Quantum Theory : Techniques & ApplicationsVibrational Motion

Achmad Rochliadi, Ph.D. Program Studi Kimia

Institut Teknologi Bandung

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Vibrational motionVibrational motion

A particle will have harmonic motion if the restoring force is propotional to its displacement.

Relation with potential energy is,

Giving,

The Schrodinger equation for the particles,

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The energy levelThe energy level

Boundary condition, allowed solution are for those:

The permitted energy are :

Energy different between two level are

Zero point energy of harmonic oscillator

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Energy of harmonic oscilator

The energy levels of a harmonic oscillator are evenly spaced with separation. Even in its lowest state, an oscillator has an energy greater than zero.

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The wave functionThe wave function

Particle having harmonic motion is trapped in a symmetrical well with the energy climb as x2. Its follow a Gausian function.

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The form of the wave functionThe form of the wave function

A solution to the Schrodinger equation of the harmonic oscilation will have the form of

The precise form of the wavefuncions are

With is a Hermite polynomial.See Followin table

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Hermite polynomials

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The wave function for the ground stateThe wave function for the ground state

For the ground state, the lowest energy state, the wave function is

The probability density is

The wavefunction for the first exited state

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The wave function for the ground stateThe wave function for the ground state

Wavefunction and probability distribution for ground state

Wavefunction and probability distribution for 1st exited state

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The wave function for the ground stateThe wave function for the ground state

Wavefunction for the 5 first states

Probability distribution for the first 5 states up to 20th

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Thank You

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Lecture Outline

The structure & spectra of hydrogenic atoms• The structure of hydrogenic atoms• Atomic orbital and their energies• Spectroscopic transitions and selection rules

The structures of many-electron atoms• The orbital approximation• Self-consistent field orbital

The spectra of complex atoms• Impact on astrophysics : Spectroscopy of star• Quantum defects & ionization limits• Singlet and triplet coupling• Spin-orbit coupling• Term symbol and selection rules

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Role of quantum mechanics in hidrogen like atom

Quantum mechanics is used to describe :• Electronic structure of atom• Arrangement of electron around nucleus

Types of atom : • Hydrogenic atom : one-electron atom or ion, H, He+, Li+, O7+

• Many-electron atom (polyelectronic atom), all neutral atom other than H

Schrodinger equation can be solved exactly forhydrogenic atoms

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Lecture OutlineLecture Outline

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Topic 1

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Simple harmonic oscilation

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