A particle performs harmonic motion if it experiences a restoring force proportional to its displacement, x, from a given point. i.e. F = -k x. k is called the force constant.
Since the force, F, is related to the potential energy, V, by the expression F = -dV/dx, we may write the following expression for the potential energy of a particle undergoing harmonic motion:
The potential energy of a harmonic oscillator can thus be seen to be parabolic:
The Schrodinger equation for a particle confined by a potential of this form (i.e. a particle that undergoes harmonic motion, a harmonic oscillator) is as follows:
Though the method is somewhat complex, it is entirely possible to solve this equation, and the properties of the solutions turn out to be very simple indeed. The permitted energy levels are:
where the quantum number ν can take values 0, 1, 2, 3….
It follows that the separation between adjacent levels is given by:
which is independent of ν.
This implies that the energy levels are uniformly spaced, like the rungs of a ladder. The separation between levels, ω, is negligibly small for large objects (with a large mass – note ω is inversely dependent upon m½) , but is significant for microscopic objects of small mass such as atoms.
Since the minimum value of ν is 0, it follows that a harmonic oscillator has a zero-point energy:
The physical reason for this is the same as that for the particle in a box. The particle is confined by potential walls, so the uncertainty in its position cannot be infinite. Thus to avoid a breach of the uncertainty principle, it can never have zero uncertainty in its linear momentum, and can thus never be stationary.
The wavefunctions that satisfy the Schrodinger equation for the harmonic oscillator are rather complex in form. It is thus helpful to consider the similarities between the harmonic oscillator and the particle in a box, to see if we can anticipate any of the properties of these solutions from the known properties of the solutions for the particle in a box.
In both situations, the particle is confined to a symmetrical well by potential energy walls that rise steeply from zero (and at sufficiently large displacements reach infinity). However, for the harmonic oscillator the potential increases as x2, a much gentler increase than the instantaneous increase for the particle in a box. We should thus expect that the particle will penetrate slightly (tunnel) into the walls. (The gentler increase of the potential with displacement can be taken as an indication that the wall is in some sense ‘softer’ than that for the particle in a box, and thus penetration into regions where classically the particle would not be allowed to exist can occur.)
|There is little to be gained by writing out the explicit forms of the wavefunctions of a harmonic oscillator, it is sufficient to know that the appearance of the wavefunctions is similar to those of a particle in a box. In fact, the harmonic oscillator wavefunction with ν = a has the same number of maxima, minima and nodes as the wavefunction for a particle in a box with n = a+1. The differences in the wavefunctions lie in the shape of the curves, those for the harmonic oscillator being less symmetric than those for the particle in a box. The first three wavefunctions for the harmonic oscillator are shown below, labeled with the appropriate value of ν:|
The probability density is based upon the square of these functions. At low quantum numbers, the particle is most likely to be found near the equilibrium position (X = 0), contrary to the classical expectation that it is most likely to be found at the amplitudes of its motion.
However, as the quantum number increases, the probability density builds up at the amplitudes – this is another example of the correspondence principle, where classical properties emerge from quantum mechanics in the limit of high quantum numbers.
(The reason that a classical particle is most likely to be found at the amplitude of the motion is that at these points, the energy of the particle has been entirely converted to potential energy, it has no kinetic energy. Thus the particle is instantaneously stationary at these points, and consequently it spends more time here than at any other point and is most likely to be found here.)