Can you define quantum mechanics in just one sentence?

[Demystifier]

I would like to challenge you all to define quantum mechanics.
An ideal definition should contain only one sentence and should cover all aspect of quantum mechanics. If you cannot construct an ideal definition, you can try with a good but not ideal one which covers most but not all aspects of quantum mechanics, and/or contains more than one sentence.

Here is my try:

Definition 1 (probability amplitude):
Probability amplitude is a complex number the squared absolute value of which is equal to probability.

Definition 2 (quantum mechanics):
Quantum mechanics is a branch of physics which studies probabilities of different measurement outcomes, for systems for which the probability can be most easily calculated from a probability amplitude.

 

[lightarrow]

What about listing QM postulates?

 

[ZealScience]

Aren't both definitions some of the postulates of quantum mechanics?

 

[ash_bang]

If I as a complete physics numpty cannot understand your definition, then what is the point in defining something ...

It would be nice, however to see a definition from the community.

If nothing else it would perhaps initiate the discussion to do so.

Ash!

 

[dextercioby]

Hrvoje is part of community.

As for the sought definition, it's merely:

Quantum Mechanics is the theory whose predictions follow logically from the following set of axioms:

1.
2.
3.
4.
5.
...

 

[Demystifier]

What about listing QM postulates?

Quantum Mechanics is the theory whose predictions follow logically from the following set of axioms:

1.
2.
3.
4.
5.
...

It's certainly legitimate, but probably not ideal for a definition. By definition, a mean a concise statement that could be written e.g. in a science dictionary or at the beginning of a Preface in a quantum-mechanics textbook. As such, a definition should be precise but still not too technical. It's of course difficult to achieve, but that's precisely why this thread should be challenging.

 

[Demystifier]

Aren't both definitions some of the postulates of quantum mechanics?

No. For example, even a probability in classical statistical mechanics can be expressed in terms of probability amplitudes, but that's not the easiest way to formulate classical statistical mechanics.

 

[jambaugh]

OK, Here's my attempt...

• As with CM, QM associates every system observable with a corresponding generator (in some Lie algebra) of a group of transformations (potential symmetries of the system) a la Noether's theorem.
  
  
• As with CM, QM defines the dynamics of a system by the action of one of these generators, the Hamiltonian.
  
  
• Where in CM the most general system description is a a probability distribution on a manifold of states (state space typically phase space), in QM the system description is a linear functional defining expectation values on the Lie algebra of observables, satisfying appropriate consistency conditions. (e.g. $\langle 1\rangle = 1$ "the system exists" so we expect it to exist), and as such transforms dually to the observables under the action of the Hamiltonian or other generators of transformations in the system's Lie algebra.
  
• and finally, as with CM, in QM the act of incorporating knowledge of a specific observed value should lead to our modifying the system description so that immediate subsequent measurement is predicted to produce the same value with certainty.

I think this paradigm is sufficient to generate all of QM once one carries out a program of empirical experiments to see which Lie algebra corresponds to a given physical system. I don't mention representations of the Lie algebra because that is just an embedding into a larger Lie algebra.

Some Details:
We begin in both cases with a probabilistic description and then see what logical certainties may be represented. We can describe a classical system in a given state using a Dirac delta distribution over the state manifold.

The linear functional for QM system description is the mapping $A \mapsto \langle A \rangle = \mathop{Tr}(\rho A)$. The maximal description is when $\rho$ is proportional to a minimal non-zero projection operator in some representation of the Lie algebra in question.

 

[ZealScience]

No. For example, even a probability in classical statistical mechanics can be expressed in terms of probability amplitudes, but that's not the easiest way to formulate classical statistical mechanics.

But in classical mechanics, they don't use vector space, right? I think phase space has nothing to do with complex vectors which you mentioned in your definition.

 

[lightarrow]

QM is the theory in which fields' energy is not continuous but an integer number of hv.

 

[dx]

It is a mathematical formalism that generalizes classical mechanics, adapted to describe physical phenomena in which h (Plank's constant) cannot be neglected.

 

[Demystifier]

But in classical mechanics, they don't use vector space, right?

They don't, but they could if they wanted to. See e.g.
http://xxx.lanl.gov/abs/quant-ph/0505143 [Found.Phys.Lett. 19 (2006) 553-566]
http://xxx.lanl.gov/abs/0707.2319 [AIPConf.Proc.962:162-167,2007]

 

[Demystifier]

QM is the theory in which fields' energy is not continuous but an integer number of hv.

It's true, but for many aspects of QM this fact is not really important.

 

[Demystifier]

It is a mathematical formalism that generalizes classical mechanics, adapted to describe physical phenomena in which h (Plank's constant) cannot be neglected.

