Quantum physics in three sentences?

Let's try something more general:

In quantum mechanics a 'realistic' description using entities (like particle trajectories with well-defined position and momentum) known from our everyday's experience is no longer applicable and becomes wrong. Instead one has to refer to an abstract mathematical formalism from which the classical world does 'emerge' in some situations, but which shows genuine (and weird) quantum behavior in other situations (double slit, entangled particles a la EPR / Bell). A typical example for the genuine quantum behavior of 'quantum objects' (which are neither particle nor wave) is the double slit experiment where the abstract formalism tells us that one single, indivisible entity like an electron sniffs out all possible classical trajectories simultaneously, i.e. 'walks through both slits' simultaneously, interferes with itself and shows partially destructive interference.

 

1. Wave function is a complex function of the space position x and time t.
2. A linear combination of physical wave functions is a physical wave function itself.
3. The squared absolute value of the wave function is the probability density for a particle to be found at the position x at time t.

 

A small ammendment to point 2. <Within a coherent subspace of the physical Hilbert space, a linear combination of physical wave functions is a physical wave function itself>.

 

The quantum mechanics of wave functions and the double-slit experiment involves the union of that which we used to think were as different as night and day: the discrete and the continuous-- particle and wave. The union is accomplished by the quantization, in bundles of the Planck constant, of a dynamical quantity known as action. This quantization allows us to associate with any particle momentum a wavelength, which is a property of waves, and with any wave period a quantum of energy, which is a property of particles. These associations allow us to treat what we used to think of as disjoint wave and particle properties as two aspects of the same animal, like the trunk and the tail of a single elephant-- they allow us to recognize, rather belatedly, why particles and waves were always the same thing in different clothes.

 

1. The state of any physical system is represented by a vector in an infinite dimensional complex space, with each component representing a particular state the system can be in.

2. The state vector of any physical system evolves according to Shrödinger's equation.

3. A measurement on a physical quantity (energy, momentum, position, spin, etc.), represented by Hermitian operators, yields one of its eigenvalues, with the probability of that value being given by the square of the projection of the state vector on the given eigenvector.

(sorry, a math free explanation is not possible)

 

What is a coherent subspace?

 

The Hilbert space of states, assumed infinite dimensional and separable, is generally decomposed into a direct othrogonal sum of closed subspaces, each corresponding to discrete spectral values of the operators describing the so-called <superselection rules>. For example charge in case of a Dirac equation/field. The linear superposition between an eigenfunction with positive charge (positron) and negative charge (electron) makes no sense physically, so that a quantum state with electrons and positrons is not described by a linear combination, but by a tensor product.

An eigenspace of an operator providing a superselection rule which corresponds to an eigenvalue of that operator is called a coherent subspace. It is a separable Hilbert space in its rights, if endowed with the scalar product inherited from the full space. Linear combinations of vectors only within the same (sub)space make sense (example above), not between vectors from different subspaces, corresponding to different eigenvalues.

This is off the top of my head. As far as I remember, I think the introductory chapter from Streater & Wightman has a story on this. Also Galindo & Pascual's text on QM, or Fonda & Ghirardi's text on symmetries of quantum systems.

Needless to say, in case superselections rules are missing, the whole Hilbert space is a coherent subspace (quanta of the KG field, or spinless Galilean particle).

 

Thanks dextercioby. I have an offtopic comment, but I will open a new thread on it.

 

Caveat: QM works mathematically, everybody agrees on that. However, when you start to talk and describe what 'happens', you always run into the 'dilemma' of interpretations, and my guess is that some will have scathing comments on my 'talk' below ;). Nevertheless, as of today no interpretation is proven better than any other...


Quantum Mechanics for Dummies

• Quantum mechanics describes the very small microscopic world, where things behave differently from what we are used to in the macroscopic world of 'classical' objects.
  
  
• One famous example is the Double-slit experiment, which demonstrates that matter and energy can display characteristics of both waves and particles.
  
  
• Waves in quantum mechanics are fundamental and described mathematically by the wavefunction and the Schrödinger equation, which results in some uncertainty when calculating the exact outcome for one single particle, the outcome is therefore probabilistic in its nature.

 

I would say that DevilsAvocado is the best 3 sentences so far (I will not attempt writing 3 sentences because I cannot top it). The others are good though.

If I could add one thing it would be that quantum mechanics approximates very well to classical mechanics when considering large systems, e.g. QM agrees with Newton's Laws and GR when you apply QM on a macroscopic and large scale.

I would also like to say that it is just a theory and is not necessarily the "truth"... maybe we will find a better theory in the future which is more generalized than QM. But it is the best theory we have as of now...

 

1. Quantum Mechanics is an attempt to predict the outcomes of experiments on systems so small that our intuitive notions of particle, wave, position, momentum and even space and time might not be applicable.

