Understanding Energy Levels and Photons in Quantum Harmonic Oscillators, Slides of Mechanics

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QUANTUM

HARMONIC

OSCILLATO

R

W H A T I S

Q U A N T U M

H A R M O N I C

O S C I L L A T O

R?

H T T P S : / / W W W. Y O U T U B E. C O M / W A T C H

? V = 5 P 1 9 H R O Y 9 V K

SAMPLE PROBLEM 1: VIBRATION IN

A SODIUM CRYSTAL

A sodium atom of mass vibrates with simple harmonic motion in a crystal. The potential energy increases by when the atom is displaced from its equilibrium position. (a) Find the angular frequency, according to Newtonian mechanics. (b) Find the spacing of adjacent energy levels in electron volts. (c) If an atom emits a photon during a transition from one vibrational level to the next lower level, what is the wavelength of the emitted photon? In what region of the electromagnetic spectrum does it lie?

𝑈 = 0. 0075 𝑒𝑉 ¿ 1. 2 × 10

− 21 𝐽 (^) 𝑥 = 0_._ 014 × 10 − 9 𝑚 𝑈 = 1 2 𝑘 𝑥 2 𝑘 = 2 𝑈 𝑥 2 = 2 ( 1_._ 2 × 10 − 21 𝐽 ) 0_._ 014 × 10 − 9 𝑚 = 12_._ 2 𝑁 / 𝑚

SAMPLE PROBLEM 1: VIBRATION IN

A SODIUM CRYSTAL

A sodium atom of mass vibrates with simple harmonic motion in a crystal. The potential energy increases by when the atom is displaced from its equilibrium position. (a) Find the angular frequency, according to Newtonian mechanics. (b) Find the spacing of adjacent energy levels in electron volts. 𝜔 = √ 𝑘 𝑚 ¿ √ 12_._ 2 𝑁 / 𝑚 3_._ 82 × 10 − 26 𝑘𝑔

¿ 𝟏. 𝟕𝟗 × 𝟏𝟎

𝟏𝟑 𝒓𝒂𝒅 / 𝒔

ℏ 𝜔 ¿(^1.^^054 ×^^10

− 34 𝐽 ∙ 𝑠 )( 1_._ 79 × 10 13 𝑠 − 1 ) ℏ 𝜔 = 1_._ 88 × 10 − 21 𝐽

SAMPLE PROBLEM 1: VIBRATION IN

A SODIUM CRYSTAL

A sodium atom of mass vibrates with simple harmonic motion in a crystal. The potential energy increases by when the atom is displaced from its equilibrium position. c) If an atom emits a photon during a transition from one vibrational level to the next lower level, what is the wavelength of the emitted photon? In what region of the electromagnetic spectrum does it lie? 𝜆 = 𝟏. 𝟎𝟓 × 𝟏𝟎 − 𝟒 𝒎 = 𝟏𝟎𝟓 𝝁 𝒎

SAMPLE PROBLEM 2: VIBRATIONAL

ENERGIES OF THE HYDROGEN

CHLORIDE MOLECULE

The HCl diatomic molecule consists of one chlorine atom and one hydrogen atom. Because the chlorine atom is 35 times more massive than the hydrogen atom, the vibrations of the HCl molecule can be quite well approximated by assuming that the Cl atom is motionless and the H atom performs harmonic oscillations due to an elastic molecular force modeled by Hooke’s law. The infrared vibrational spectrum measured for hydrogen chloride has the lowest-frequency line centered at. a) What is the spacing between the vibrational energies of this molecule? b) What is the force constant k of the atomic bond in the HCl molecule?