Light : Wave - particle duality

[Pushoam]

Interference pattern made by light shows the wave nature of light and photoelectric effect shows
particle nature of light. So, what is light?

According to the photoelectric effect, light consists of photons with energy E and momentum $\vec p$.
According to the interference pattern, we don't know where an individual particle will reach on the screen.

So, considering one-dimensional case, what I understood is that there is a wave - function $\psi (x,t)$ associated with each photon such that $|\psi|^2 dx$ tells us the probability of finding the particle in the region dx at time t. So, if a particle reaches x at time t then it can be said that the particle has momentum $\vec p$ and energy E at x and t. Thus, it is meaningless to say that the particle has momentum $\vec p$ and energy E without specifying its position and time.
Thus, what we can say is the probability of finding the particle with momentum $\vec p$ and
energy E in dx at time t. Thus, the probability of finding the energy of particle to be E in region dx at time t should be given as E $|\psi|^2 dx$.

I don't understand why the author is emphasizing that $\psi$ is associated with each particle, not a collection of particles.

 

[vanhees71]

To understand our present understanding about what light is (I'd more carefully say about how we describe the phenomenon "light") you have to learn Quantum Electrodynamics. There's no other way to understand it. I'm not aware of any popular-science book which gets the issue wrong. There are some myths and theories are quoted which are outdated for more than 90 years by now. Some of the misconceptions stated in your posting are:

(1) There is no wave-particle dualism since the discovery of modern quantum theory in 1925/26 but one abstract scheme called quantum theory to mathematically describe all phenomena known today except gravity. We have no satisfactory quantum theory of gravity, but everything else is pretty well described.

(2) Photons are not very particle like. Particularly you cannot define a position for them (as is possible for all massive quanta), and it doesn't make much sense to talk about a particle which has not a position observable. According to up-to-date electrodynamics a photon is a single-quantum Fock state.

(3) In modern quantum theory the photoeffect on the level of Einstein's famous paper is fully explained by the socalled semiclassical approximation as is also the Compton effect, i.e., the description where the electromagnetic field is treated as a classical external field and only the charged particles are quantized using the Schroedinger (or Dirac) equation.

(4) For relativistic particles and photons the concept of a wave function as used in non-relativistic quantum mechanics, is a bad concept, because as soon as you have particles with interactions at relativistic energies particle number is no longer conserved, i.e., you must use a formalism that describes the possibility to create and/or destroy particles. The most convenient way for this is quantum field theory (also a bit unfortunately called "second quantization").

(5) The author is in some sense right about the meaning of quantum states of single particles (represented in non-relativistic quantum mechanics by a wave function, if you deal with what's called pure states only). On the one hand, as the name suggests, a single-particle state refers to a system with one particle (which might move freely or under the influence of some external fields). On the other hand all physics there is described by the state (e.g., the wave function) are probabilities for the outcome of measurements on this particle, i.e., you can only check the validity of the predictions made by quantum theory about the particle, if you prepare many particles independently in the same state and make the corresponding measurements on each of these particles. In other words, you have to prepare an ensemble. In this sense the state for the single particle refers to a "preparation procedure" which "prepare" the particle in the particular state, described by some wave function but it's physical meaning is probabilistic and thus only refers to an ensemble of many single particles and in this sense the quantum state always refers to such an ensemble.

It should be noted that (5) is according to what's called the "minimal statistical interpretation" (also called "ensemble interpretation") and that there are many other (non-minimal) interpretations, of which I'm sure other contributers to this forum will tell you. For me personally, I could never make sense about any of these other interpretations, but that's rather a metaphysical or philosophical opinion than a scientific one. From a physics point of view, there's no evidence whatsoever about any meaning of the quantum formalism beyond the ensemble interpration. Also the question whether or not you consider quantum theory as a "complete description" is a metaphysical standpoint rather than science.

 

[ZapperZ]

View attachment 229837


Interference pattern made by light shows the wave nature of light and photoelectric effect shows
particle nature of light. So, what is light?

According to the photoelectric effect, light consists of photons with energy E and momentum $\vec p$.
According to the interference pattern, we don't know where an individual particle will reach on the screen.

So, considering one-dimensional case, what I understood is that there is a wave - function $\psi (x,t)$ associated with each photon such that $|\psi|^2 dx$ tells us the probability of finding the particle in the region dx at time t. So, if a particle reaches x at time t then it can be said that the particle has momentum $\vec p$ and energy E at x and t. Thus, it is meaningless to say that the particle has momentum $\vec p$ and energy E without specifying its position and time.
Thus, what we can say is the probability of finding the particle with momentum $\vec p$ and
energy E in dx at time t. Thus, the probability of finding the energy of particle to be E in region dx at time t should be given as E $|\psi|^2 dx$.

I don't understand why the author is emphasizing that $\psi$ is associated with each particle, not a collection of particles.

First of all, if you are quoting a source, and asking questions about what was written in the source, then you need to properly cite the source. This should always be the practice here in this forum.

Secondly, while there are books and articles now that tend to use the photoelectric effects as the definitive evidence that light consists of "photons", new developments have indicated that while the standard photoelectric effect (not photoemission in general) that is in the same vein as the Millikan's experiment is a compelling support for this picture, it is no longer a slam-dunk evidence. There are already many discussions in this forum on this.

Thirdly, I don't quite understand the issue you have the ability to describe single-particles. The interference pattern that we see is really a single-particle interference pattern. The evidence for this is that we can use single-photon sources and let one photon goes through the slits one at a time, and over large accumulation, you get these interference pattern. So why can't the wavefunction be the description of each particle?

