Classical%20Harmonic%20Oscillator

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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 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.
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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


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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.

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Kinetic energy
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Energy in Classical oscillator
E T V ½
kA2 .. How ???? Total energy is
constant i.e. harmonic oscillator is a
conservative system
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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.

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Solution of Schr?dinger Equation for Quantum
Harmonic Oscillator
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It is only possible if
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Electromagnetic Spectrum
Near Infrared
Thermal Infrared
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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/?)

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Stretching Frequencies

• Frequency decreases with increasing atomic
  weight.
• Frequency increases with increasing bond energy.
  

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IR spectroscopy is an important tool in
structural determination of unknown compound
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IR Spectra Functional Grps
Alkane
C-C
-C-H
Alkene
Alkyne
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IR Aromatic Compounds
(Subsituted benzene teeth)
CC
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IR Alcohols and Amines
O-H broadens with Hydrogen bonding
CH3CH2OH
C-O
N-H broadens with Hydrogen bonding
Amines similar to OH
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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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Hermite polynomial

• Recurrence Relation A Hermite Polynomial at one
  point can be expressed by neighboring Hermite
  Polynomials at the same point.

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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

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Wave functions of the harmonic oscillator
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Potential well, wave functions and probabilities
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• 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 ????

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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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?x ?ph/2
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