I am reading a David Bohm book on quantum theory. He says the idea that light is both a particle and a wave is incompatible:

(1) we know light has particle-like properties through the photoelectric effect

(2) we know light also has wave like properties because of slit experiments.

He then explains why they are incompatible.

But what is a "particle"? What is a "wave"? What do these terms mean precisely? I know what they mean loosely. A particle means something occupying a spatial position. A wave is like a density or something defined with peaks, troughs, and nodes over a spatial region. But I want something more rigorous and more accurate than these loose definitions so that I know what I mean when I use the term.

I'd like the definitions stated like we state axioms in math, clearly and specifically. In math, I say a vector space is closed under addition if $x,y\in V$ implies $x+y \in V$. Those are very specific claims.

Can someone do something similarly clear and specific with wave and particle?

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    $\begingroup$ A lot of comments deleted. Physics SE is not a place to fix the internet. $\endgroup$ Jan 1, 2015 at 1:21
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    $\begingroup$ A "point particle" in classical mechanics is the concept that one can approximate the motion of a real, extended object by the motion of its center of mass. If you stick to this ontology, it makes no sense to say that the photoelectric effect implies particle properties. A "wave" in classical mechanics is the collective motion of a solid, liquid or gas. In classical electrodynamics it's the propagation of energy in a vacuum field. The slit experiments do not show any resemblance to that kind of wave motion because single quanta show interference effects without there being a "collective". $\endgroup$
    – CuriousOne
    Jan 1, 2015 at 1:44
  • $\begingroup$ @CuriousOne well they do. Otherwise the term "wave-particle duality" wouldn't have appeared. $\endgroup$
    – Ruslan
    Jan 1, 2015 at 1:52
  • $\begingroup$ @Ruslan: The term wave-particle duality appeared because otherwise very smart people were highly confused. It's a historic artifact of that confusion, just like "aether" is the result of another. If you apply the correct classical definition of "point particle" as the approximation that neglects internal degrees of freedom and moments of inertia of the motion of extended real objects, then you can find no logical way of using it in the context of the photo effect. There are no extended objects there that need to be approximated. $\endgroup$
    – CuriousOne
    Jan 1, 2015 at 1:59
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    $\begingroup$ Related: physics.stackexchange.com/q/46237 $\endgroup$ Jan 1, 2015 at 8:00

3 Answers 3


The concepts "particle" and "wave" started from classical physics and from the everyday use of the terms, to begin with. A particle of dust got into one's eye, and the sea had huge waves.

Physics came into its reign when mathematics was seriously used to model observations.

For classical physics "particle" means an entity with small mass and a center of mass tracked at coordinates (x,y,z) at time t. Solutions of kinematic differential equations described the trajectory with accuracy determined by experimental errors.

For classical physics, waves are modeled by sinusoidal functions, i.e. functions that were the solution of "wave equations", could describe the behavior of sea waves, sound waves, and finally electromagnetic waves. Classically a wave is a variation of a measurable quantity like energy, or electric field, in space at a given time t, and the theoretical models were very successful in describing the observations of periodic energy distributions in bulk matter, and even in empty space ( electromagnetic waves).

Then quantum mechanics became necessary, from the discreteness of atoms, the black body radiation spectrum, the photoelectric effect it was finally understood that there were regions in the variables measured that displayed a quantization of energy.

It so happens that the equations that successfully describe the quantum mechanical state of matter were diferential equations with sinusoidal solutions, i.e. wave equations, like the Schrodinger equation. The solutions for the hydrogen atom were able to explain the spectral series adhoc assigned by the Bohr model, IF the postulate was assumed that the wavefunction squared did not represent the energy of the electron at (x,y,z) at time t, but a probability distribution. i.e. if one accumulated with the same boundary conditions a large number of measurements and plotted the (x,y,z) at time t distributions one would know how probable it would be to find the electron at that location.

