Dear Anil,

Please read the following text:

Classical vs. Quantum Mechanics:

The Harmonic Oscillator in One Dimension

The harmonic oscillator is the model system of model systems. We study it here to characterize differences in the dynamical behavior predicted by classical and quantum mechanics, stressing concepts and results. More details and mathematical formalism can be found in textbooks [1,2]. Our model system is a single particle moving in the x dimension connected by a spring to a fixed point. Its potential energy is , where k is the spring constant. Stiff springs are described by large k's.

Classically, this oscillator undergoes sinusoidal oscillation of amplitude and frequency , where E is the total energy, potential plus kinetic. In equilibrium at tempertureT, its average potential energy and kinetic energy are both equal to  KBT/2; they depend only on temperature, not on the motion's frequency.

Quantum mechanically, the probability of finding the particle at a given place is obtained from the solution of Shrödinger's equation, yielding eigenvalues and eigenfunctions . For the one dimensional harmonic oscillator, the energies are found to be , where is Planck's constant, f is the classical frequency of motion (above), and n may take on integer values from 0 to infinity. The turn out to be real functions involving the Hermite polynomials. From equation 1, only the ground state ( ) is populated as the temperature . The energy does not go to zero but to . The corresponding zero-point motion is a quantum mechanical phenomenon. Classically, there is no motion as . Thus, we expect that quantum mechanics predicts more motion than classical mechanics, especially at low temperature.

http://cmm.cit.nih.gov/intro_simulation/node3.html

The classical harmonic oscillator will have a probability density that peaks at the edges of its motion where it moves most slowly. It will have a lower but non-zero probability density in the centre in the centre where it moves most quickly.

In the quantum harmonic oscillator there will be:

1) Discrete energy levels.

2) The wave like aspect of the particle will show itself and the probability density will wave up and down. Each successive increase in energy level will result in a probability density with a shorter wavelength and an extra peak. Note that the troughs between the peaks go to zero and there may be many places where the particle will not be found.

Interestingly the lowest energy level, that has just one peak, will have its peak right in the centre, a point that corresponds to a trough in the classical regime.

As the energy gets higher, the many peaks will begin to look like the classical regime with the highest peaks at the edges. There will still be peaks and troughs but supposedly they will become too narrow to be noticeable.

You can see a picture of this here:

http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

3) Quantum tunnelling will show itself. The probability density will remain non-zero in regions where, from a classical point of view, the particle should not have enough energy to get to.

Source(s):

Harmonic Oscillator:

http://en.wikipedia.org/wiki/Harmonic_oscillator

Quantum Harmonic Oscillator:

http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

Hoping this will be helpful,

Rafik

In QFT the - standard - electromagnetic field is a superposition of suitable 3-dimensional oscillator states |n(1),n(2),n(3)>. It is a matter of taste whether you see a single photon |n> = exp(ikx) as a classical or quantal field.

@ Rafik Karaman : "probability of finding the particle", "more motion than classical mechanics". Good demonstration of how much corpuscularist is the standard believing, taught everywhere.

Much, much work will be needed to obtain they open their eyes on such widely spread illusions.

The notion of "position" requires that you are in the macroscopic domain, clearly above the inferior horizon of validity of the standard and scholar geometry we were taught at school.

Suppose that the "oscillating thing" would be an electron, where could you find a something smaller than an electron to inform you on where is this electron, how small it is, and what is its color ? On standard videos for the atoms, usually the electrons are green and orbiting balls, and the nucleous is in magenta, looks like a strawberry...

When I observe the waves breaking at the shore, from the top of the cliff, the light from the moon does not perturb the waves. The light impulsions are very negligeable compared to the inertias of these gravity waves. But to contemplate the "motion" of an oscillating electron ? How do you plan to experimentally prove your statements on "motion" ?

Not so easy to find real harmonic oscillators in microphysics. Except the electromagnetic ones, they are all anharmonic, in molecules and in condensed matters. That is why any condensed matter dilates with temperature, any phonon is anharmonic.

What is new with the quantic domain is the basic level : 1/2 . h.\nu

But we can not associate any observable "motion" to it. Only an energy and a frequency. Let's take the exemple of a molecule of gas that can absorb or emit an infrared photon at their resonance(s) : CO_2, O_3, H_2O, CO. The simplest is the CO : diatomic so only one mode, extension-compression and the main frequency, depending of the isotopes, is 2170 cm^-1. Without this rest-frequency, this molecule could not emit backward-in-time waves, for handshaking a transaction with a potential emitter at its resonance frequency.

Another exemple is a dye molecule, with a delocalized electron, oscillating from one atom or bond to another. So there is already an active rest-frequency, shouting loudly that the molecule can swallow a photon at the resonance, and evacuate the energy by phonon means. Even in the visible range, the captured photons are much larger than the absorbing molecule, so the absorption demands a convergence of the wave on its target.

I have no proof, but I suspect that at the mesoscopic scale, you still will not have a behaviour approaching the macroscopic oscillator. What you need is real massive objects, say cilia in a cochlea.

The occupation number of its energy levels: high occupation numbers that describe a large number of quanta, behave, rather, classically, low occupation numbers, that describe a small number of quanta, whose ``shot noise fluctuations'' can be resolved, behave quantum mechanically. 

The temperature, similarly, sets a scale. So for kT >> hbar w, where w is the fundamental frequency, the number of quanta involved is very large and the behavior is classical; for kT ~ hbar w, quantum effects become significant.