It's true, but it explains nothing. Someone who does not already know what QM is, after reading this will still have no idea what it is.
At least, one should add one additional definition, which defines the Planck constant.

 

[jambaugh]

...
At least, one should add one additional definition, which defines the Planck constant.

For Goka's definition yes. But given it is an empirically determined quantity one need not define its value. It should be sufficient to state or imply that some observables are quantized with one's definition.

To All: I think the "meta-question" should be hashed out here. What do we want in a definition?

• Should it be sufficient to reconstruct QM given sufficient empirical experimentation?
• Should it include foundational experimental results (e.g. Einstein's photo-electric effect).
• Should it be sufficient to allow an educated layman to understand QM? [tall order!]
• Should it be axiomatic? Operational? Minimalistic?

----------
To my thinking, one must imply or explicitly invoke:

• Born's probability formula or the equivalent,
• The Eigen-Value Principle or some equivalent expression of observable spectra,
• Hamiltonian dynamics.

I don't see these included without pulling out the heavy duty algebra which will get a bit esoteric for the layman.

I'd also suggest practicing with "Define classical mechanics" as a warm up.

 

[lightarrow]

QM is the theory in which fields' energy is not continuous but an integer number of hv.

It's true, but for many aspects of QM this fact is not really important.

For example? (Just in order to clarify it to myself).

 

[Demystifier]

For example? (Just in order to clarify it to myself).

For example, superposition and entanglement.

 

[Demystifier]

To All: I think the "meta-question" should be hashed out here. What do we want in a definition?

• Should it be sufficient to reconstruct QM given sufficient empirical experimentation?
• Should it include foundational experimental results (e.g. Einstein's photo-electric effect).
• Should it be sufficient to allow an educated layman to understand QM? [tall order!]
• Should it be axiomatic? Operational? Minimalistic?

That's a good question. I would say it should be minimalistic, but by that word I probably do not mean the same as you do.

 

[TGlad]

Quantum mechanics studies the behaviour of nature's smallest particles which statistically follow the combined distribution of all possible paths, leading to the appearance of electric, magnetic and nuclear force fields, and ultimately giving rise to the fundamental properties of matter.

 

[jetwaterluffy]

Quantum mechanics is the mechanics of quanta.

 

[bbbeard]

I would like to challenge you all to define quantum mechanics.

Quantum mechanics is the branch of physics which describes natural phenomena for which the action is on the order of Planck's constant or smaller.

To me, that's the essence of quantum mechanics. It takes several library shelves to fill in the details afterward. But that's the first sentence.

BBB

 

[Mordred]

Although the Plancks constant is fundamental to quantum mechanics. I don't feel a definition should be based on it.

 

[bbbeard]

Although the Plancks constant is fundamental to quantum mechanics. I don't feel a definition should be based on it.

I disagree. The quantization of action -- which arises because Planck's constant has a non-zero value -- is pretty much the whole point of quantum mechanics, at least from the standpoint of a theoretical physicist.

BBB

 

[jambaugh]

That's a good question. I would say it should be minimalistic, but by that word I probably do not mean the same as you do.

I don't think we'd disagree on the word, only on the constraints we'd invoke within which we'd minimize.

Here's another attempt:

Quantum mechanics is the physics of systems with continuous symmetry and finite information content.

 

[Demystifier]

Quantum mechanics is the branch of physics which describes natural phenomena for which the action is on the order of Planck's constant or smaller.

First, there are macroscopic quantum effects, like superconductivity, for which the action is much larger.

Second, there are many time-independent quantum effects (for example, entanglement) for which the value of action is irrelevant.

 

[jetwaterluffy]

The quantization of action -- which arises because Planck's constant has a non-zero value -- is pretty much the whole point of quantum mechanics, at least from the standpoint of a theoretical physicist.

In other words, QM is the mechanics of quanta.

 

[bbbeard]

I don't think we'd disagree on the word, only on the constraints we'd invoke within which we'd minimize.

Here's another attempt:

Quantum mechanics is the physics of systems with continuous symmetry and finite information content.

Wouldn't that just as easily include equilibrium thermodynamics? How much information is in a mole of gas at STP, from a thermodynamic standpoint?

And aren't there quantum systems which don't have a continuous symmetry -- for example, most of lattice gauge theory, which only recovers Poincare symmetry in the limit of infinite lattices?

BBB

 

[bbbeard]

First, there are macroscopic quantum effects, like superconductivity, for which the action is much larger.

