What is the difference between classical and quantum law?

[ndung200790]

Please teach me this:
I think that the classical physical laws are the relations between expectation values of macroscopic objects(the values are taken average on ''quantum values'').The quantum physical laws in one sense are the relations between operators.Then there are a great difference between classical and quantum laws.So I do not understand why they say that the classical things are the leading order of aproximation of quantum things(e.g they say in leading order of aproximation the electromagnetic potential is square(e)/square(momentum)(in momentum space) when they considering renormalization group(considering the flow of electric charge under changing renormalization condition)).
Thank you very much in advance.

 

[jrlaguna]

Classical and quantum mechanics use different formalisms, you're right. In QM it is very usual to use operators for magnitudes which are described by just numbers in classical mechanics. But the macroscopic world is classical (ok, more or less) (otherwise, QM would have been discovered by Newton), so in some limit, QM should give classical mechanics, right? But his limit has proved to be extremely non-trivial, and there are still lots of discussions about it. In some cases, it is straightforward. For example, a single particle in a potential. If the mass grows a lot, you reach the classical regime, where the expected values of operators become closer and closer to the classical results, and fluctuations become negligible. But even in this case there are subtleties.

An analogy may help. Consider a physical theory for a classical particle subject to some deterministic forces and a random noise. For example, brownian motion. Then, as you increase the mass of the particle, the noise is less and less effective, so you may say you're reaching the "deterministic limit".

Hope it was useful...

 

[ndung200790]

So,now I think that the quantum physics is ''broader''(meaning involve) than classical physics,because some macroscopic phenomena are the results of quantum physics(e.g superconductivity,superfluidity...).Then classical physics is not enough to describe the macroscopic world.But in the ''narrow'' domain of classical physics,the classical is the limit of quantum physics.But do anyone please prove this for me?Does the fluctuation become small in fluctuation theory when the mass of object is large?Does the spectral of physical value(spectral of operators) ''shrinks'' to one determinate value when the mass is large?

 

[Jacques_L]

Please teach me this:
I think that the classical physical laws are the relations between expectation values of macroscopic objects(the values are taken average on ''quantum values'').The quantum physical laws in one sense are the relations between operators.Then there are a great difference between classical and quantum laws.So I do not understand why they say that the classical things are the leading order of aproximation of quantum things(e.g they say in leading order of aproximation the electromagnetic potential is square(e)/square(momentum)(in momentum space) when they considering renormalization group(considering the flow of electric charge under changing renormalization condition)).

Please choose : do you mean the difference between micro- and macrophysics domains and laws, or between micro- and macrophysics formalisms currently in use ?
The two questions are not equivalent.

Strictly speaking :
in microphysics, any wave or any quanton comes from one emitter and converges into one absorber. One or very few successful transactions before thermalization.

Intermediate level for bosons : the emitter may be collective, as in lasers, whereis a whole medium as a gas in a cavity, or as a laser diod, acts as synchronous emitter for a lot of synchronous photons; and/or the travel may be collective-and-united so the photons have time and (astronomical) distance to synchronize themselves in frequency an phase, and it is used in interferometry astronomy.
Intermediate level too : supraconductivity and superfluidity. All involving the gregar properties of bosons.

Strict macrophysics : lots of differents emitters, lots of different absorbers, lots, lots, separated from the microphysics by the Avodadro constant. No more any common Broglie or Dirac-Schrödinger frequency.

So are the different physical domains.
What are the formalisms in use is another story.

 

[A. Neumaier]

So,now I think that the quantum physics is ''broader''(meaning involve) than classical physics,because some macroscopic phenomena are the results of quantum physics(e.g superconductivity,superfluidity...).Then classical physics is not enough to describe the macroscopic world.But in the ''narrow'' domain of classical physics,the classical is the limit of quantum physics.But do anyone please prove this for me?Does the fluctuation become small in fluctuation theory when the mass of object is large?Does the spectral of physical value(spectral of operators) ''shrinks'' to one determinate value when the mass is large?

Expectations of commutators are a factor of hbar smaller as the expectations of the macroscopically relevant quantities. Thus as soon as these quantities are large enough, one can neclect the commutators and regard the operators as approximately commuting. This gives the classical limit.

The spacing of discrete spectra is also O(hbar), hence in the classical limit, all spectra (possible periods of trajectories) become continuous.

 

[Naty1]

Classical and quantum mechanics are in general low energy, weak field approximations...Nothing works at big bang and black hole singularities where gravity dominates and is so powerful that space and time "disappear".

