Loading...

PPT – Classical Harmonic Oscillator PowerPoint presentation | free to download - id: 7900fe-YjM4Z

The Adobe Flash plugin is needed to view this content

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.

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

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.

Kinetic energy

Energy in Classical oscillator

E T V ½

kA2 .. How ???? Total energy is

constant i.e. harmonic oscillator is a

conservative system

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.

Solution of Schr?dinger Equation for Quantum

Harmonic Oscillator

It is only possible if

(No Transcript)

(No Transcript)

Electromagnetic Spectrum

Near Infrared

Thermal Infrared

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

Stretching Frequencies

- Frequency decreases with increasing atomic

weight. - Frequency increases with increasing bond energy.

52

IR spectroscopy is an important tool in

structural determination of unknown compound

IR Spectra Functional Grps

Alkane

C-C

-C-H

Alkene

Alkyne

15

IR Aromatic Compounds

(Subsituted benzene teeth)

CC

16

IR Alcohols and Amines

O-H broadens with Hydrogen bonding

CH3CH2OH

C-O

N-H broadens with Hydrogen bonding

Amines similar to OH

17

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

(No Transcript)

Hermite polynomial

- Recurrence Relation A Hermite Polynomial at one

point can be expressed by neighboring Hermite

Polynomials at the same point.

(No Transcript)

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

Wave functions of the harmonic oscillator

Potential well, wave functions and probabilities

- 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 ????

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.

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.

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.

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.

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.

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.

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.

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.

(No Transcript)

(No Transcript)

(No Transcript)

?x ?ph/2

(No Transcript)