@Stam Nicolis: eventually you may explain to me the difference between 1 classical photon and 1 QFT photon, with E=hxnu. I want to know whether the physical object 1 photon exists in two versions: classical and quantum mechanical. (similar to the Coulomb problem)

Very curious. A.S. 

@Anton Schober. Maybe no physical "object".

W. Lamb had bursted "There is no such thing as a photon !"

For us, transactionnist physicists, here is the definition :

A photon is a successful transaction between three partners :

An emitter,

An absorber,

and the vacuum or medium between them (a nicol for instance...).

This transaction transfers a quantum h from the emitter to the absorber by electromagnetic means. The transfer obeys to Maxwell's equations, so the width of the Fermat's spindle is limited and an approximate value has been published.

The physics of the handshake before a transaction succeeds has not yet been well established : when de Broglie conjectured the law of harmony of phases (1923), and established the phase velocity of an electron (c2 / v), he meant a one component electron, and for 1928 we know that it has four components, so the broglian noise everywhere must be by advanced and by retarded Dirac-de- Broglie waves.

(How do you visualize a photon? - ResearchGate. Available from: https://www.researchgate.net/post/How_do_you_visualize_a_photon/15)

@Jacques Lavau: "How do you visualize a photon?"

I visualize nothing: I prepare the well defined physical state: Energy density = 1/2(E²+B²) = h bar omega (=1 photon (E=hnu) /vol). First, from the solar constant I know, there pass a rather precise number of photons (Maxwell distr.) through square meter per second. I controll this energy flux by heating a black absorber. Next I polarize this junk by reflecting it on two plane window glasses. Then I reduce the flux down to 1 photon/time unit. which is simple, and then the question: is this now a "classical" or a QFT photon?

have a nice day

Please explain what could be a "classical" photon.

@Jacques Lavau 

a "classical photon" is a physical object (="state") which is fixed by its energy momentum 4-vector k = (E=h bar omega; p, with |p|=E/c) and its spin s = +-p/|p|. You did not ask but the QFT photon is just |1), in obvious notation.

By the expression for the energy, alone, one realizes immediately that the term ``classical photon'' doesn't make sense, since classical means that all terms proportional to powers of hbar are neglected. In addition, photons, when they can be counted, don't follow classical, Boltzmann, statistics, but Bose-Einstein statistics (as was found by Bose).  Bose-Einstein statistics can be approximated by Boltzmann statistics, in the classical limit, that is when kT >> hbar w. This is textbook stuff, also. 

``classical photon'' is only a word for the physical state p = (E;p=E/c) and spin 1. Next, what are terms proportional powers of h bar? exp(i h bar (kx - omega t))??? 

very curious. A.S.

But beyond the math steno (shorthand), what is its length ? Length of coherence ? What is its width and the evolution of the width along the journey ? What is its precision in frequency ? What is the evolution of its polarity ? How do you predict its behaviour in front of a diffraction grating ? Can you dimension and design the grating device ? Why and how this one absorber has been chosen by this one emitter ? Why and how this one emitter has been chosen by this one absorber ?

Beyond the math steno, how a large photon can converge on a tiny molecule, and be absorbed by it ?

Lots of political hype nowadays on the convergence of some infrared photons on CO_2 or H_2O molecules, you see ? ...

As I understand, Schrodinger equation solutions describe almost classical picture for n >> 1, where n - level number.

@ Minas K. Balyan. Please be more precise about the final and initial conditions you silently mean.

Hic jacet lepus.

 No.

Hic jacet lepus : Here the hare lies.

 Do not hesitate to list the postulates you use, and those you attribute to other parties.

Only overt postulates may be evaluated and eventually criticized ; surrepticious postulates may escape any investigation.

In this discussion I have not spotted any mention of coherent states  https://en.wikipedia.org/wiki/Coherent_states   These are giving direct and straightforward answer to the posed question. Also, I have not spotted Landau levels for the charged particle placed in the constant magnetic field. Quantum mechanically such a system is harmonic oscillator. As for the photons...The answer really depends on the boundary conditions. Recall Max Planck treatment of the black body radiation. Electrons in synchrotron emit light which is NOT quantized! 

 @ Minas K. Balyan. Not even able to explicit the surrepticious postulates you silently use ? Not even able to differentiate the macrophysic domain from the individual quantic one ?

I'll help you, here is a bilingual list of the surrepticious postulates we do not more accept :

http://jacques.lavau.perso.sfr.fr/Physique/postulates.html

@ Arkady Kholodenko : "Electrons in synchrotron emit light which is NOT quantized"

Sure, an electron in a synchrotron does not oscillate between two stationnary states, so it is not very bothered by the frequency of the photonic transaction. However, the absorber may be more concerned by a resonant frequency, depending on its own physics.

I have the impression that something is missing in the previous discussion

a) I agree with Christian that the expression non quantized light is an oximoron

b) The radiation emitted by electrons in a bending magnet as it happens in synchrotron radiation, can be understood as the radiation field generated by a classical current

c) in this case the current radiates coherent (namely quasi-classical states)

@Christian Baumgarten. Much more simply : an "emittors only" theory can not predict which frequencies will be emitted by the ESRF ring.

However the absorbers can not read the Göttingen-København litterature, so they resolve the problem without you.

I don't quite understand the question.  When it comes to the correspondence between the classical harmonic oscillator (SHO) and the quantum harmonic oscillator: this is sussed out in optics.  The ground state of the quantum SHO has no correspondence with the classical SHO but if you have to take excited states of the quantum SHO, take linear combinations of them to get rid of the "ringing" which occurs in these excited states and then take a limit to achieve the classical limit where the particle or SHO pendulum spends most of its time near the critical points.