Marcella has a QM description of interference effects that you might want to look into.

Zz.

 

[vanhees71]

Ad 2) That's a problem of bad textbooks! Again: The photoeffect on the level as described in Einstein's 1905 article is fully described by the semiclassical approximation. The most simple evidence for the necessity of a photon picture of electromagnetic radiation is BTW also discovered by Einstein in 1916, i.e., the necessity to assume spontaneous emission in addition to induced emission and absorption to get the correct Planck formula for the black-body spectrum. The semiclassical picture cannot explain spontaneous emission. Nowadays it's pretty clear, because spontaneous emission is of course included in the quantization of the em. field in QED.

Ad 3) The double-slit experiment with particles (or photons) is indeed a good example to explain this quite subtle (in this forum even often controversial) issue. I can, of course, only argue within the ensemble interpretation. On the one hand you are right in saying that the interference pattern is a "single-particle/single-photon interference pattern", because what you can do to get it (and this experiment has been done in the lab, and it's not so easy for photons, because to prepare true single-photon Fock states is not that trivial; we are only used to it for some decades now thanks to the development of non-linear quantum optics thanks to the applicability of lasers and thus parametric downconversion to produce single-photon states) is to use single particles at a time. You have to prepare this single particle with a rather sharp momentum and then observe its position on a screen sufficiently far from the double slit. Of course each single particle just occurs as a single point on the screen never as an interference pattern, but only an ensemble of equally prepared particles can leave a distribution of such dots according to the probability distribution predicted by QT. So at the same time the quantum state (in this case a wave packet with pretty well determined momentum) refers on the one hand to one particle in the sense of the preparation procedure used to prepare this particle in the quantum state, but on the other hand it refers to an ensemble needed to observe the predicted probability distribution for the location of the particle on the screen, where it is observed.

The quoted Marcella article should not be quoted to a beginner in learning quantum mechanics. It's full of fundamental mistakes. One mistake (a very big sin also in some sloppily written textbooks!) is to claim that generalized momentum eigenstates (plane waves in the position representation) are representatives of physical quatum states of a particle. It's not, but its a distribution living outside the Hilbert space (of square integrable wave functions). It should be clearly distinguished between generalized functions (distributions) living in the dual of an appropriate subspace, which is the domain and co-domain of the self-adjoint operators representing observables like position and momentum. This socalled "nuclear space" is smaller than the Hilbert space and thus its dual larger, i.e., there are more elements in the dual, among them the generalized eigenstates of the self-adjoint operators with the spectral value in the continuum of the spectrum of these operators (the plane waves as momentum eigenstates are an important example). Of course, a beginner doesn't need the full formalism of "Rigged Hilbert Spaces", but nevertheless also in a "naive" treatment one should be "as simple as possible but not simpler" in explaining this issue!

 

[Nugatory]

I don't understand why the author is emphasizing that $\psi$ is associated with each particle, not a collection of particles.

There are other experiments that more clearly show why it's natural to think of it that way. The double-slit experiment done with single particles may be most convincing at an introductory level.

 

[YoungPhysicist]

(1) There is no wave-particle dualism since the discovery of modern quantum theory in 1925/26 but one abstract scheme called quantum theory to mathematically describe all phenomena known today except gravity. We have no satisfactory quantum theory of gravity, but everything else is pretty well described.

Wait a sec, so,the concept of wave-partice duality is outdated for 90 years?

 

[Nugatory]

Wait a sec, so,the concept of wave-particle duality is outdated for 90 years?

Yes, give or take a half-decade.

Like Schrödinger's neither dead nor alive cat, and the idea that it matters whether a conscious observer is involved, wave-particle duality has become one of those things that "everyone knows" but isn't so.

If you want to understand what's happened since 1930, you might try "Sneaking a look at God's cards" by Giancarlo Ghiradi (who sadly has recently died) and David Lindley's book "Where does the weirdness go?". These aren't real textbooks, but they're way better than most non-serious explanations and you can get through them without any college-level math.

(You will see the term "wave-particle duality" used in newer papers, but it means something completely different, involving the implications of the commutation relationship between position and momentum).

 

[Delta2]

Best explanation I got on this in some other threads is that photons, (and more generally all the "particles" of the standard model) are neither particles neither waves, they are quantum objects. Depending on the phenomena we are examining these quantum objects appear to have properties of a classical particle, or a classical wave. But they are neither classical particles or classical waves, they are just quantum objects.

 

[bhobba]

Wait a sec, so,the concept of wave-partice duality is outdated for 90 years?

Its one of a number of myths in QM:
https://arxiv.org/abs/quant-ph/0609163

Starting out the following is probably a better way to look at QM than what some textbooks meant for beginning students say:
https://www.scottaaronson.com/democritus/lec9.html

Those textbooks usually follow a semi historical approach and as you progress its sort of expected you figure out for yourself the early ideas like wave-particle duality no longer apply.

Thanks
Bill

 

[phinds]

... I'm not aware of any popular-science book which gets the issue wrong.

@vanhees71 I assume that what you said here is the opposite of what you meant, yes?

 

[YoungPhysicist]

@vanhees71 I assume that what you said here is the opposite of what you meant, yes?

I thought about the same thing lol.

 

[vanhees71]

@vanhees71 I assume that what you said here is the opposite of what you meant, yes?

Indeed. Of course I wanted to say "I'm not aware of any popular-science book which gets the issue right." One can also add philosophy books to that very sad empirical finding.