What is the difference between a wave of water and qm wave?

[g.lemaitre]

I get the feeling that when we say that electrons sometimes are particles or a wave depending on whether or not we measure it that what we mean by wave is not what first comes to people's mind. When we measure the electron it becomes a particle, if not then it is a wave. When I think of wave I think of something that goes up and down at a regular rate. Adding to my confusion is that I saw a documentary once where Susskind said how can particle be a wave and they showed a picture of a literal ocean wave. I think what is meant by wave is a probability cloud and the particle could be anywhere in that cloud once you decide to measure it. It just so happens that when you shoot electrons through a double slit that they appear on the other side in a pattern that looks like waves. To say that an electron is either a particle or a wave, I think wave is not the best use of English. I think a better formulation would be the electron in a definite location if you measure it, otherwise its being is spread out over space.

 

[Simon Bridge]

The water wave is a wave of something - it is made up of water molecules (and anything floating/dissolved in it). The amplitude, for example, is always real.

A QM wave is a wave of probability - statistics.
The amplitude may be complex in that it may have a component multiplied by the square-root of minus one.

It is the probability that is spread over space, not the electron.
A bit like a dice-roll value is spread over six possible values until you measure it - in which case it shows only a single number ... or a blind man investigating the elephant in the famous story cannot know before he touches the elephant whether he'll get the trunk or the tail end - but you wouldn't say that the elephant has trunk-tail duality.

Electron measurements show properties associated with classical particles or classical waves ... but the electron is neither. It is itself.

 

[vanhees71]

This is again a subtle question on the interpretation of quantum mechanics, and one should warn that what you get is an answer is an opinion from the physicist trying to answer the question. The short answer, in my opinion, is that an electron can neither be described as a classical particle nor as a classical wave. There is no such thing as a wave-particle dualism, which was part of the "old quantum mechanics", discovered by Planck, Einstein, Bohr, and Sommerfeld between 1900 and 1915 and which has only survived until the discovery of "modern quantum mechanics" in 1925/26 by Heisenberg, Born, Jordan, Schödinger and (in the most comprehensive form) by Dirac and clarified in its mathematical structure by John von Neumann shortly thereafter.

To make clear, where my opinion on how to answer this question: I'm a proponent of the Minimal Statistical Interpretation (or Ensemble Interpretation), and I'll answer the question from the point of view of this interpretation of the quantum theoretical formalism. According to this interpretation the wave function of a particle (which has a well defined physical meaning only in non-relativistic quantum mechanics, by the way) it's modulus squared $|\psi(t,x)|^2$ gives the probability distribution to find an electron at a position $x$ when looking for the particle there at time $t$, provided the particle is prepared (or observed) to be in a state, described by this wave function.

This is a very abstract definition of the quantum mechanical state, and it implies that we cannot know more about the particle, concerning its position, than the probability distribution to find it at a certain position at a certain time.

Now it's always a good question in physics to ask, how we can verify such a theoretical claim by observations/experiments. A probability, in my frequentist opinion, practically can only be checked on a sufficiently large ensemble of single electrons, all prepared in the same way such that its state is described by the wave function $\psi(t,x)$. This preparation must be uncorrelated for the different members of the ensemble such that there are no correlations produced somehow, which would need a more detailed description of the statistical ensemble. Then I can simply put a detector at $x$ and cound how many electrons of the ensemble are found within the detector's volume $\mathrm{d} x$. Then this number divided by the total number of prepared electrons and further divided by the volume should give (within statistical errors, which can be made arbitrarily small by preparing and detecting a large enough number of electrons) the probability, given by the wave function, namely $|\psi(t,x)|^2$.

Now, this answers your question: Within the Minimal Statistical Interpretation of quantum theory, there is no wave-particle duality, which makes no sense in the first place. A single electron in general can not be described adequately by a classical particle, i.e., a very small ("pointlike") body running around on a trajectory with a certain momentum (or velocity for that matter) as we are used to from every-day experience with macroskopic little bodies like billiard balls.

At the same time an interpretation of the quantum-mechanical wave function as a classical field, whose modulus squared is in some sense the number density of particles in the sense of classical continuum theories like hydrodynamics, is also not tenable: First of all in the very foundations of quantum mechanics, the Schrödinger wave function refers to the description of a single particle and not of an assembly of many particles. Within quantum theory, on the fundamental level such many-body systems are described by wave functions which are functions of time and the positions of each particle, i.e., an N-body wave function has n position vectors as arguments. To go over to an approximative description of a many-body system one has to use quantum-statistical methods, involving the description of the system by a Statistical operator and some "coarse graining" formalism to come to a classical phase-space picture or further to a hydrodynamical description of the many-body system that can be interpreted within the realm of classical (continuum) mechanics. It is of course an important point that such an approximation is possible when the appropriate conditions are met, but it does not help for the interpretation of the single-particle wave function.

The single-particle wave function thus of course fulfills a dynamical equation of motion (the Schrödinger equation for a non-relativistic particle in a potential, for example) with wave-like solutions. However, this does not imply that one can interpret it as a classcial field with wavelike solutions. It's only the probability distribution, describing an ensemble of equally but independently and uncorrelatedly prepared single particles that behave like the modulus squared of such a "wave field".

Such intepretational problems, you don't have within classical continuum mechanics, where, e.g., the velocity field describing water waves on a pond, have a directly observable and even measurable meaning.

It is important to get this radical distinction between the classical picture of nature's behavior and the picture we have gained in the discovery of the quantum nature of it. In my opinion, within quantum theory, there is no way to resolve this tension between an immediate intuitive picture we have in classical (deterministic) physics and the intrinsically probabilistic picture we have in quantum physics, which is inherently and irreducibly probabilistic.

Whether there is a deterministic description in accordance with observation or not, leading to quantum mechanics as a certain approximation, is unknown today. If so, according to all findings on the issues related with entanglement ("Bell's inequality"), such a theory must be a quite complicated theory that is nonlocal in both space and time. I'm not sure whether this would make our understanding of nature easier than quantum theory. As long as such a theory is not found, we have to live with quantum theory, which is in fact the most successful physical theory ever, and there are no known contradictions to its predictions yet! It seem as if nature is as "weird" as described by quantum theory ;-)).

 

[Bill_K]

I think what is meant by wave is a probability cloud and the particle could be anywhere in that cloud once you decide to measure it.

I agree.

 

[Simon Bridge]

...an electron can neither be described as a classical particle nor as a classical wave. ... I think what is meant by wave is a probability cloud and the particle could be anywhere in that cloud once you decide to measure it.

I believe that is the generally accepted understanding.

Trouble here comes when people try to describe QM in terms of the, presumed familiar, classical mechanics ... which won't work. The different opinions are, as observed, in Feild theory... where the "particle" and "wave" descriptions form a kind of chicken-and-egg effect. But these are not classical particles and waves that are being discussed and most people won't encounter them before postgrad-college level. (Or has that changed?)