2. Several different properties of a quantum system may be measured but it is never possible to simultaneously measure all of them.

3. Depending on the preparation of the quantum system some of its properties may be predicted with certainty but, in general, the predictions of the outcomes of measurements are probabilistic but have tremendous experimental confirmation.

Not perfect but it is non-mathematical.

 

By way of ‘gentle’ debate, rather than argument, just over a generation ago, Feynman suggested that nobody understood QM and now we appear to aspire to explain some of its central concepts to ‘an idiot’ in just 3 sound-bites. Clearly, there is hope for me :uhh: yet, although Heisenberg had a go at summarising the double slit experiment and the wave function collapse in just 1 sentence:

“The path of a particle comes into existence only when we observe it."​

I also like the summary in post #10, but possibly the is in the detail, if not in the avocado, as per bullet 3:

People might also like to review the following thread ‘https://www.physicsforums.com/showthread.php?t=541962"’, which wasn’t restricted to 3 sentences, as a yardstick as to whether any consensus was reached on its definition, let alone its interpretation. For example, do quantum waves have any physical existence?

 

According to the standard formalism of QM, there's no more distinction between particles and waves, as these 2 concepts, as I said before, actually pertain to classical physics, namely the mechanics of point particles and waves (including electromagnetism). So <quantum waves> as a concept does not exist.
The fundamental concepts of QM are: (quantum) system, states and observables of a system and virtual statistical ensembles. The rest is essentially mathematics.

 

Cool, I love "one-liners", less is more. In 1935 Erwin Schrödinger published http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=1737068" defining the term "entanglement":

"I would not call [entanglement] one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought."​

 

Agreed, but if you’re going to tell 'an idiot' that the "QM world" does not exist, I suspect this 'idiot' would 'object' in some way:

– Okay... I dunno... but... why should I spend my time on something that doesn’t exist...??? :uhh:​

This is of course more 'contemporary' and closer to the 'truth' than post #10. I don’t know how to get around this 'problem of language'... I get a slight feeling that the 'contemporary abstraction' of QM is so abstract, that when 'an idiot' tries to make a figurative picture in his head... he’s immediate lost.

This is from "Quantum Mechanics - A Modern Development", by Leslie E. Ballentine:

Postulate 1a. To each dynamical variable there is a Hermitian operator whose eigenvalues are the possible values of the dynamical variable.

Postulate 2a. To each state there corresponds a unique state operator, which must be Hermitian, nonnegative, and of unit trace.​

I have an idea what Ballentine is talking about, but do I get a deeper conceptual understanding from this? Well, unless I read the book, and understand the math, I’m afraid the answer is No...

I’m not trying to raise an 'argument' here, I’m just curious. What’s your answer, if 'an idiot' asks:

– Okay, "virtual statistical ensembles" sounds cool, but in case I’m not an idiot, we have single electrons in this Double-slit experiment, and afaict, we could have eons between each, so where’s your "ensemble" then??

http://www.youtube.com/watch?v=FCoiyhC30bc&hd=1
https://www.youtube.com/watch?v=FCoiyhC30bc ​

 

I somewhat do not get your point. If you perform such a double slit experiment it does not matter whether you perform one measurement with 12876 (non-interacting) electrons or 12876 measurements with one electron. Both experiments are basically giving the same ensemble.


For my take at the question of the begin of this topic, let me cite Neil David Mermin:
"My complete answer to the late 19th century question "what is electrodynamics trying to tell us?" would simply be this: Fields in empty space have physical reality; the medium that supports them does not.
Having thus removed the mystery from electrodynamics, let me immediately do the same for quantum mechanics: Correlations have physical reality; that which they correlate, does not."

 

I like Mermin's quote, but to correct it a little, all we can really say is that we do not yet have any reason to attribute reality to either the medium of the electromagnetic fields, or what is being correlated in quantum mechanics. That is not quite the same as being to assert that either are unreal, these are only 100 year-old theories! So what we are noticing is, physics is a process of assigning reality as needed, and in the mean time, we must adopt a stance of complete agnosticism-- or repeat the errors of our predecessors. Thus I view the various interpretations that attempt to assign reality to what is being correlated as "peeks" into a potential next theory, rather than as being able to say anything about reality currently. Mermin seems to adopt the ensemble interpretation, which is like saying "thou shalt adopt no interpretation until its time." That's a fine approach, but many physicists like to attempt a "sneak peek" all the same.

 

Yes, this is asking "what can we really know about a single particle". If the answer given by the ensemble interpretation is "nothing", this is formally correct, but seems a bit unsatisfying. Physics has never really been about knowing things, there was always some small chance that the whole experiment could blow up in our face. We are seeking effective knowledge in physics, not absolute knowledge. We are perfectly happy with idealizations, we call that "knowledge" in physics all the time! This causes considerable consternation to mathematicians, who seem to have a much more exacting definition of what it means to "know."