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What are the main differences between a quantum and classical system? How does one can distinguish them?

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closed as too broad by CuriousOne, ACuriousMind, Martin, Kyle Kanos, John Rennie May 13 '15 at 6:21

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ This question might be a bit too broad... $\endgroup$ – hft May 12 '15 at 21:46
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    $\begingroup$ The main difference between quantum systems and classical ones seems to be that quantum systems actually exist, while classical systems are only an approximation of the behavior of quantum systems in certain limits (e.g. large mass/energy, high temperature, long timescales). Beyond that your question is way too general, at least for my taste. What do you want to know in detail? $\endgroup$ – CuriousOne May 12 '15 at 21:46
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    $\begingroup$ @hft It's broad but a darn good question, IMO. $\endgroup$ – DanielSank May 12 '15 at 21:50
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    $\begingroup$ Again, ALL systems are quantum systems. The only thing for you to decide is whether you care for the quantum effects, or not. That is a choice made by the precision of the experiment that you set up. If it's precise enough, you will observe the quantum nature, if it isn't, then you may not. Some quantum phenomena, like the quantum behavior of light (or superconductivity or even the existence of matter), can not be eliminated on any level of precision of observation. That is exactly what Max Planck found out when he came up with a working explanation for the black body spectrum. $\endgroup$ – CuriousOne May 12 '15 at 21:55
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    $\begingroup$ @DanielSank So darn good that no one is answering it... $\endgroup$ – hft May 12 '15 at 23:26
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What are the main differences between a quantum and classical system? How does one can distinguish them?

An experimentalist's answer, in other words, why do we need two different theories.

In a classical system if we do experiments with massive bodies , they have in principle well measured (x,y,z) positions at time t. From falling apples to the planetary system an elegant mathematically theory was developed called classical mechanics, modeling our everyday world .

In studying the wave properties of light the theory of classical electrodynamics was developed, complementing classical mechanics describing and predicting new experimental situations.

With thermodynamics and statistical mechanics the tool box of theories for describing physical reality was considered complete. There even were statements of prominent physicists of the nineteenth century that now only engineers would be needed, physics was complete.

Then the worm turned, because experimental observations appeared that were not predicted and could not be theoretically explained within the classical system, and the mathematical theory of quantum mechanics had to be invented.

What were these observations that separate a classical from a quantum mechanical system?

1) Black body radiation, which could not be explained classically and had to assume that the electromagnetic radiation came in quanta, units, carrying energy h*nu

2)The photoelectric effect which also needed quanta ( packages of energy) for the electrons emitted from materials

3)The light spectra from atoms, both emission and absorption, which instead of showing the classical continuous behavior showed absorption and emission lines, again showing discrete packages, quanta , of energy . For hydrogen these lines even followed mathematical series.

Trying to understand a model of how electrons could revolve around nuclei , incompatibility with the classical EM theory was found . There was no reason that the electrons would always stay in fixed orbits around the nucleus. Fixed orbit solutions with classical electromagnetism could be found, but there was no theoretical reason once disturbed that the electron would not fall into the nucleus diminishing its charge by one.

The constraint was imposed by Bohr, that the orbits had to be fixed/quantized with the Bohr atom. It could explain the Balmer series, but it was an ad hoc model, not a theory. The time came for the Schrodinger equation to enter the stage which allowed the development of a mathematical theoretical model that explained not only simple atoms , but set the stage for the further theoretical study of elementary particle interactions. It was complemented with the Born rule, which is the postulate that allows to relate theoretical predicted numbers to data

We are now at the stage where just these two frameworks , the classical and the quantum mechanical , are sufficient to explain observations and predict new phenomena successfully, mainly for large dimensions classically, and for small dimension quantum mechanically. The real size is determined , as stated in other answers, by the value of h_bar and the mathematical relations that constrain the solutions, which give rise to the Heisenberg uncertainty principle. Sometimes, as in superconductivity, quantum mechanical effects can exist in large dimensions, but in general the framework is the underlying one for small dimensions. As dimensions grow large the quantum mechanical mathematical formulations lead at the limit consistently to the classical formulations.

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The main difference between classical and quantum physics is the fact that observables (Hermitian operators whose eigenvalues determine the possible values of physical quantities of the system) do not commute. For classical systems commutativity is trivially satisfied.

A direct consequence of this is the uncertainty principle and the fact that via Bell's theorem there is no local realism.

If it wasn't for this simple but important fact, quantum mechanics could essentially be reduced to a classical (albeit stochastic) theory.

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$\newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}}$ $\newcommand{\ket}[1]{\left| #1 \right>}$

Quanta vs. Continuous

Quantum, as the name suggests, deals with quanta ie packages of something. For example for a given particle and given potential the energy is quantised. Eg. for quantum mechanical SHO the energy levels are $E_n=\hbar \omega (n+\frac{1}{2})$. Notice that the only allowed energies in this system are $E_0= \hbar \omega \frac{1}{2}$, $E_1= \hbar \omega \frac{3}{2}$, $E_2= \hbar \omega \frac{5}{2}$ etc. However there shouldn't be any confusion that QM forbids energies like $E=\hbar \omega \frac{1}{4}$ or any energy for that matter. For this specific potential and for a given $\omega$ however this is not a possible energy eigenvalue.

Position and Momentum vs. Wave Function

In classical mechanics a system is completely described by a point on the phase space, which usually consists of the momentums and positions of the objects involved. In QM an isolated system of one or more particles is completely characterised by a complex-valued function called the wave function and usually denoted with greek capital letter psi $\Psi$ or with lower case psi $\psi$. Note that the information about the position and the momentum of the particle is “encoded” in this wave function. Some times the system is described by a vector $\ket \psi$ called ket in a Hilbert space, which is of course mathematically equivalent to the wave function.

