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Quantum Harmonic Oscillator

Page history last edited by Alexis Jensen 13 years, 5 months ago

The Merge of Classical and Quantum Mechanics

-Quantum Harmonic Oscillator-

Alexis Jensen, Weiwei Wu, Henry Kuang, Victoria Washington


Classic Mechanics


     The study of classical and quantum mechanics reveals that the universe only displays classical elements of physics when the scale of the matter is brought to a relatively large, everyday scale. Classic mechanics include the first set of laws proposed by Isaac Newton in his studies of mechanics. The ultimate distinguishing factor between classical and quantum mechanics is that classical mechanics is predetermined: it follows the idea that the motion of particles can be accurately predicted.

Certain concepts of classic mechanics can be transferred to quantum mechanics. One such case is the harmonic oscillator. In classical mechanics, the harmonic oscillator is a model for any object that follows a repeated motion. This can include springs, pendulums, and sin waves. The implications of the transfer from classic mechanics to quantum mechanics carries the harmonic oscillator with it, making it a simple, repeating function into a group of probabilities.


Quantum Mechanics


     The field of quantum mechanics began to address this striking difference between classical and quantum physics – that quantum physical properties are not continuous and can only occupy discrete values. Central to the development of this field was Max Planck’s research regarding black body radiation.  Planck observed that light shined on a pure sample was only emitted and absorbed in discrete amounts, which contradicted the standing theory that light was an electromagnetic wave (which would expect light to be emitted and absorbed over a continuous spectrum).  From this research he identified Planck’s constant, h, that governs much of atomic structure and behavior, such as the oscillating nature of atomic bonds and electron behavior.

As mentioned above, because atomic bonds are not rigid structures, but rather involve vibrational motion and mobile electrons, a harmonic oscillator can be used to model their behavior.  If bonds behaved like classic harmonic oscillators, energy of the electrons within the bond could have any energy and that the endpoints of the oscillators (where it changes direction) would have the highest probability density of locating the oscillator since this is where the electrons must “slow down” and change directions.  This is actually not the case.  The oscillator can only take discrete energy values and the center of the oscillator has the highest electron density probability.   The explanation of this phenomena stems from the de Broglie wavelengths and Schrodinger’s electron wave functions.  It was proposed that an electron (or rather, the probability of finding electrons) exists as a spherical standing wave function.  Hence, since wave functions must be continuous, there are only discrete energies which the wave function can occupy.

One analogy to show the difference between classical mechanics and quantum mechanics is a light bulb compared to a strobe light.  When viewing motion under a normal light, it appears continuous.  However, if the same movement were looked at under a strobe light it would appear discrete and jerky.  This is because only certain times (in this case time rather than energy is the “quantized” value) are illuminated.  However if you were to increase the frequency of the flashes from the strobe light, it would flash fast enough that the human eye would not be able to distinguish the change and the movement would again appear continuous. 


Classical and Quantum Merge


     When thinking of nature on the atomic level, it is necessary to consider the quantum mechanical model. However on a larger scale, classical mechanics have a better fit. Even though the underlying principles of both quantum and classical mechanisms have their differences, they must also at some point overlap because the matter that makes up our world consists of atoms. This concept of the merging of the quantum and classical models was introduced by Bohr and is called the "correspondence principle" in which both classical and quantum behaviors are considered. The greatest difference between the two models is noticed at lower quantum numbers or lower energy levels (n values). It is at higher quantum values, that quantum and classical models begin to look very similar.

A direct application of the correspondence principle is in the function of the quantum harmonic oscillator. As the name suggests, there is a spring relationship between two atoms., just like in a classical harmonic oscillator; which is simply the model to study the vibrational motion in the classical world.  However, in the classical harmonic model, the probability function, which is the square of the wave function, gives greatest value when the two atoms are at the ends of the oscillator. In contrast, the quantum mechanical model has the greatest probability of finding the two atoms at the midpoint. Also, in the quantum harmonic oscillator, the energy levels are at discrete and evenly spaced values, whereas the classical harmonic model has continuous and differentiable energy levels.. The quantum model produces a potential diagram (see Figure to right) that reasonably fits the quadratic function. For both the classical and quantum mechanical models, the natural frequencies of harmonic oscillators for diatomic molecules are the same. Natural frequencies are associated with the rhythm of the moving oscillator and found by dividing the number of cycles per second of time. However, it is because of the quantum harmonic oscillator that we are able to consider non-diatomic systems. When we climb up to higher energy states in the quantum oscillator model by increasing the energy level, the most probable position of the particle will shift away from the center and towards the edges of potential energy diagram (the two vertical lines in the above figure). The distances between the peaks decrease. This will make overall trend of the probability begin to look like the classical probability model, which is shown by the purple quadratic in the figure above.

     Another thing worth mentioning here is that the classical probability density function is contained between the vertical lines while the quantum probabilities extend out of the classical limit. This is known as quantum tunneling. As the energy level increases, the region outside the classically forbidden region decreases. Therefore, it will look more like classical model.













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