I think if you look in any decent text that deals with superconductivity you will find actions that are normalized by hbar. The first one I pulled off the shelf was Sakurai's Advanced Quantum Mechanics -- see pp 17-18. I think you are mistaking the extensivity of quantum phenomena with their microscopic explanations. The expression for the action generically includes terms like N*hbar, which is of the order of Planck's constant no matter how large N is; it's literally O(hbar).

Ferromagnetism is another example. Consider how to describe the interaction of two dimestore magnets. You can take a strictly classical approach and describe the fields in terms of the dipole strength, with Planck's constant never appearing in the formulation. But if you want to explain the origin of ferromagnetism, you have to study the Hamiltonian, which has spin coupling terms, and of course the resulting quantum statistical mechanics requires us to formulate the action integral to find the partition function (via Wick rotation). Again Planck's constant shows up.

Second, there are many time-independent quantum effects (for example, entanglement) for which the value of action is irrelevant.

Yes, but if you want to discuss the quantum mechanics of entanglement, Planck's constant will show up in the action. If not, it's classical mechanics.

BBB

 

[Ken G]

I think Demystifier's goal is to have a definition that is the first thing someone would learn about QM, rather than a definition like jambaugh's, which is more like a synthesis of what QM is that appears once you already know it quite well. The latter is useful for people who know a lot of QM but haven't seen a good synthesis of what they know, the former is good for answering a newbie question like "what is quantum mechanics anyway?"

Personally, I think the former type should focus on what quantum mechanics accomplishes for us, rather than what it is. That's generally what non-experts want to know about anything, if someone asks what a car is they want something like "a wheeled and motorized machine we use to get from place to place along roads", not a description of an internal combustion engine. So along those lines, I would offer that quantum mechanics is the theory we use to describe the behavior of systems [edit out: individual particles and systems of small numbers of particles] when they are only weakly interacting with their macroscopic environments [edit: where by "environment" I just mean "anything not being tracked as part of the self-consistent behavior of the system"]. One way to characterize the weakness of the interaction is a timescale comparison-- quantum mechanics is what we need to use when the behavior we are interested in plays out over a timeframe up to the time between significant interactions with the macroscopic environment, as compared to classical mechanics which plays out in timescales much longer than the interaction time.

Of course, the place where h appears is in the issue of what counts as a "strong" interaction.

 

[Neandethal00]

QM is a branch of physics which approximates physical properties of particles
using probability instead of costly (or our inability to) exact determination of the properties.

 

[Mordred]

Quantum mechanics is the study of all particle properties including their momentum, energy levels and interactions at the atomic scale and below.

Why limit the scale of quantum mechnics to the Planck constant. Each time science believes we undertand the fundamental particles, we discover something smaller with lower energy levels. Instead define the upper boundary where cladsical physics takes over. That way you allow for future discoveries.

 

[jimgraber]

Are you familiar with Scott Aaronson's "definitions":
QM is physics with negative probabilities. or
QM is physics with a quadratic norm in place of a linear norm. ?

Here is a reference:
http://www.scottaaronson.com/democritus/lec9.html

Best
Jim Graber

 

[jambaugh]

Wouldn't that just as easily include equilibrium thermodynamics? How much information is in a mole of gas at STP, from a thermodynamic standpoint?

Given it has a continuous energy spectrum which in principle can be measured (classically) to arbitrary precision it has infinite information content.

And aren't there quantum systems which don't have a continuous symmetry -- for example, most of lattice gauge theory, which only recovers Poincare symmetry in the limit of infinite lattices?

The discrete symmetry of lattice gauge theory is an approximation of the continuous symmetries one is modeling. LGT is an abstraction of e.g. finite elements methods for solving PDE's. But note that even in LGT the gauge symmetry is still continuous e.g. SU(3).

I think the "Quantum" in Quantum Mechanics is best expressed in simple terms as the finiteness of the amount of information which can be encoded in the system. Of course since "all is quantum" we will see the same limits to information content in classical systems once pragmatic considerations are taken into account. That's just QM peeking its head out from under the floorboards.

 

[Ken G]

Are you familiar with Scott Aaronson's "definitions":
QM is physics with negative probabilities. or
QM is physics with a quadratic norm in place of a linear norm. ?

That's extremely insightful stuff. Excellent.

 

[Ken G]

Of course since "all is quantum" we will see the same limits to information content in classical systems once pragmatic considerations are taken into account. That's just QM peeking its head out from under the floorboards.

Indeed, I would argue that QM gives us no more reason to expect the "finiteness" limitation you are talking about than did classical mechanics-- after all, classical mechanics never said that reality encoded infinite information, it just said we can process finite amounts of information by ignoring how much "more than that" information was actually there. We only run into trouble when we extrapolate our theories farther than we have any right to.