As an example, this is similar to taking a census of the population of a city by age, and gauging how probable it would be that the first person you meet will be 8 years old. The wave function's function is just that, to give probabilities mathematically which are checked experimentally, and have been very accurate.

The "wave" part confused and continues to confuse people, because they think that the quantum mechanical entity, the electron for example, is spread out according to the solution of the Schrodinger equation. This is a misunderstanding, as the double slit interference experiments show with incoming single electrons:

dbl slit single electron

Note the top photo, where the electron impinges on the screen, it is one whole electron . The probability pattern accumulated though shows clearly the interference effect that is expected by the sinusoidal form of the wave functions describing the electron when it hits the slits and goes through one or the other.

The "particle" facet of the electron is that it appears as a point at ( x,y,z_0) of the screen, and the "wave" facet is the probability distribution displayed in its trajectories.

If one is becoming a physicist it is simple to accept this fact, that the microcosm behaves differently than the macroscopic world we are used to. Bohm was stuck on classical frameworks and tried to derive the quantum mechanical probabilities from an underlying classical description. He succeeded in reproducing the same results as the usual quantum mechanical solutions, but afaik his model is complicated and limited, and cannot be extended into second quantization where the ball game has gone now.

  • $\begingroup$ Is Bohm out of date? I am wary of reading people with views that aren't current enough or not mainstream enough. Would you suggest I read someone else to learn more about quantum theory? If so, any recommendations? $\endgroup$ Jan 1, 2015 at 20:07
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    $\begingroup$ Yes , Bohm is both out of date and not mainstream, although I believe there exist people carrying on those lines of research. For mainstream try the recommendations on this site physics.stackexchange.com/questions/33215/… $\endgroup$
    – anna v
    Jan 1, 2015 at 20:18
  • $\begingroup$ @annav : that's not true, no, not at all. Bohm's interpretation is considered a courageous trial to get rid of the "collapse" postulate. If this trial is good, that's another question. There are serious researchers ready to admit that there exists, in the microscopic world, a preferred frame. I had long talks with such people. And there were experiments (Gisin's group) trying to find whether a preferred frame can exist for quantum particles. But the experiments were non-conclusive. Thus, as far as we can't disprove the existence of a preferred frame, we can't rule out Bohm's interpretation. $\endgroup$
    – Sofia
    Jan 2, 2015 at 1:29
  • $\begingroup$ @Sofia I am saying that it is not main stream, and also mentioning that there are people still working on it. From the various ways one can expand even a simple mathematical function, it is always possible to have much more complicated ways of arriving to the same prediction. There is no reason to adopt the most complicated path and a lot of reasons to keep the simpler one. $\endgroup$
    – anna v
    Jan 2, 2015 at 4:36
  • $\begingroup$ @annav, let me tell you from my contacts. Nobody likes the collapse. The people are not ready to accept it. It's not mathematics here, it's physics. The QM is unable to explain the measurement. QM is not a complete theory, as simple as that, it needs something from outside, and that something is not at all understood. About Bohm's interpretation, it is not a more complicated way. Its mathematics is simple, plausible, many people believe in it. And it is a very well developed mathematics, there are many books on it. Unfortunately, it is at odds with the relativity. $\endgroup$
    – Sofia
    Jan 2, 2015 at 14:31

A particle is simply a piece of matter.

A wave is a type of oscillatory motion (vibrating up and down in crests and troughs).

Not too long ago in the history of science, scientists used to think that a moving particle moves in strictly straight line (provided that no other force is acting on it). They also used to think that electromagnetic radiation do not contain any matter and are just an expression of energy. Planck then gave his famous law that every moving object in fact moves as a wave (with oscillations) and not exactly a straight line. The heavier the object, the lesser the oscillation (and greater the wavelength). Electromagnetic radiation are in fact extremely light particles (called photons) which travel through space in oscillatory motion.


A particle is the smallest vibration of a quantum field.A wave is like a density or something defined with peaks, troughs, and nodes over a spatial region.


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