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## Classical Harmonic Oscillator

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### Classical Harmonic Oscillator Let us consider a particle of mass m attached to a spring At the beginning at t = o the particle is at equilibrium, that is no ... – PowerPoint PPT presentation

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Title: Classical Harmonic Oscillator

1
Classical Harmonic Oscillator
Let us consider a particle of mass m attached
to a spring
At the beginning at t o the particle is at
equilibrium, that is no force is working at it
, F 0
In general, according to Hookes Law
F -k x i.e. the
force proportional to displacement and pointing
in opposite direction and where k is the force
constant and x is the displacement.
Classically, a harmonic oscillator is subject to
Hooke's law.
2
Newton's second law says
F ma Therefore,
The solution to this differential equation is of
the form where the angular frequency of
oscillation is ? in radians per
second Also,
? 2p?, where ? is
frequency of oscillation

3
Potential Energy
• The parabolic potential energy V ½ kx2 of a
harmonic oscillator, where x is the displacement
from equilibrium.
• The narrowness of the curve depends on the force
constant k the larger the value of k, the
narrower the well.

4
Kinetic energy
5
Energy in Classical oscillator
E T V ½
kA2 .. How ???? Total energy is
constant i.e. harmonic oscillator is a
conservative system
6
Quantum Harmonic Oscillator
In classical physics, the Hamiltonian for a
harmonic oscillator is given by where µ
denotes the reduced mass The quantum
mechanical harmonic oscillator is obtained by
replacing the classical position and momentum by
the corresponding quantum mechanical operators.

7
Solution of Schr?dinger Equation for Quantum
Harmonic Oscillator
8
It is only possible if
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11
Electromagnetic Spectrum
Near Infrared
Thermal Infrared
12
IR Stretching Frequencies of two bonded atoms
What Does the Frequency, ?, Depend On?
• frequency
• k spring strength (bond stiffness)
• ? reduced mass ( mass of largest atom)
• is directly proportional to the strength of the
bonding between the two atoms (? ? k)
• is inversely proportional to the reduced mass
of the two atoms (v ? 1/?)

51
13
Stretching Frequencies
• Frequency decreases with increasing atomic
weight.
• Frequency increases with increasing bond energy.

52
14
IR spectroscopy is an important tool in
structural determination of unknown compound
15
IR Spectra Functional Grps
Alkane
C-C
-C-H
Alkene
Alkyne
15
16
IR Aromatic Compounds
(Subsituted benzene teeth)
CC
16
17
IR Alcohols and Amines
CH3CH2OH
C-O
Amines similar to OH
17
18
Question A strong absorption band of infrared
radiation is observed for 1H35Cl at 2991 cm-1.
(a) Calculate the force constant, k, for this
molecule. (b) By what factor do you expect the
frequency to shift if H is replaced by D? Assume
the force constant to be unaffected by this
substitution. 516.3 Nm-1 0.717
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20
Hermite polynomial
• Recurrence Relation A Hermite Polynomial at one
point can be expressed by neighboring Hermite
Polynomials at the same point.

21
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22
Quantum Mechanical Linear Harmonic Oscillator
It is interesting to calculate probabilities
Pn(x) for finding a harmonically oscillating
particle with energy En at x it is easier to
work with the coordinate q for n0 we have

23
Wave functions of the harmonic oscillator
24
Potential well, wave functions and probabilities
25
• Energy levels are equally spaced with separation
of h?
• Energy of ground state is not zero, unlike in
case of classical harmonic oscillator
• Energy of ground state is called zero point
energy
• E0 h?/2
• Zero point energy is in accordance with
Heisenberg uncertainty principle
• Show harmonic oscillator eigenfunctions obey the
uncertainty principle ????

26
Difference from particle in a box
• P.E. varies in a parabolic manner with
displacement from the equilibrium and therefore
wall of the box is not vertical.
• In comparison to the hard vertical walls for a
particle in a box, walls are soft for harmonic
oscillator.

27
Difference from particle in a box
• Spacing between allowed energy levels for the
harmonic oscillator is constant, whereas for the
particle in a box, the spacing between levels
rises as the quantum number increases.
• v0 is possible since E will not be zero.

28
Classical versus Quantum
• The lowest allowed zero-point energy is
unexpected on classical grounds, since all the
vibrational energies, down to zero, are possible
in classical oscillator case.

29
Classical versus Quantum
• In quantum harmonic oscillator, wavefunction has
maximum in probability at x 0. Contrast
bahaviour with the classic harmonic oscillator,
which has a minimum in the probability at x 0
and maxima at turning points.

30
Classical versus Quantum
• Limits of oscillation are strictly obeyed for the
classical oscillator. In contrast, the
probability density for the quantum oscillator
leaks out beyond the classical limit.

31
Classical versus Quantum
• The probability density for quantum oscillator
have n1 peaks and n minima. This means that for
a particular quantum state n, there will be
exactly n forbidden location where wavefunction
goes to zero. This is very different from the
classical case, where the mass can be at any
location within the limit.

32
Classical versus Quantum
• At high v, probability of observing the
oscillator is greater near the turning points
than in the middle.
• At very large v ( 20), gaps between the peaks in
the probability density becomes very small. At
large energies, the distance between the peaks
will be smaller than the Heisenberg uncertainty
principle allows for observation.

33
Classical versus Quantum
• The region for non-zero probability outside
classical limits drops very quickly for high
energies, so that this region will be
unobservable as a result of the uncertainty
principle. Thus, the quantum harmonic oscillator
smoothly crosses over to become classical
oscillator. This crossing over from quantum to
classical behaviour was called Correspondence
Principle by Bohr.

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37
?x ?ph/2
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