But both quantum theory and GR gives good insights, although complementary ones, for example in the vicinity of the black hole horizon...far from the singularity.


from Albert Messiah, QUANTUM MECHANICS:

"In classical mechanics the evolution in time of physical systems is described by dynamic variables with well defined values at every instant. It became evident around 1900 phenomena on the atomic and sub atomic scale do not fit this framework. The first series of experiments forcing a revision of the wave theory of Maxwell-Lorentz was the photoelectric effect and Compton scattering. The appearance of quanta in the photoelectric effect and the instantaneous and discontinuous transfer of energy to certain electrons exposed to radiation (instead of the continuous transfer of energy to all electrons as postulated classically) required the introduction of light quanta.."

I think "well defined values at every instant" means that is what was assumed at the time...nobody for thousands of years likely thought about "uncertainty", wave particle duality, and forces as exchanges of quanta of energy. "physical systems" originally meant macroscopic, I think, then experimental phenomena at small scales didn't fit.

 

[Naty1]

I was reviewing my own notes and came across this...an interesting view of classical versus quantum:

The following quote is from Roger Penrose celebrating Stephen Hawking’s 60th birthday in 1993 at Cambridge England...this description offered me a new insight into quantum/classical relationships:

Either we do physics on a large scale, in which case we use classical level physics; the equations of Newton, Maxwell or Einstein and these equations are deterministic, time symmetric and local. Or we may do quantum theory, if we are looking at small things; then we tend to use a different framework where time evolution is described... by what is called unitary evolution...which in one of the most familiar descriptions is the evolution according to the Schrödinger equation: deterministic, time symmetric and local. These are exactly the same words I used to describe classical physics.

However this is not the entire story... In addition we require what is called the "reduction of the state vector" or "collapse" of the wave function to describe the procedure that is adopted when an effect is magnified from the quantum to the classical level...quantum state reduction is non deterministic, time-asymmetric and non local...The way we do quantum mechanics is to adopt a strange procedure which always seems to work...the superposition of alternative probabilities involving w, z, complex numbers...an essential ingredient of the Schrödinger equation. When you magnify to the classical level you take the squared modulii (of w, z) and these do give you the alternative probabilities of the two alternatives to happen...it is a completely different process from the quantum (realm) where the complex numbers w and z remain as constants "just sitting there"...in fact the key to keeping them sitting there is quantum linearity...

From:THE FUTURE OF THEORETICAL PHYSICS AND COSMOLOGY: Celebrating Stephen Hawking 60th birthday, Gibbons,Shellard and Rankin

(The first few talks are understandable to laymen...but after that the math and presentations are very advanced...too much for me...such as methods and concepts of doing perturbation calculations)

 

[dx]

In brief, classical laws express the changes that take place continually in the state of a system with time. Quantum laws express the discontinuous changes in systems in terms of the continuous changes in the probabilities of various possible results of measurements.

 

[Jacques_L]

In brief, classical laws express the changes that take place continually in the state of a system with time.

Mmmh...
Land slidings are not at all continuous phenomena. When a swedish clay defloculates at the passage of a train, and slides on 200 m, with a slope of 4 %, where is the continuity ? ...
The cracks in Earth crust under shear stress, that are perceived as earthquakes and tsunamis are not at all continuous phenomena.
The electronics is full of catastrophic events and flip-flops, or avalanches, or triggerings, or burnings.
In mechanics, ruptures of metals or ceramic or mineral materials are not so continuous.

 

[dx]

In classical physics, the state of a system changes continuously in time; i.e. the state variables P_i and Xⁱ are continuous functions of the time t. These continuous changes are expressed as differential equations (Hamilton's equations).

Quantum theory however, arose from the recognition that the laws of nature in fact have a peculiar discontinuous character, which in fact makes it impossible to separate the behavior of an atomic system from its interaction with the measuring instruments; an independent reality can be ascribed neither to the phenomena nor to the measuring devices. Thus we can no longer specify the state of a system as in classical mechanics in terms of P and X. I won't go into details here, but the state of a system in quantum theory is given in terms of the statistics of measurements attainable on the system. The discontinuous character is taken into account by laws which express the changes taking place (continuously in time) in the probabilities of various results of measurements.

 

[Jacques_L]

In classical physics, the state of a system changes continuously in time; i.e. the state variables P_i and Xⁱ are continuous functions of the time t. These continuous changes are expressed as differential equations (Hamilton's equations).

Here you have described a certain kind of formalism. But this peculiar formalism does not represent the bulk of macrophysics.

 

[dx]

The 'peculiar formalism' which you refer to is exactly the logical frame of classical physics based on the ideas of cause and effect represented with spacetime pictures, and gives an exhaustive description of all physical phenomena in which quantum effects play no role, i.e. what you refer to as macrophysics.

The individual character of quantum effects however prevents a causal-spacetime account of these phenomena and the quantum formalism is a rational generalization of classical mechanics which takes into account the complementarity between spacetime coordination and the claim of causality.