The Harmonic Oscillator, страница 2


Just from symmetry:

                                  odd                                                         odd

Average displacement & average momentum = 0

IR spectroscopy     Þ       H.O. selection rules


Intensity of vibrational absorption features          


Vibrational transition


1)       Dipole moment of molecule must change as molecule vibrates       Þ      

HCl can absorb IR radiation, but N2, O2, H2 cannot.

2)       Only transitions with            allowed (selection rule).

                     (Prove for homework.)



Classical turning points


Classical H.O.:        Total energy     

oscillates between K and U.

Maximum displacement x0 occurs when






all the energy is potential.

  is the “classical turning point”

The classical oscillator with energy ET can never exceed this displacement, since if it did it would have more potential energy than the total energy.



At high n, probability density begins to look classical, peaking at turning points.

Non-zero probability at x > x0!

Prob. of (x > x0, x < -x0):

“Complementary error function”

tabulated or calculated numerically

Prob. of (x > x0, x < -x0) = erfc(1) = 0.16

Significant probability!

The oscillator is “tunneling” into the classically forbidden region. This is a purely QM phenomenon!

Tunneling is a general feature of QM systems, especially those with very low mass like e- and H.


Even though the energy is less than the barrier height, the wavefunction is nonzero within the barrier! So a particle on the left may escape or “tunnel” into the right hand side.

Inside barrier:      


Solutions are of the form          


If barrier is not too much higher then the energy and if the mass is light, then tunneling is significant.

Important for protons (e.g. H-bond fluctuations, tautomerization)

Important for electrons (e.g. scanning tunneling microscopy)

Nonstationary states of the QM H.O.

System may be in a state other than an eigenstate, e.g.

      with     (normalization),  e.g.         

Full time-dependent eigenstates can be written as



System is then time-dependent:

          What is probability density?

Probability density oscillates at the vibrational frequency!


What happens to the expectation value <x>?

<x>(t) oscillates at the vibrational frequency, like the classical H.O.!

Vibrational amplitude is  


Relations among Hermite polynomials

Recall H.O. wavefunctions


                     Normalization                   Gaussian

Hermite polynomial

Generating formula for all the Hn:

A useful derivative formula is:

Another useful relation among the Hn’s is the recursion formula:

Substituting    above gives

Use these relations to solve for momentum <p>(t)

<p>0 = 0


<p>1 = 0


To solve integral   use relations among Hn’s


Average momentum also oscillates at the vibrational frequency.