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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
9
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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
O-H broadens with Hydrogen bonding
CH3CH2OH
C-O
N-H broadens with Hydrogen bonding
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
19
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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
38
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