therefore by putting above equation in equation (4), we get de-Broglie wavelength equation for material
particle as:
λ = h/√2mE – – (5)
2. de-Broglie wavelength for an accelerated electron:
Suppose an electron accelerates through a potential difference of V volt. The work done by electric field
on the electron appears as the gain in its kinetic energy
That is E = eV
Also E = p2/2m
Where e is the charge on the electron, m is the mass of electron and v is the velocity of electron, then
Comparing above two equations, we get:
eV= p2/2m
Or p = √2meV
Thus by putting this equation in equation (4), we get the the de-Broglie wavelength of the electron as
λ = h/√2meV= 6.63 x 10-34/√2 x 9.1 x 10-31 x1.6 x 10-19 V
λ=12.25/√V Å
This is the de-Broglie wavelength for electron moving in a potential difference of V volt.
Uncertainty Principle:
In classical physics it is generally assumed that position and momentum of a moving object can be
simultaneously measured exactly i.e. no uncertainties are involved in its description. But in microscopic
world it is not possible. It is found that however refined our instruments there is a fundamental limitation
to the accuracy with which the position and velocity of microscopic particle can be known
simultaneously. This limitation was expressed by a German physicist Werner Heisenberg in 1927 and
known as 'Heisenberg's uncertainty principle'.
In microscopic particles we can observe two type of uncertainties viz.
Uncertainty in position and momentum
Uncertainty in energy and time
STATEMENT:
It is impossible to determine both position and momentum of an electron simultaneously.
If one quantity is known then the determination of the other quantity will become impossible
MATHEMATICAL REPRESENTATION:
Let
∆x = uncertainty in position
∆P = uncertainty in momentum
According to Heisenberg's uncertainty principle:
The product of the uncertainty in position and the uncertainty in momentum is in
the order of an amount involving h, which is Planck’s constant

P x x h/4
i
v x x ≥ h/4пm ----------- (ii)
RESULTS OF UNCERTAINTY PRINCIPLE:
It is impossible to chase an electron around the nucleus.
The principle describes the incompleteness of Bohr's atomic theory.
According to Heisenberg's uncertainty principle there is no circular orbit around the nucleus.
Exact position of an electron can not be determined precisely.
SIGNIFICANCE:
Heisenberg's uncertainty principle is not applicable in our daily life. It is only applicable on micro objects
i.e. subatomic particles.
The reason why the uncertainty principle is of no importance in our daily life is that Planck's constant 'h'
is so small (6.625 x 10-34joule-seconds) that the uncertainties in position and momentum of even quiet
small (not microscopic objects) objects are far too small to be experimentally observed. For microscopic
phenomena such as atomic processes, the displacements and momentum are such that the uncertainty
relation is critically applicable.
Introduction to Schrodinger Wave Equation:
Schrödinger Wave Equation:
1) Schrodinger wave equation is given by Erwin Schrödinger in 1926 and based on dual nature of
electron.
(2) In it electron is described as a three dimensional wave in the electric field of a positively charged
nucleus.
(3) The probability of finding an electron at any point around the nucleus can be determined by the help
of
Schrodinger wave equation which is,
∂2Ψ/∂x2 + ∂2Ψ/∂y2 + ∂2Ψ/∂Z2 + 8π2m (E-V)Ψ /h2 = 0
Where x,y, and z are the 3 space co-ordinates, m = mass of electron, h = Planck’s constant, E = Total
energy, V = potential energy of electron, Ψ = amplitude of wave also called as wave function, ∂ = for an
infinitesimal change.
The Schrodinger wave equation can also be written as,
∇2Ψ + (8π2m/h2) (E-V) Ψ = 0 Where ∇ = laplacian operator.
Physical significance of Ψ and Ψ2
(i) The wave function Ψ represents the amplitude of the electron wave. The amplitude Ψ is thus a function
of space co-ordinates and time i.e. Ψ = Ψ (x, y, z.....t)
(ii) For a single particle, the square of the wave function (Ψ2) at any point is proportional to the
probability of finding the particle at that point.
(iii) If Ψ2 is maximum than probability of finding e- is maximum around nucleus and the place where
probability of finding e- is maximum is called electron density, electron cloud or an atomic orbital. It is
different from the Bohr’s orbit.
(iv) The solution of this equation provides a set of number called quantum numbers which describe
specific or definite energy state of the electron in atom and information about the shapes and orientations
of the most probable distribution of electrons around the nucleus.
Derivation of Schrodinger’s wave equation:
Where, ψ’(x) = ∂ψ/∂x and
ψ’’(x) = ∂2ψ/∂x2
Now wavelength λ and momentum p of the wave
are related to each other by the
following equation called de Broglie wavelength
equation,

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