Deterministic vs. Probabilistic

Classical mechanics describes a deterministic system, which means that what will happen at the next instant of time is completely determined, where as QM describes a probabilistic system. The norm squared of the wave function $|\psi|^2$ gives the probability density. Notice that in QM a wave function can be in superposition of more than one states ie. $\psi=\alpha \psi_1+\beta \psi_2$ then the probability that the particle is in state $\psi_1$ is given by $|\alpha|^2$ and the probability that the particle is in state $\psi_2$ is given by $|\beta|^2$. Since the total probability must add to one the wave function is usually normalized in such a way that $|\alpha|^2+|\beta|^2=1$.

Mathematical Differences

There is also mathematical difference between them. CM almost always deals with real numbers. You can see here and there some complex numbers but they are mainly computational tools, which have no physical meaning. On the other hand QM is governed by imaginary numbers, which have actual physical meanings. In particular the wave function is a complex valued function, which can be interpreted as probability amplitude.

Equations of Motions

The equation of motion for classical physics is a nice second order ordinary differential equation known as Newtons Law:

$$\vec F = m \ddot a$$

Sometimes you also see it in the following form:

$$m \ddot a = - \nabla V$$

The equation of motion for QM is a second order in space and first order in time partial differential equation known as the Schrödinger equation:

$$i\hbar\pdv{}{t}\Psi(\vec r,t)=\frac{\hat { \mathbf {p}}^2}{2m}\Psi(\vec r, t)+V(\vec r)\Psi(\vec r, t)=-\frac{\hbar^2}{2m} \nabla^2 \Psi(\vec r,t)+V(\vec r) \Psi(\vec r,t)$$

or in matrix form as:

$$i\hbar\pdv{}{t}\ket{\Psi(\vec r,t)}=H\ket{\Psi(\vec r,t)}$$

Orders of Magnitude

The QM can be considered as the real world since it describes the world of atoms and what you see around you is made of atoms. The classical mechanics is usually an approximation of QM. For example you can take the mass of the particle to be large or the energy level of the system to be large and you usually arrive at classical predictions. Notice that QM deals with energies of the order $10^{-34}$ Joule and masses of the order $10^{-23}$ kg. This should give you a feeling of how small the scales are.

In general you can make the assumption that if you are dealing with particles like electrons, protons photons or whatever you should consider the system to be a quantum mechanical system. However, in most applications quantum objects can be considered as classical object for convenience.

There are probably more differences but these are the major ones that come to my mind.

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    $\begingroup$ Energy isn't quantized in such a naive way. Atoms and light, for instance, have a continuous spectrum and they can emit photons of any energy. Size is also not a criteria for quantum systems. We have built superconducting magnets with many cubic meters of volume and light retains its quantum nature over intergalactic distances. This has actually been measured with interferometric experiments. In the "small" world particles like electrons and protons can be treated classically for many applications, like in plasma physics or in the interaction with particle detectors. $\endgroup$ – CuriousOne May 12 '15 at 22:15
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    $\begingroup$ Please get your facts right. The OP is not being helped by anybody making completely false statements about quantum mechanics for the sake of simplification. If you don't know how to write correctly about physics, then don't write about physics. Fair? $\endgroup$ – CuriousOne May 12 '15 at 22:19
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    $\begingroup$ I can have any $\omega$ and any energy in that dispersion relation for photons. That's NOT a quantization condition. The quantization in the square well is a side effect of the classical potential, which, strictly speaking, doesn't exist, either, moreover, it lets you create ANY energy eigenstates you like by picking the potential. It's an 80 year old intermediate step towards quantum field theory that misses what this is really about. $\endgroup$ – CuriousOne May 12 '15 at 22:27
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    $\begingroup$ To be honest with you, I think your effort shows that you are really trying to understand the subject for yourself, but I also think you are way too invested in the details right now to put the broader question of the OP in the right context. I could hand you a three pound magnet and none of its magnetic properties and hundreds of J of magnetic field energy would have ANY explanation in classical mechanics. You can't just throw a small energy or mass around and say "That's QM!". No, it isn't. Every solid body, every molecule of air around you is QM! $\endgroup$ – CuriousOne May 12 '15 at 23:57
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    $\begingroup$ There's a large number of details in this answer which are incorrect or misleading. However, its main drawback is that it misses the point completely: you are doing very little to clarify (and quite a bit to further confuse) on how and why exactly QM is different from classical mechanics, in a fundamental way, and in terms that the OP can understand. $\endgroup$ – Emilio Pisanty May 13 '15 at 0:16
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In a classical system (as we have studied), the behavior of a body can be exactly determined. For example, in projectile motion, the position, velocity, and acceleration of a system can be determined exactly by knowing only the time $t$ plus your initial conditions. In terms of size a classical system usually deals with macroscopic systems.

In a quantum system, you are bound to uncertainties, and therefore, to a range of values in determining the motion of the body, as stated by Heisenberg Uncertainty Principle. In addition, we are dealing with particles and Waves, and the energies that these waves exhibit.

But quantum systems can also be treated classically. For example, in geometric Optics, we focus on the path that light travels. However, if we focus on the particles of light that were in motion, we are dealing with quantum systems of an ensemble of particles.

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    $\begingroup$ "In a quantum system... we are dealing with particles and Waves [sic], and the energies that these waves exhibit." In classical systems we also deal with particle and waves. This does not differentiate between quantum and classical systems. $\endgroup$ – hft May 13 '15 at 5:39

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